![Series on
Software
GRAPH DRAWING
AND APPLICATIONS
FOR SOFTWARE AND
KNOWLEDGE ENGINEERS]
Kozo Sugiya
•
iirflh^
World Scientific](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-5-320.jpg)
![2 Graph Drawing and Applications for Software and Knowledge Engineers
elements of diagrams) are distinct. In the same way as with sentences or mathematical
equations, those that principally draw out the meaning are called diagram languages
[Izuhara, Yoshida & Atsumi (86)]. Diagram languages are divided into four basic types:
matrix, net, region and coordinate type. These are two-dimensional expressions of the
mathematical concepts of matrix, graph, set, and coordinate respectively. In real diagrams
it is common to see a composite of these basic types. Figure 1.1.1 shows some examples
of the basic types and a composite type. The reader is requested to interpret meanings of
each diagram.
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Figure 1.1.1. Basic and composite diagram languages: (a) Matrix, (b) Net,
(c) Region, (d) Coordinate, and (e) A composite of net and region.](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-19-320.jpg)
![Introduction 3
Amongst these diagram languages, the structural grammar of matrix and coordinate
types are relatively straightforward, and the research in representation methods is
advancing. In comparison, the structural grammar and drawing methods for net and
region diagram languages are relatively underdeveloped. This is despite the fact that they
are very important for the visual representation of the structure of information or systems,
and are commonly used in everyday life. The directed and undirected graphs dealt with in
this book are of the net type, whilst the compound graphs are a composite of net and
region types. Thus this book deals with automatic graph drawing methods as part of the
research field of the visual representation of the structure of information and systems.
1.2 The Importance of Automatic Graph Drawing Methods
Graphs are convenient mathematical concepts for abstractly expressing the
relationships between elements of some kinds of systems or information. Some specific
examples of graphs we commonly use in research or practice include: system diagrams,
organizational charts, flowcharts and process diagrams. In this way, graphs are abstract
models for the structure of the real world, and are one powerful form of expression for
abstract concepts. They are widely used as fundamental models in system engineering,
information engineering, software engineering and a variety of other fields such as
information visualization for Internet environment [Chen (99.01)] and Bioinformatics
[Schwikowski, Uetx & Fields (00), Becher & Rojas (01) etc.]. With the spread of
workstations and personal computers using high performance graphics in these fields, the
demand for visually manipulating and displaying graphs has greatly increased, and there
is growing interest in graph visualization tools and drawing algorithms.
It is well recognized through experience that when graphs are visualized and
displayed as diagrams, they are easier to understand and remember. However, the
usefulness of the visualization of a graph is largely strongly dependent on its layout. In
other words, a good drawing gives a high possibility of quickly and accurately
communicating the meaning of a diagram, but a bad drawing gives a high possibility of
confusion or misunderstanding.
In this way, for the visualization of graphs to function as a useful method for the
communication of concepts, it is necessary to produce good diagrams; however, the
generation of good diagrams is not so easy. Drawing by hand is time consuming (unless
the graph is small), and doesn't produce very good results. Even with small graphs, there](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-20-320.jpg)
![o—o
CHAPTER 2
A FRAMEWORK FOR AUTOMATIC GRAPH
DRAWING METHODS
This chapter describes a fundamental framework for automatic graph drawing,
and in the next chapter a variety of automatic graph drawing methods are outlined in
accordance with this framework. It is necessary to consider five aspects of the
framework: drawing object, drawing conventions, drawing rules, priority relationships
and the features of drawing algorithms [Sugiyama 89b]. Drawing conventions and
drawing rules can be collectively called aesthetic criteria. These aspects are all explained
in what follows.
2.1 Drawing Objects
Until now, research has mainly dealt with simple graphs (that is, graphs without self loops
7](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-24-320.jpg)
![Framework for Automatic Graph Drawing Methods 15
2.4 Priorities Between Conventions and Rules
The above convention and rules are not independent of each other, and there are
competitive relationships and dependency relationships between them. For example,
Figure 2.4.1 illustrates a competitive relationship between the "minimization of edge
crossings " rule and the "clearly show symmetry" rule. In (a) there are no crossings but
symmetry except horizontal is lost, whilst in (b) symmetry is clear but there are five edge
crossings [Kamada (88)]. The choice between these 2 drawings depends on the purpose
of the drawing, and cannot be done indiscriminately. In the development of graph
drawing algorithms it is necessary to set priorities between convention and rules.
(a) (b)
Figure 2.4.1. An example of a competitive relationship between drawing rules:
(a) symmetry but no crossings, (b) symmetry clearly shown, with 5 crossings [Kamada
(88)].
Priority relationships are individually set for each algorithm or application, but it
is also interesting to consider general priority relationships concerning conventions and
rules. This can be used as a guide when building drawing algorithms, and as criteria for
comparison of the priority relationships for convention and rules of existing drawing
algorithms.
It is possible to consider several priority relationships that generally materialize.
For example, for drawing convention and drawing rules, which is generally higher in
priority order? From their definition drawing convention must always be fulfilled, whilst](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-32-320.jpg)
![22 Graph Drawing and Applications for Software and Knowledge Engineers
1
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(b) vertical format (c) tip-over convention
Figure 3.1.2. Towformats and the tip-over convention [Lin (93)].
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(a) tip-over convention (b) inclusion convention
Figure 3.1.3. Tow conventions [Lin (93)].](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-39-320.jpg)
![24 Graph Drawing and Applications for Software and Knowledge Engineers
A
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c
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y- a i p 1
(a)
(b)
(c)
Figure 3.1.4. Connected undirected graphs and free trees: (a) a connected graph,
(b) its 2 - connected elements, and (c) afree tree [Eades (91)].
Eades (91) presented a linear tree drawing algorithm using concentric circle layout
and proved that the drawing rule of no crossing will always be achieved, and sought to
bound the ratio of lengths of the largest edge and the shortest edge, for uniform edge
length. As a modification of that algorithm, he proposed a linear time algorithm that
placed all the leaves on the outermost concentric circle. Also, Manning & Atallah (89)
proposed an algorithm for concentric circle layout that clearly shows symmetry. Other
similar methods include Bernard (81), Read (86), and Esposito (88).
A number of methods that simulate mechanical or thermodynamical systems may
be used to draw graphs according to the principle that the system seeks a stable condition,
in which energy is locally minimized. These methods are known as force-directed
methods. The forces acting between vertices are attractive when neighboring vertices are](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-41-320.jpg)
![Outlines of Automatic Graph Drawing Methods 25
fan-shaped wedge
Figure 3.1.5. Concentric circle drawing of afree tree [Eades (91)].
far apart, repulsive when they are too close together, and again repulsive when they are
not neighbors. There are several different models, varying with the assignment of
attractive and repulsive forces: Eades (84), Kamada & Kawai (89), Fuchterman &
Reingold (91). and Davidson & Harel (96).
These methods were originally devised for drawing general undirected graphs, so
they are outlined in Section 3.3 and detailed in Sections 4.3 and 4.4. The problem with
drawing trees using force-directed methods is that depending on the initial placement of
vertices, it is not always possible to attain no crossings.
There are several other drawing methods for free trees. H-tree drawing is a method
of orthogonal grid layout for binary trees, and it is particularly for complete binary trees
as it produces an orderly drawing as shown by Figure 3.1.6 (a). Methods presented by
Eades (91) show that even for incomplete binary trees, a drawing such as Figure 3.1.6 (b)
can be attained. The garden drawing method is a modification of the orthogonal grid
layout, in which the longest path is drawn as a horizontal line, and each sub-tree is drawn
near this line (see Figure 3.1.7). The name comes from the fact that the resulting picture
looks like plants growing in a garden.](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-42-320.jpg)
![26 Graph Drawing and Applications for Software and Knowledge Engineers
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(b) a general binary tree
Figure 3.1.6. H-tree drawing method [Eades (91)].](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-43-320.jpg)
![Outlines of Automatic Graph Drawing Methods 27
u LU
Figure 3.1.7. Garden drawing method [Eades (91)].
3.2 Directed Graph Drawing Methods
Drawing methods for directed graphs use two important classifications. These are
the existence of cycles and the possibility of planar drawing.
Graphs with no cycles are called acyclic directed graphs, and those with cycles are
called general directed graphs. For drawing directed graphs, it is often convenient if the
direction of all the edges flow in one direction (for example top to bottom, or left to right).
This is called monotone drawing, and it is only possible with acyclic directed graphs.
Also, for general directed graphs in which a monotone drawing is impossible, there is
interest in drawing so that the number of edges that appear against the flow of direction of
the majority of edges is minimized.
There are two important problems with planar drawing. The first is concerned with
planar monotone drawings of acyclic directed graphs. There has been a great deal of
theoretical research into this problem recently, with many results. The second problem is
concerned with planar drawings of the underlying undirected graph of a general directed
graph. This is the same as the problem of planar drawings of undirected graphs, and is
discussed in Section 3.3.
3.2.1 Acyclic Directed Graphs
Acyclic directed graphs are widely used to express hierarchical structures. PERT
graphs, ISA hierarchies, subroutine call graphs, and Hasse diagrams are all examples.
These graphs are usually drawn in a monotone fashion, that is, all edges follow the same](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-44-320.jpg)
![28 Graph Drawing and Applications for Software and Knowledge Engineers
direction. For general directed graphs, one should compute a minimum size set of
feedback edges; the direction of these is reversed, so changing it into a acyclic directed
graph. Thus drawing methods for acyclic directed graphs are also important as drawing
methods for general directed graphs.
Upward (Downward) Drawing
A planar drawing of a directed graph is an upward (downward) drawing if all edges
are drawn as curves that monotonically increase (decrease) in a vertical direction. Of
course if a directed graph has an upward (downward) drawing, then it is acyclic.
i p«-- - t — f — i - — - « - - + — + -
+ - — ^ i - / - i — ~ ^ H + • — < i
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(a) (b)
(c)
Figure 3.2.1. Three types of upward drawing: (a)free upward drawing, (b) orthogonal
grid upward drawing, and (c) parallel line upward drawing [Di Battista et al (88)].
Di Battista & Tamassia (88) proposed an efficient algorithm for producing three
types of upward drawing of acyclic directed graphs. Figure 3.2.1 shows the features of
these drawings schematically. As drawing conventions, Figure 3.2.1(a) uses (free,](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-45-320.jpg)
![Outlines of Automatic Graph Drawing Methods 31
b c
(a) (b)
c b a b a
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Figure 3.2.3. Steps of hierarchical drawing, (a) given directed graph, (b) layering
and normalization, (c) order determination, (d) coordinate determination.
Figure 3.2.4. Hierarchical drawing with straight line routing [Carpono (80)].](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-48-320.jpg)
![32 Graph Drawing and Applications for Software and Knowledge Engineers
Figure 3.2.5. Hierarchical drawing in which the number of bends has been reduced
(this is the same graph as in Figure 3.2.2) [Rowe et al (87)].
Messinger (88) proposed a divide and conquer method aimed at speeding up the
STT method. There are 3 steps to this method (see Figure 3.2.6):
1. Partition the large graph into subgraphs.
2. Compute the drawing for each subgraph using the STT method.
3. Now consider the edges between subgraphs. Determine the overall layout, and
route the edges between subgraphs.
The smaller the subgraphs, the less machine time, but the more difficult the
composition (the third step) becomes. In this method there is not enough consideration
given to the composition step. One possible solution to this problem is to use the drawing
method for compound graphs discussed in Chapter 4.
For hierarchical drawing, Itumi & Kogure (88) proposed a new drawing rule to add
to the structural rules. This new rule is elimination of crossing among outlines, and they
proposed an algorithm to achieve this. In Figure 3.2.7 (a) the set of descendants of vertex
a and the set of descendants of vertex b intersects in the dotted line area, making it
difficult to see. The elimination of crossing among outlines eliminates this area of
intersection, as shown in Figure 3.2.7 (b).
The STT method, its modifications and extensions are discussed in greater detail in
Chapter 4.
The minimization of the number of crossings in a hierarchical graph, even if it is a
2-layer graph and the position of the vertices of the first layer are fixed, is NP-complete](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-49-320.jpg)
![Yet the mind is not discontented and unfed. This is no
boundless solitude of ocean where one may take a kind of pleasure
To float for ever with a careless course
And think himself the only being alive.
It is not an end but a beginning that we have reached. These are
the elements—pure earth and wind and sunlight—out of which
beauty and joy arise, original and ancient, for ever young. Their
presence restores us not to the Middle Ages, not to the days of Mr.
Doughty’s heroic princes and princesses of Britain,[4] not to any dim
archæologist’s world of reeking marsh and wood, of mammoth and
brutish men, but to a region out of space and out of time in which
life and thought and physical health are in harmony with sun and
earth, fragrant as the flowers in the grass, blithe as the grasshopper,
swift as the hares, divine; and out of it all arises a vision of the man
who will embody this thought, a man whom human infelicity,
discontented with the past, has placed in a golden age still farther
back, for the sufficient reason that in every age he has been a
dream, and our dreaming is of the dawn or the night, always
disappointed but undaunted by the day that follows. And so no
storied valley or hillside is richer in humanity than this coombe. It is
one of the countless Edens where we are in contact not with the
soldier and ploughman and mason that change the surface of the
earth, but with prophet and poet who have ever lived to trace to
Nature and to the early ages the health and vigour of men. There is
the greatest antiquity of all, peace and purity and simplicity, and in
the midst is the mother Earth, the young mother of the world, with a
face like Ceres before she had lost Persephone in the underworld. In
fact, so blessed is this solitary hall that after climbing out it is
mournful to see the rabbit-worn tunnels and the Roman camp on the
ridge.](https://image.slidesharecdn.com/38232-250413225015-941d4a94/85/Graph-Drawing-and-Applications-for-Software-and-Knowledge-Engineers-Kozo-Sugiyama-52-320.jpg)
Graph Drawing and Applications for Software and Knowledge Engineers Kozo Sugiyama
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- 5. Series on Software GRAPH DRAWING AND APPLICATIONS FOR SOFTWARE AND KNOWLEDGE ENGINEERS] Kozo Sugiya • iirflh^ World Scientific
- 6. This page is intentionally left blank
- 7. GRAPH DRAWING AND APPLICATIONS FOR SOFTWARE AND KNOWLEDGE ENGINEERS
- 8. SERIES ON SOFTWARE ENGINEERING AND KNOWLEDGE ENGINEERING Series Editor-in-Chief S K CHANG {University of Pittsburgh, USA) Vol. 1 Knowledge-Based Software Development for Real-Time Distributed Systems Jeffrey J.-P. Tsai and Thomas J. Weigert (Univ. Illinois at Chicago) Vol. 2 Advances in Software Engineering and Knowledge Engineering edited by Vincenzo Ambriola (Univ. Pisa) and Genoveffa Tortora (Univ. Salerno) Vol. 3 The Impact of CASE Technology on Software Processes edited by Daniel E. Cooke (Univ. Texas) Vol. 4 Software Engineering and Knowledge Engineering: Trends for the Next Decade edited by W. D. Hurley (Univ. Pittsburgh) Vol. 5 Intelligent Image Database Systems edited by S. K. Chang (Univ. Pittsburgh), E. Jungert (Swedish Defence Res. Establishment) and G. Tortora (Univ. Salerno) Vol. 6 Object-Oriented Software: Design and Maintenance edited by Luiz F. Capretz and Miriam A. M. Capretz (Univ. Aizu, Japan) Vol. 7 Software Visualisation edited by P. Eades (Univ. Newcastle) and K. Zhang (Macquarie Univ.) Vol. 8 Image Databases and Multi-Media Search edited by Arnold W. M. Smeulders (Univ. Amsterdam) and Ramesh Jain (Univ. California) Vol. 9 Advances in Distributed Multimedia Systems edited by S. K. Chang, T. F. Znati (Univ. Pittsburgh) and S. T. Vuong (Univ. British Columbia) Vol. 10 Hybrid Parallel Execution Model for Logic-Based Specification Languages Jeffrey J.-P. Tsai and Bing Li (Univ. Illinois at Chicago) Forthcoming titles: Acquisition of Software Engineering Knowledge edited by Robert G. Reynolds (Wayne State Univ.) Monitoring, Debugging, and Analysis of Distributed Real-Time Systems Jeffrey J.-P. Tsai, Steve J. H. Yong, R. Smith and Y. D. Bi (Univ. Illinois at Chicago)
- 9. GRAPH DRAWING AND APPLICATIONS FOR SOFTWARE AND KNOWLEDGE ENGINEERS Kozo Sugiyama Japan Advanced institute of Science and Technology V f e World Scientific I M NewJersey London »Sinc NewJersey• London • Singapore • Hong Kong
- 10. Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. GRAPH DRAWING AND APPLICATIONS FOR SOFTWARE AND KNOWLEDGE ENGINEERS Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-02-4879-2 Printed in Singapore by Mainland Press
- 11. To Sachiko, Taichiro, Mihoko, and the spirits of my parents
- 12. 6—O O PREFACE This book outlines the basic concepts, methods and applications of automatic graph drawing, which is one of the most important tools for visual interfaces. Graphs are a convenient mathematical concept for pictorially expressing the structural relationships between elements of a specific system. Some examples of graphs that we often use in research or work include: organization charts, process diagrams, system diagrams, semantic nets, entity-relationship diagrams, and flowcharts. Automatic graph drawing is a method in which a computer is used to draw a diagram automatically from the given graph, so that it is easily understandable. Therefore, it is a useful method for visualizing the structural relationships in information or systems. Although there was some early research into automatic graph drawing methods from the 1930's, research began in earnest in the 1980's with the spread of workstations and personal computers, and the realization of the importance of visual interfaces. The first international workshop on automatic graph drawing methods was held in Rome (June 1992). Even though there are many unresolved issues still remaining, the achievements of research efforts are considerable, and further developments are expected towards practical applications in such diverse fields as software engineering, knowledge engineering, project management, network design, and visual interfaces.
- 13. viii Preface Diagrams are particularly useful in stimulating and understanding human ideas. In the near future, it will be possible to draw the structure of information or systems as diagrams in whatever way humans want to, using computers. When we are able to redraw diagrams at will, they will become more than just the expression or communication of information, and are going to be valuable tools of thought. Thus it can be expected that visual interface tools such as CASE tools, hypertext, and groupware will become even more valuable. The first and second chapters of this book deal with the thinking behind automatic graph drawing methods, and the basic concepts. Chapter three outlines a variety of algorithms that show how research has progressed, and Chapter four gives greater detail on a few typical algorithms for those who want to know more. Chapter five discusses developments that broaden the uses of automatic graph drawing methods. Chapters six and seven introduce several simple applications, and a higher level application for creativity support. Chapter eight discusses the author's views on the future of automatic graph drawing methods. This book was completed in three stages: first the book was written in Japanese (Kozo Sugiyama: Automatic Graph Drawing Methods and Applications Visual Human Interfaces, The Society of Instrument and Control Engineers, 1993, 158p. It received the award as a distinguished book at the 30th anniversary competition of the Society), and then it was translated into English by Ruth Eades and then revised considerably by the author. Kozo Sugiyama Ishikawa Prefecture, November 2001
- 14. Acknowledgments This book is based on our translation project funded by the Japan Society for the Promotion of Science (JPSP). Members of the project consisted of Ms. Ruth Eades (translator), Professor Peter Eades (expert), and myself (author). This was the best team indeed! First of all I would like to thank Ms. Ruth Eades for her considerable effort to translate and Professor Peter Eades at University of Sydney for his valuable advice and help in writing and publishing this book. Special thanks to Doctor Kazuo Misue at Fujitsu Laboratories Limited for his participation in a series of studies described in this book and for supplying hardcopies of figures found in this book. IX
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- 16. 6—o o CONTENTS Preface vii Acknowledgments ix Chapter 1: Introduction 1 1.1 Diagram language interfaces 1 1.2 The importance of automatic graph drawing methods 3 1.3 The aims of this book 5 1.4 The structure and terminology of this book 6 Chapter 2: A Framework for Automatic Graph Drawing Methods 7 2.1 Drawing objects 7 2.2 Drawing conventions 9 2.3 Drawing rules 11 2.4 Priorities between conventions and rales 15 2.5 Drawing algorithms 16 Chapter 3: Outlines of Automatic Graph Drawing Methods 19 3.1 Tree drawing methods 19 xi
- 17. xii Contents 3.2 Directed graph drawing methods 27 3.3 Undirected graph drawing methods 37 3.4 Other drawing methods 45 3.5 Automatic graph drawing methods at a glance 48 Chapter 4: Details of Automatic Graph Drawing Methods 49 4.1 General tree drawing methods 49 4.2 General directed graph drawing methods 59 4.3 General undirected graph drawing methods 81 4.4 Generic graph drawing method by the magnetic spring model 91 4.5 Drawing methods for compound graphs 102 Chapter 5: Extensions of Automatic Graph Drawing Methods 127 5.1 Drawing methods using curves 127 5.2 Fisheye views 134 5.3 Dynamic drawing methods 142 5.4 Other extensions 145 Chapter 6: A Variety of Applications 147 6.1 The uses of automatic graph drawing methods 147 6.2 Application examples 149 Chapter 7: Applications for Creativity Support 169 7.1 What is creativity support 170 7.2 Creativity support based on empirical creativity methods 171 7.3 Diagrammatic idea organizer D-ABDUCTOR 179 7.4 Evaluation of D-ABDUCTOR 188 Chapter 8: Conclusion 195 Bibliography 197 Index 211
- 18. 0—0 INTRODUCTION 1.1 Diagram Language Interfaces Amongst visual interfaces, interfaces using diagrams are some of the most basic, and have the advantages of being widely useable, handy, approachable and easily understood. In fact, most personal computers and workstations have some sort of function for visual representation or editing, presented as a user-friendly interface. In engineering or information technology and other fields, such diagrams as system structure diagrams, plans, operations process charts, and logic circuit diagrams are often used in everyday work, and are used as a common language between designers and users. Furthermore, anyone faced with a difficult problem has had the experience of using diagrams to stimulate thought. In this way, diagrams are currently used widely for the expression and communication of information and as tools of thought. The history of diagrams is long, and since prehistoric times humans have developed many kinds of diagrams to express various matters and to stimulate thought. In many of these diagrams, units of meaning (individual elements of diagrams) and syntactic grammar (the rules of arrangement of 1
- 19. 2 Graph Drawing and Applications for Software and Knowledge Engineers elements of diagrams) are distinct. In the same way as with sentences or mathematical equations, those that principally draw out the meaning are called diagram languages [Izuhara, Yoshida & Atsumi (86)]. Diagram languages are divided into four basic types: matrix, net, region and coordinate type. These are two-dimensional expressions of the mathematical concepts of matrix, graph, set, and coordinate respectively. In real diagrams it is common to see a composite of these basic types. Figure 1.1.1 shows some examples of the basic types and a composite type. The reader is requested to interpret meanings of each diagram. / J A o X o B o o X c X o o p o o o (a) M c (b) /TN .'Tf JDL (d) (e) Figure 1.1.1. Basic and composite diagram languages: (a) Matrix, (b) Net, (c) Region, (d) Coordinate, and (e) A composite of net and region.
- 20. Introduction 3 Amongst these diagram languages, the structural grammar of matrix and coordinate types are relatively straightforward, and the research in representation methods is advancing. In comparison, the structural grammar and drawing methods for net and region diagram languages are relatively underdeveloped. This is despite the fact that they are very important for the visual representation of the structure of information or systems, and are commonly used in everyday life. The directed and undirected graphs dealt with in this book are of the net type, whilst the compound graphs are a composite of net and region types. Thus this book deals with automatic graph drawing methods as part of the research field of the visual representation of the structure of information and systems. 1.2 The Importance of Automatic Graph Drawing Methods Graphs are convenient mathematical concepts for abstractly expressing the relationships between elements of some kinds of systems or information. Some specific examples of graphs we commonly use in research or practice include: system diagrams, organizational charts, flowcharts and process diagrams. In this way, graphs are abstract models for the structure of the real world, and are one powerful form of expression for abstract concepts. They are widely used as fundamental models in system engineering, information engineering, software engineering and a variety of other fields such as information visualization for Internet environment [Chen (99.01)] and Bioinformatics [Schwikowski, Uetx & Fields (00), Becher & Rojas (01) etc.]. With the spread of workstations and personal computers using high performance graphics in these fields, the demand for visually manipulating and displaying graphs has greatly increased, and there is growing interest in graph visualization tools and drawing algorithms. It is well recognized through experience that when graphs are visualized and displayed as diagrams, they are easier to understand and remember. However, the usefulness of the visualization of a graph is largely strongly dependent on its layout. In other words, a good drawing gives a high possibility of quickly and accurately communicating the meaning of a diagram, but a bad drawing gives a high possibility of confusion or misunderstanding. In this way, for the visualization of graphs to function as a useful method for the communication of concepts, it is necessary to produce good diagrams; however, the generation of good diagrams is not so easy. Drawing by hand is time consuming (unless the graph is small), and doesn't produce very good results. Even with small graphs, there
- 21. 4 Graph Drawing and Applications for Software and Knowledge Engineers are inherent limits to drawing by hand, for example when it is necessary to repeatedly revise the diagram after it is drawn. For this reason, interest has been shown in automatic graph drawing methods using computers, but in this case it is necessary to solve the extremely difficult problems in human cognition and computational complexity (there is a combinatorial explosion). The problem of generating good diagrams is mathematically a problem of optimuml embedding of labeled graphs in a plane. It is similar to problems such as the layout of integrated circuits. However, these problems differ considerably in purpose, optimality criteria and constraints, as shown in Table 1.2.1, and algorithms for circuit layout are not suitable for generating good diagrams. The generation of good diagrams is concerned with only small-scale graphs, but in order to fulfill diverse demands related to human cognition, such as readability and aesthetics, much more careful consideration is required. In this book, these problems are called automatic graph drawing method problems based on cognitive criteria, or simply automatic graph drawing problems. Aim Object Criteria for optimality Constraints Features COGNITIVE VISIALIZATION Communication of a concept Relatively small-scale graphs Readability Aesthetics Relatively flexible A diversity of demands CIRCUIT LAYOUT Physical layout Large-scale graphs Resources (such as minimization of area) Reliability (such as minimal crossings) Usually strict Uniform demands Table 1.2.1. Comparison of the generation of "good" diagrams and the layout of circuits. To develop automatic graph drawing methods the following are necessary: • Clearly determine the layout criteria usually used by humans. • The development of algorithms that satisfy these criteria (even approximately). Forefront research into graph drawing methods has been carried out since the 1930s. However, it is more recently that research has begun in earnest, with the spread of workstations and personal computers in the 1980s and the recognition of the importance of human interfaces. From research in the theory of graph algorithms, it has been discovered that in many cases of optimization of layout criteria, an efficient method may
- 22. Introduction 5 not exist. Thus based on a theoretical standpoint, the key point is to find an effective heuristic to enable the development of actual drawing methods. It is also likely to be, in the end, a problem related to the construction of expert systems that incorporate the wisdom of designers. Currently research into theoretical and heuristic methods is developing, and there are even those that are at the stage of practical use. Heuristic methods are also being used in such things as simulations by dynamical models, simulated annealing, and genetic algorithms. In this way the field of graph drawing methods, although there are still many research questions remaining, is showing considerable results, and is expected to develop further towards practical use in the future. Once it is possible to draw good diagrams automatically, troublesome operations such as redrawing the diagram can be simplified, and it is possible to develop high-level diagram editors. It is believed that, in the future, the usefulness of automatic graph drawing methods will be further enhanced, for example to be applied in dynamic diagrammatic thought support in software engineering, hypermedia and groupware. 1.3 The Aims of This Book As mentioned above, automatic graph drawing methods is a field that is currently developing, and it is also a large field, so it would be difficult to cover the whole field in this book. In this book the author has aimed to: 1. Give an overall outline of the whole field. 2. Introduce the representative drawing methods of such graphs as trees, directed graphs, undirected graphs, and compound graphs. We aim to explain the concepts and methods as plainly as possible. 3. Make the intuitive understanding of these easier by using examples and diagrams in the explanation. 4. In the choice of drawing methods for each different class, rather than those that are theoretically strong but have limited application, give preference to those that use heuristic methods that will allow the user to always find a drawing within that class for a give graph. 5. Present a wealth of references for those readers who wish to know more. Therefore this book does not equip the reader to begin programming once they have read it, but rather aims to give the reader a good understanding of the field of automatic
- 23. 6 Graph Drawing and Applications for Software and Knowledge Engineers graph drawing methods. It is hoped that this book will be an incentive for readers to take an interest in this developing field and through further study develop new methods of their own. 1.4 The Structure and Terminology of This Book Chapter 2 gives the basic framework of automatic graph drawing methods, whilst Chapter 3 uses this framework to systematically explain the current state of the research in this field, giving the reader an outline of automatic graph drawing methods. Chapter 4 explains the basic drawing methods in greater detail for those readers who wish to know more, or for those who wish to actually prepare interfaces. Drawing methods for tree graphs, directed graphs, undirected graphs mixed graphs, and compound graphs are described, with greater detail given to directed and compound graphs in particular, as these are the original results of the authors. Chapter 5 discusses several extensions in which the basic drawing methods of Chapter 4 can be used in more realistic situations. First extensions of drawing methods to use curves, and the usefulness of this extension, are discussed. Next the fisheye drawing method, in which the part being viewed is emphasized whilst the part not being viewed is de-emphasized, is explained. Furthermore extensions to dynamic drawing methods such as redrawing or incremental drawing are discussed. In Chapter 6, after reviewing the uses of automatic graph drawing methods, several simple sample applications are introduced. In Chapter 7, as an example of a high level application, the author's system for diagrammatic idea support is discussed in detail. This is an attempt to develop a high capability graphical user interface for idea support, using the KJ method as background. Its concepts and current state are described and results of evaluating the system are shown. Lastly, Chapter 8 concludes with the author's thoughts on the future research questions issues of automatic graph drawing methods. It is assumed that the reader already has knowledge of the basic concepts of graph theory and algorithms, and these are not explained or defined here. For terminology not explained or defined in this book, it would be useful to consult an introductory text on graph theory or algorithms. For example: Aho, Hopcroft and Ullman, "The design and analysis of computer algorithms", Addison-Wesley 1974.
- 24. o—o CHAPTER 2 A FRAMEWORK FOR AUTOMATIC GRAPH DRAWING METHODS This chapter describes a fundamental framework for automatic graph drawing, and in the next chapter a variety of automatic graph drawing methods are outlined in accordance with this framework. It is necessary to consider five aspects of the framework: drawing object, drawing conventions, drawing rules, priority relationships and the features of drawing algorithms [Sugiyama 89b]. Drawing conventions and drawing rules can be collectively called aesthetic criteria. These aspects are all explained in what follows. 2.1 Drawing Objects Until now, research has mainly dealt with simple graphs (that is, graphs without self loops 7
- 25. 8 Graph Drawing and Applications for Software and Knowledge Engineers or multiple edges) as drawing objects. These graphs are usually classified as trees, directed graphs, planar graphs and undirected graphs. Graphs in these classes only have adjacency edges, but recent research includes compound graphs, which have both adjacency and inclusion edges. Drawing methods for compound graphs are discussed in Section 4.5; in the current chapter, we describe basic drawing methods for simple graphs. It may not be impossible to devise a universal drawing algorithm for all classes of graphs, but it is more practical and more effective to consider drawing methods that take into account the graph-theoretic characteristics of each class. The classification of graphs used in this book is outlined in Figure 2.1.1. Graphs Trees 1— — Directed graphs Undirected graphs Compound graphs Rooted trees Free trees Binary trees n-ary trees Acyclic directed graphs General directed graphs Planar undirected graphs General undirected graphs Figure 2.1.1. A classification of graphs. Trees are divided into rooted trees with specified roots, and free trees with no specified roots. All trees are planar, that is, they can be drawn with no edge crossings. Rooted trees represent hierarchical structures such as organization diagrams, but free trees have no specified roots and there is a high degree of freedom in their placement. Directed graphs are divided into acyclic directed graphs and general directed graphs. Acyclic directed graphs often represent a hierarchy, and they are drawn with a fixed direction of flow of all edges. In general directed graphs, the cyclic and acyclic parts have different characteristics and are often drawn in different ways.
- 26. Framework for Automatic Graph Drawing Methods 9 Figure 2.2.1. Standard and compound coordinate systems. Undirectedgraphs are the most common drawing objects. They are often divided intoplanar undirectedgraphs (or simplyplanar graphs) and general undirectedgraphs. Planar graphs form a fundamental and important class of graphs, and therefore they have received a considerable amount of attention in the literature of graph theory, graph algorithms and graph drawing. 2.2 Drawing Conventions A drawing convention is a protocol for the placement of vertices and the routing of edges; effectively it consists of constraints that must be satisfied by a drawing.
- 27. 10 Graph Drawing and Applications for Software and Knowledge Engineers Drawing conventions may be divided into the placement conventions for vertices and the routing conventions for edges. Vertices are often placed according to a standard coordinate system as in Figure 2.2.1. Placement conventions for vertices corresponding to standard coordinate systems a to/are shown in Table 2.2.1. It is possible to combine these standard coordinate systems to create compound coordinate systems. For example, in Figure 2.2.1, m is a coordinate system that combines b and e. COORDINATE SYSTEM a b c d e f PLACEMENT CONVENTION Free placement (no constraints on placement) Parallel line placement (on or between parallel lines) Concentric circle placement (on or between concentric circles) Radial line placement (on or between radial lines) Orthogonal grid placement (on the integer grid). On the intersections of a polar grid. Table 2.2.1. Placement conventions for vertices. TYPE Line type Relationship with coordinate system ROUTING CONVENTION 1. Straight line routing 2. Polyline routing 3. Curve routing (routing with curves) 4. Independent of coordinate system (Routing is independent of coordinate system lines) 5. Dependent on coordinate system (Routing is parallel to or perpendicular to coordinate system lines.) Table 2.2.2. Routing conventions for edges. Conventions concerning the routing of edges consist of those concerning the types of lines, and those concerning the relationship with the coordinate system. These conventions for edge routing are shown in Table 2.2.2.
- 28. Framework for Automatic Graph Drawing Methods 11 When the placement conventions from Table 2.2.1 and the routing conventions from Table 2.2.2 are combined, we have • 6 placement conventions, • 3 line types, • 2 relationships with the coordinate system; this makes 36 possible drawing conventions formed by triples, for example: "The drawing convention for rooted trees is (parallel lines, straight lines, independent). " Such triples are called drawing conventions. 2.3 Drawing Rules It is extremely difficult to accurately define a "good" drawing of a graph. Words such as "good", "nice" or "aesthetically pleasing" are often used, but these are subjective words and strictly speaking their meaning changes according to the area of application or the user's taste. However, since a graph is an extremely simple abstract representation, despite minor differences, one can enunciate several elementary and common criteria for "good" drawings, reflecting the characteristics of graphs. Batini, Furlani & Nardelli (85) investigated some 200 diagrams in actual use. They determined many types of placement criteria used by humans, and classified these criteria according to 3 axes. Using the criteria stated by Batini, Furlani & Nardelli (85) as reference, we have constructed a list of drawing rules that encompasses most of the rules of existing drawing methods. We have classified these rules on four axes as stated in Section 2.3.2. In drawing algorithms these rules are called optimization criteria or constraint conditions. 2.3.1 Static Rules and Dynamic Rules Rules for a single drawing of a graph are called static rules. Rules which apply to a sequence of drawings, as the graph is changed and redrawn, are called dynamic rules. When one part of a drawing is corrected and redrawn, dynamic rules ensure that the relative positional relationship between diagram elements in the new drawing remains as much as possible the same as in the original diagram. This relates to some very interesting and important problems about how to preserve the human mental map of a
- 29. 12 Graph Drawing and Applications for Software and Knowledge Engineers graph, and how to preserve the continuity of cognition. Dynamic rules are addressed in Chapter 5; Table 2.3.1 lists some common static rules. 2.3.2 Semantic Rules and Structural Rules Static rules are divided into semantic rules and structural rules. Semantic rules art placement and routing rules derived from the meaning of vertices or edges; for example, the importance of a vertex or the strength of the relationship of an edge. They are given by the user, or automatically derived from information in the labels on the vertices or edges. Some important semantic rules are shown in Table 2.3.1. As well as those listed in the table, some rules specify the placement of or the relationships between a specified set of vertices, or specify the routing of or the relationships between a specified set of edges. Structural rules are placement or routing rules concerned only with the structural information (in other words, graph-theoretic features) of a given graph. The diversity of demands of human cognition leads to a wide variety of such rules. Table 2.3.1 lists a number of structural rules. Drawing rules can be classified using the following axes. A classification according to these 4 axes is noted in the right hand column of Table 2.3.1. (1) Whether the solution to a rule can be obtained uniquely (U), or not (N). (2) Whether the rule is topological (T) (specifying only the placement relationship between elements), shape-oriented (S) (specifying the direction also), or metric (M) (specifying distances as well). (3) Whether the rule applies globally, to the whole drawing (G), or locally, only to a part of the drawing (L). (4) Whether the rule is hierarchical (H), or flat (F), or both (B). The following cautionary remarks are necessary concerning Table 2.3.1: the actual meaning of each rule differs slightly between drawing objects, drawing conventions, and algorithms, and the four axes are not completely independent. To help readers understand each rule instantly, simple examples of better and worse layouts in terms of each rule are shown in Figure 2.3.1.
- 30. Framework for Automatic Graph Drawing Methods 13 TYPE Ssvnantic Rules Structural Rules DRAWING RULES 1. A specified sequence of vertices is placed on a straight line. 2. A specified sequence of vertices is placed on a specified curve. 3. Vertices are drawn with a specified size. 4. A specified set of vertices is placed at the boundary of the drawing. 5. A specified set of vertices are drawn near to each other. 6. A specified set of vertices is placed near the center. 7. An upper limit to the number of edge crossings is specified. 8. An upper limit to the number of edge bends is specified. 9. The lengths of specified edge have a specified upper limit. 1. Vertices of high degree are placed near the center. 2. Isomorphic subgraphs are always drawn identically. 3. The vertices of isomorphic subgraphs are always placed identically. 4. Hierarchical structure is clearly shown vertically or horizontally. 5. The number of edge crossings is minimized. 6. The ratio of length to breadth of the drawing area is balanced. 7. Symmetry is clearly shown. 8. The number of edge bends is minimized (using straight lines wherever possible). 9. The number of faces drawn as convex polygons is maximised. 10. Children of a vertex are symmetrically placed. 11. Crossings among outlines are eliminated, (see Figure 3.2.7) 12. The density of the placement and the routing is uniform. 13. The drawing area is minimized. 14. The total edge length is minimized. 15. The difference in sizes of vertices is minimized. 16. The average length of edges is minimized. 17. The difference between the length of contours of vertices and the length of edges is maximized. 18. The differences in edge lengths is minimized. 19. The length of the longest edge is minimized. 20. Vertices on the boundary are placed with uniform density. CLASSIFI- CATION AXES USLB USLB UMLB NTLB NTLB NTLB NTLB NSLF NMLF UTLB USGB USGB NTGH NTGB NSGB NSGB NSGB NSLH NMGH NMGB NMGB NMGB NMGB NMGF NMGF NMGF NMGF NMLF NMLF Table 2.3.1. Drawing rules.
- 31. 14 Graph Drawing and Applications for Software and Knowledge Engineers BASIC RULES STRUCTURAL RULES <f Avoid overlapping Q Q among vertices C T O I ""O Avoid overlapping Q — | among edges •V cfxr Avoid overlapping among vertices and edges % SEMANTIC RULES *v° Place specified A J ^ ^ A A A vertices on *T<JT*J^J~^J a straight line <f{Px> ''vfrtfc'Sfn6 '1 ^ ^ ^ ^ V Q a curve Q O ° ^ Drawn vertices with specified sizes o ©®o © ^ © : Place specified Q vertices at the boundary >* ...o.©i © O @ Place specified i j j ^ O O © o 0 v ™«> W o © o© © o 0 Place specified vertices near the center jo STRUCTURAL RULES Central placement of high degree vertices • ¥ ° / > Identical layout orphi subgraphs of isomorphic K^> &? Hierarchical layout Minimize edge crossings Balance of length and breadth ^ Symmetrical layout n * & Minimize edge bends U Draw faces as convex polygons o ^ ° Place childrei symmetrically "c/K Avoid crossings among outlines o 0 o o Uniform 0 X 0 placement O O Q O O OpO Minimize drawing area Minimize total edge length ( ) — H Q Minimize Q -0 ^-^ y / T difference in - " sizes of vertices n Ol Q ^& Minimize 0 ^ j y ^ average length GsJ^O 0"Jt> o f e d 8e s O^Q Figure 2.3.1. Simple examples of better (right) and worse (left) layouts.
- 32. Framework for Automatic Graph Drawing Methods 15 2.4 Priorities Between Conventions and Rules The above convention and rules are not independent of each other, and there are competitive relationships and dependency relationships between them. For example, Figure 2.4.1 illustrates a competitive relationship between the "minimization of edge crossings " rule and the "clearly show symmetry" rule. In (a) there are no crossings but symmetry except horizontal is lost, whilst in (b) symmetry is clear but there are five edge crossings [Kamada (88)]. The choice between these 2 drawings depends on the purpose of the drawing, and cannot be done indiscriminately. In the development of graph drawing algorithms it is necessary to set priorities between convention and rules. (a) (b) Figure 2.4.1. An example of a competitive relationship between drawing rules: (a) symmetry but no crossings, (b) symmetry clearly shown, with 5 crossings [Kamada (88)]. Priority relationships are individually set for each algorithm or application, but it is also interesting to consider general priority relationships concerning conventions and rules. This can be used as a guide when building drawing algorithms, and as criteria for comparison of the priority relationships for convention and rules of existing drawing algorithms. It is possible to consider several priority relationships that generally materialize. For example, for drawing convention and drawing rules, which is generally higher in priority order? From their definition drawing convention must always be fulfilled, whilst
- 33. 16 Graph Drawing and Applications for Software and Knowledge Engineers drawing rules should be fulfilled as much as possible, so in general "drawing conventions > drawing rules". Here A>B means that A has a higher degree of priority than B. From experience it can be seen that global rules must be given preference over local rules. Some general priorities that can be determined psychologically or experimentally (even if only approximately) are shown in Table 2.4.1. The drawing rules in Table 2.3.1 are sorted according to these general priorities. 1. Placement conventions > Routing conventions 2. 3. Drawing conventions Semantic rules > > Drawing rules Structural rules 4. Uniqueness (U) > Non-uniqueness (N) 5. Topological rules (T) > Shape rules (S) > Metric rules (M) 6. Global rules (G) > Local rules (L) 7. Hierarchical rules (H) > Common rules (B) > Flat rules (F) Table 2.4.1. General priorities between conventions and rules. 2.5 Drawing Algorithms We have defined drawing conventions to be constraints that must be fulfilled, and drawing rules to be goals of optimization problems. With these definitions, we can enunciate our general framework: a graph drawing algorithm consecutively solves a number of constrained optimization problems, in the priority order. The general priorities in Table 2.4.1 are useful for designing an ordered strategy for fulfilling drawing rules. Semantic rules are often stated as constraints, and the core of a graph drawing algorithm is a method for achieving the structural rules. The most important priority is relationship 5 in Table 2.4.1. An appropriate general strategy is: 1. In order to decide the relative position relationships between diagram elements, the topological rules are fulfilled. 2. Next the shape rules, concerning the ratio of length to breadth and the directional relationships between diagram elements, are fulfilled. 3. Lastly, in order to determine the lengths or distances the metric rules are fulfilled.
- 34. Framework for Automatic Graph Drawing Methods 17 For example, the general directed graph drawing algorithms and general undirected graph drawing algorithms presented in Chapter 4 adopt this strategy. Apart from trees or planar graphs, there are few efficient methods to fulfill rules, and it is often difficult to obtain an optimal solution. Therefore various heuristic methods have been developed. There has been research into the performance of heuristics used in heuristic methods, and there are some that have theoretical guarantees. Algorithms can be divided into the following types: (a) Algorithms that use graph theory and graph algorithms. (b) Heuristic algorithms. (c) Those that use force directed models (such as the ring and spring model), thermodynamics models (simulated annealing), bioinformatics model (genetic algorithm) and other simulations. (d) Hybrid algorithms combining different types of algorithms mentioned above. (e) AI algorithms such as " layout by example." A great variety of graph drawing methods have been proposed and investigated. Although these drawing methods may not have been developed according to the framework described here, this framework can help to understand the characteristics of graph drawing problems and the current state of research. In the next chapter this framework is used to describe existing drawing methods. At the end of the next chapter, there is a table showing how the features of typical graph drawing methods fit into our framework.
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- 36. o—o OUTLINES OF AUTOMATIC GRAPH DRAWING METHODS In this chapter we outline some typical drawing methods. The principal aim is to explain the various styles of drawing that are currently possible; this simplifies the description of technical problems. In order to understand the features of each drawing method, explanatory figures and reference documents are used as much as possible. Readers with particular requirements should be able to find a drawing method suitable to their problem by reading this chapter. For technical details, read Chapter 4 or one of the reference documents introduced. Most of the methods described in this chapter can be found in Walker (90), Eades (91), Sugiyama (88b, 89b), Eades & Sugiyama (90), Eades & Tamassia (89), Fruchterman & Reingold (91), Di Battista, Eades, Tamassia & Tollis (94), and Herman, Melancon & Marshall(OO). 3.1 Tree Drawing Methods Trees are divided into rooted trees, with specified roots, andfree trees, with no root 19
- 37. 20 Graph Drawing and Applications for Software and Knowledge Engineers specified. Rooted trees are also often divided into those limited to less than 2 child vertices, or binary trees, and those with no such limitation, or general trees. 3.1.1 Rooted Trees Rooted trees are often used to express hierarchical structures such as family trees, organization charts, search trees, decision trees and so on. The drawing convention {parallel lines, straight lines, independent) is appropriate for drawing rooted trees. Further, the following are used (see Figure 3.1.1): 1. Place the root at the highest level, and in order place the descendants on lower parallel lines {layering). 2. Do not allow edges to cross {no crossing). 3. In each layer, separate neighboring vertices by at least distance d {least separation). 4. Place parent vertices at the median center of child vertices {balance). 5. Minimize the width of the drawing {minimum width). 6. Draw isomorphic subtrees congruently, and axially isomorphic ones in a mirror image {identical). Drawing rules 1 to 3 are easily attain and so they are used in all. Various algorithms have been suggested for rules 4 to 6. layer 1 layer 2 layer 3 layer 4 symmetry 'f , ,. . * J J least distance minimum width Figure 3.1.1. Drawing of a rooted tree.
- 38. Outlines of Automatic Graph Drawing Methods 21 Binary Trees There has been a great deal of research into drawing methods of binary trees. In particular, ordered binary trees, where the left and right children are distinguished, have received a considerable amount of attention. In this case, the order of siblings within each layer is given. Wetherell & Shannon (79) proposed two algorithms (I and II). The difference between them is in the priority of the "balance" and "least width" rules above. In algorithm I "balance" is always achieved but "least width" is not always possible. Algorithm II achieves "least width" by weakening the conditions for "balance" (so that parent vertices are to be placed between child vertices, but not necessarily centered). Vaucher (80) independently proposed a similar algorithm to achieve "balance" and "least width". The algorithm proposed by Reingold & Tilford (81) is an improved version of Wetherell & Shannon's algorithm. This algorithm introduces the "identical drawing" rule, and gives this rule higher priority order than the "least width" rule. With this method, a symmetrical drawing can be achieved for a symmetrical tree. Supowit & Reingold (83) have investigated the smallest width drawing problem of binary graphs with consideration to the drawing rules of "balance", "least width" and "identical drawing". They model the problem using linear programming, and show that it can be solved in polynomial time; however, it can produce a drawing that is wider than necessary. They also show that if the orthogonal grid convention is imposed, then the problem becomes NP-hard. Note that none of these drawing methods for binary trees have been able to completely achieve the drawing rules of "balance", "least width" and "identical drawing". General Trees General trees are ordered «-ary trees: the number of children for each vertex is at most n, and the left-right order of the children is specified. Walker (90) investigated extensions of binary tree drawing algorithms to general trees. He proposed a linear time algorithm for arbitrary general trees that would completely fulfill the drawing rules (1) to (6) and draw within the smallest necessary width. The algorithm is discussed in detail in Section 4.1. Other Convention The convention for drawing a rooted tree as shown in Figure 3.1.1 is called the
- 39. 22 Graph Drawing and Applications for Software and Knowledge Engineers 1 b 1 1 e a 1 1 f 1 c 1 d (a) horizontal format •- f a b i r e 1 f (b) vertical format (c) tip-over convention Figure 3.1.2. Towformats and the tip-over convention [Lin (93)]. 1 * 1 - r 1 r 1 1 C 1 1 H 1 1 " 1 1 r 1 —L_fJ f L T J 1 £T 1 1 s 1 I, 1 1 h 1 ,• 1 ^ ' 1 7- 1 1 k 1 1 ' 1 1 ' 1 m 1 1 ™ 1 1 n 1 1 n j 1 b 1 1 I ^ 1 hJ " 1 e 1 1^ 1 1 A i 1 i 1 L^ - 1 'i 1 (a) tip-over convention (b) inclusion convention Figure 3.1.3. Tow conventions [Lin (93)].
- 40. Outlines of Automatic Graph Drawing Methods 23 classical convention. The classical convention demands that vertices of the same layer share a horizontal line, so that if there are many vertices of the same layer, a large width is required. To overcome this shortcoming other types are proposed by Lin (93): the tip-over convention and the inclusion convention. The tip-over convention combines the horizontal and vertical formats presented in Figure 3.1.2. The inclusion convention use a positional relation containment between the boxes between the boxes to represent the geometric structural relation between a parent and its children, as in Figure 3.1.3. In both conventions minimizing the size of drawing is considered; see Lin (93). 3.1.2 Free Trees A free tree has no specified root, and is not intrinsically hierarchical. They often have a layout other than the parallel lines layout, and it is common for vertices to be placed independently of coordinate system. Straight line routing is used. The drawing rules are as follows: 1. Edges are not allowed to cross {no crossing) 2. The placement of vertices avoids clustering (smallest separation). 3. Edge lengths are approximately equal (uniform edge lengths). 4. Where there is symmetry, this must be clearly shown (symmetry). Although there is not a great deal of research into drawing free trees for their own sake, there is interest in drawing methods of free trees because this can be the foundation of drawing methods for general undirected graphs. As shown in Figure 3.1.4, a connected undirected graph may be split into 2-connected components, and a drawing algorithm for free trees is useful for connecting the drawings of each 2-connected element. By slightly amending algorithms for rooted trees mentioned previously, concentric circle layout can be carried out. That is, one vertex is chosen as a root, and placed in the center of the drawing area, and the other vertices are placed on concentric circles with the root as a center. The sub-tree below a vertex is placed in afan-shaped wedge (see Figure 3.1.5). There are several variations of this algorithm, depending on the method of choosing the root, the diameter of the circles, the size of the fan-shaped wedge and so on. Usually the graph-theoretic center (the one or two vertices for which the length of a longest path to a leaf is minimized) is chosen as the root.
- 41. 24 Graph Drawing and Applications for Software and Knowledge Engineers A . B Ni • — - » c K—T ' y- a i p 1 (a) (b) (c) Figure 3.1.4. Connected undirected graphs and free trees: (a) a connected graph, (b) its 2 - connected elements, and (c) afree tree [Eades (91)]. Eades (91) presented a linear tree drawing algorithm using concentric circle layout and proved that the drawing rule of no crossing will always be achieved, and sought to bound the ratio of lengths of the largest edge and the shortest edge, for uniform edge length. As a modification of that algorithm, he proposed a linear time algorithm that placed all the leaves on the outermost concentric circle. Also, Manning & Atallah (89) proposed an algorithm for concentric circle layout that clearly shows symmetry. Other similar methods include Bernard (81), Read (86), and Esposito (88). A number of methods that simulate mechanical or thermodynamical systems may be used to draw graphs according to the principle that the system seeks a stable condition, in which energy is locally minimized. These methods are known as force-directed methods. The forces acting between vertices are attractive when neighboring vertices are
- 42. Outlines of Automatic Graph Drawing Methods 25 fan-shaped wedge Figure 3.1.5. Concentric circle drawing of afree tree [Eades (91)]. far apart, repulsive when they are too close together, and again repulsive when they are not neighbors. There are several different models, varying with the assignment of attractive and repulsive forces: Eades (84), Kamada & Kawai (89), Fuchterman & Reingold (91). and Davidson & Harel (96). These methods were originally devised for drawing general undirected graphs, so they are outlined in Section 3.3 and detailed in Sections 4.3 and 4.4. The problem with drawing trees using force-directed methods is that depending on the initial placement of vertices, it is not always possible to attain no crossings. There are several other drawing methods for free trees. H-tree drawing is a method of orthogonal grid layout for binary trees, and it is particularly for complete binary trees as it produces an orderly drawing as shown by Figure 3.1.6 (a). Methods presented by Eades (91) show that even for incomplete binary trees, a drawing such as Figure 3.1.6 (b) can be attained. The garden drawing method is a modification of the orthogonal grid layout, in which the longest path is drawn as a horizontal line, and each sub-tree is drawn near this line (see Figure 3.1.7). The name comes from the fact that the resulting picture looks like plants growing in a garden.
- 43. 26 Graph Drawing and Applications for Software and Knowledge Engineers h f T ? " T '' 1, 11 • 1> ,, • .( 1 • ii 1 1 1 1 • 1 i • 1 , 1 1 • ( > • • • • • • i i 1 i i i 1 1 i i i • 1 i i i • 1 1 1 1 1 • • 11 • 1, T T t T T T i • 1 i> • 1. II • *< i • 1 1 1 1 1 1 1 , 1 1 , 1 1 1 1 < • • • • • • , X 1 . i 1 1 i i i 1 • • < • 1 1 1 1 1 1 1 (a) a complete binary tree i . n • a ' ' ' i • • » • 1 » 1 , « 1' • • 1 1 • • • • 1 1 > m 1 1 • • 1 » - • ' (b) a general binary tree Figure 3.1.6. H-tree drawing method [Eades (91)].
- 44. Outlines of Automatic Graph Drawing Methods 27 u LU Figure 3.1.7. Garden drawing method [Eades (91)]. 3.2 Directed Graph Drawing Methods Drawing methods for directed graphs use two important classifications. These are the existence of cycles and the possibility of planar drawing. Graphs with no cycles are called acyclic directed graphs, and those with cycles are called general directed graphs. For drawing directed graphs, it is often convenient if the direction of all the edges flow in one direction (for example top to bottom, or left to right). This is called monotone drawing, and it is only possible with acyclic directed graphs. Also, for general directed graphs in which a monotone drawing is impossible, there is interest in drawing so that the number of edges that appear against the flow of direction of the majority of edges is minimized. There are two important problems with planar drawing. The first is concerned with planar monotone drawings of acyclic directed graphs. There has been a great deal of theoretical research into this problem recently, with many results. The second problem is concerned with planar drawings of the underlying undirected graph of a general directed graph. This is the same as the problem of planar drawings of undirected graphs, and is discussed in Section 3.3. 3.2.1 Acyclic Directed Graphs Acyclic directed graphs are widely used to express hierarchical structures. PERT graphs, ISA hierarchies, subroutine call graphs, and Hasse diagrams are all examples. These graphs are usually drawn in a monotone fashion, that is, all edges follow the same
- 45. 28 Graph Drawing and Applications for Software and Knowledge Engineers direction. For general directed graphs, one should compute a minimum size set of feedback edges; the direction of these is reversed, so changing it into a acyclic directed graph. Thus drawing methods for acyclic directed graphs are also important as drawing methods for general directed graphs. Upward (Downward) Drawing A planar drawing of a directed graph is an upward (downward) drawing if all edges are drawn as curves that monotonically increase (decrease) in a vertical direction. Of course if a directed graph has an upward (downward) drawing, then it is acyclic. i p«-- - t — f — i - — - « - - + — + - + - — ^ i - / - i — ~ ^ H + • — < i '' ' *t r ^ ^ - — x X / ' ' ' (a) (b) (c) Figure 3.2.1. Three types of upward drawing: (a)free upward drawing, (b) orthogonal grid upward drawing, and (c) parallel line upward drawing [Di Battista et al (88)]. Di Battista & Tamassia (88) proposed an efficient algorithm for producing three types of upward drawing of acyclic directed graphs. Figure 3.2.1 shows the features of these drawings schematically. As drawing conventions, Figure 3.2.1(a) uses (free,
- 46. Outlines of Automatic Graph Drawing Methods 29 straight lines, independent), Figure 3.2.1(b) uses {orthogonal grid, polylines, independent), and Figure 3.2.1(c) uses {parallel lines, straight lines, dependent). Algorithms for expressing symmetry and isomorphic shape subgraphs in planar upward drawing are presented in Di Battista, Tamassia & Tollis (89). Bertolazzi, Di Battista, Liotta & Mannino (94) give an efficient algorithm for determining whether 3-connected directed graphs with planar upward drawings. Hierarchical Drawing The most frequently used monotone drawing is hierarchical drawing. As with rooted trees, vertices and edge bends are placed on horizontal lines called layers drawn at regular intervals (see Figure 3.2.2). Directed graphs in which vertices are assigned to layers are called hierarchical graphs. The drawing conventions {parallel lines, straight lines, independent), alternatively {parallel lines, polylines, independent) are used, and the usual drawing rules include the following: 1. Edge crossings are avoided {minimization of edge crossings) 2. On each layer, neighboring vertices are separated by at least a certain distance {minimum separation). 3. Parents are placed at the barycenter of their children {balance). 4. Neighboring vertices in the immediately upper (or lower) layer are placed nearby {proximity). layer 1 layer 2 layer 3 layer 4 layer 5 layer 6 layer 7 layer 8 layer 9 Figure 3.2.2. Hierarchical drawing.
- 47. 30 Graph Drawing and Applications for Software and Knowledge Engineers The minimization of the number of edge crossings is a basic drawing rule for hierarchical drawing. An efficient algorithm for producing planar hierarchical drawings of hierarchical graphs can be seen in Di Battista & Nardelli (86, 89). Sugiyama, Tagawa & Toda (79, 81) and Sugiyama (82, 87) proposed the basic method (called STT method or Sugiyama method) for hierarchical drawing. This is a method for hierarchically drawing general directed graphs, and when there are cycles, the strongly connected component is substituted for the representative vertex, or the feedback edge is sought and directed in the opposite direction, turning it into a hierarchical graph. The algorithm consists of 4 steps: layering, normalization, vertex order determination, vertex location determination (see Figure 3.2.3). Normalization allows the simplification of the problem by changing the original edge routing problem into a placement problem. Even with this simplification, if each step is formalized as an optimization problem, then the minimum feedback arc set problem (NP complete), the edge-crossings minimization problem (NP-complete), and the quadratic programming problem must be solved, and heuristic methods such as the barycentric method for the reduction of the number of edge crossings, or the priority method for the determination of coordinates, have been developed. Variations of this approach can be found in: Carpano (80), Rowe, Davis, Messinger, Meyer, Spirakis & Tuan (87), Gansner, North & Vo (88), Messinger (88), Itumi & Kogure (88), and Sander (96). Carpano (80) developed an iterative method for the reduction of crossings in 2-layer graphs, and applied it to the hierarchical drawing of directed graphs. This was extended and applied also to &-layer graphs. The main feature of this method is that all edges are routed as straight lines (see Figure 3.2.4). Rowe et al. (87) improved the STT method and developed a browser for directed graphs. Changes can be seen in such areas as the way that the smallest feedback edge set is sought, and layering is formally attained, and the routing convention where the minimization of the number of bends is emphasized and diagonal line routing is considered (see Figure 3.2.5). This method is aimed at applications in databases, and diagrams of references between modules in large scale computer programs and so on. Gansner, North & Vo (88) amended the following points of the STT method. They improved the efficiency of the layering algorithm, adopted the median method (Eades & Wormald (86)) instead of the barycentric method for determining the order, and a linear programming method instead of the quadratic programming method for determining coordinates. They also developed a drawing tool called DAG, with postscript output, and B-spline edges, and so on.
- 48. Outlines of Automatic Graph Drawing Methods 31 b c (a) (b) c b a b a I A4s (c) h* i • •_/' (d) Figure 3.2.3. Steps of hierarchical drawing, (a) given directed graph, (b) layering and normalization, (c) order determination, (d) coordinate determination. Figure 3.2.4. Hierarchical drawing with straight line routing [Carpono (80)].
- 49. 32 Graph Drawing and Applications for Software and Knowledge Engineers Figure 3.2.5. Hierarchical drawing in which the number of bends has been reduced (this is the same graph as in Figure 3.2.2) [Rowe et al (87)]. Messinger (88) proposed a divide and conquer method aimed at speeding up the STT method. There are 3 steps to this method (see Figure 3.2.6): 1. Partition the large graph into subgraphs. 2. Compute the drawing for each subgraph using the STT method. 3. Now consider the edges between subgraphs. Determine the overall layout, and route the edges between subgraphs. The smaller the subgraphs, the less machine time, but the more difficult the composition (the third step) becomes. In this method there is not enough consideration given to the composition step. One possible solution to this problem is to use the drawing method for compound graphs discussed in Chapter 4. For hierarchical drawing, Itumi & Kogure (88) proposed a new drawing rule to add to the structural rules. This new rule is elimination of crossing among outlines, and they proposed an algorithm to achieve this. In Figure 3.2.7 (a) the set of descendants of vertex a and the set of descendants of vertex b intersects in the dotted line area, making it difficult to see. The elimination of crossing among outlines eliminates this area of intersection, as shown in Figure 3.2.7 (b). The STT method, its modifications and extensions are discussed in greater detail in Chapter 4. The minimization of the number of crossings in a hierarchical graph, even if it is a 2-layer graph and the position of the vertices of the first layer are fixed, is NP-complete
- 50. Another Random Scribd Document with Unrelated Content
- 51. of Elizabeth, of Richard Plantagenet, of Harold, of the earliest bards. We, too, like Taliesin, have borne a banner before Alexander, have been with our Lord in the manger of the ass, have been in India, and with the “remnant of Troia,” and with Noah in the ark, and our original country is “the region of the summer stars.” And of these many folds in our nature the face of the earth reminds us, and perhaps, even where there are no more marks visible upon the land than there were in Eden, we are aware of the passing of time in ways too difficult and strange for the explanation of historian and zoologist and philosopher. It is this manifold nature that responds with such indescribable depth and variety to the appeals of many landscapes. We come to a huge, flat-bottomed, grassy coombe, smooth as a racecourse, that winds out of the cornland into the heart of the Downs. It is like the bed of a river of great depth. At its entrance beeches clothe either side; but presently they cease, and up the steep juniper slopes go the paths of hares, of the herds and flocks of earliest ages and of the men and women and children also, whose children’s children’s children have forgotten them though not perhaps their philosophy. The grass of the slope is mingled with small sweet herbage, the salad burnet rosy-stemmed, the orange bird’s-foot trefoil, the purple thyme, the fine white flax, the delicatest golden hawk-bit, and basil and marjoram, and rosettes of crimson thistles, all sunny warm and fragrant, glittering and glowing or melting into a simmering haze, musical with grasshoppers and a- flutter with blue butterflies, so that the earth seems to be a thick- furred, genial animal. At length the windings shut out the plain, and the coombe is a green hall roofed by the hot blue sky. Its walls are steeper than ever, and the burrowings of the rabbits have streaked the grasses with long splashes—like those made by sea-birds on rocks—of white chalk. The curves of these walls are like those of the flight of the swifts that dive overhead. Here there are no human paths, no sign of house, of grave, of herd, of cultivation. It is the world’s end, and the rabbits race up and down as in a dream of solitude.
- 52. Yet the mind is not discontented and unfed. This is no boundless solitude of ocean where one may take a kind of pleasure To float for ever with a careless course And think himself the only being alive. It is not an end but a beginning that we have reached. These are the elements—pure earth and wind and sunlight—out of which beauty and joy arise, original and ancient, for ever young. Their presence restores us not to the Middle Ages, not to the days of Mr. Doughty’s heroic princes and princesses of Britain,[4] not to any dim archæologist’s world of reeking marsh and wood, of mammoth and brutish men, but to a region out of space and out of time in which life and thought and physical health are in harmony with sun and earth, fragrant as the flowers in the grass, blithe as the grasshopper, swift as the hares, divine; and out of it all arises a vision of the man who will embody this thought, a man whom human infelicity, discontented with the past, has placed in a golden age still farther back, for the sufficient reason that in every age he has been a dream, and our dreaming is of the dawn or the night, always disappointed but undaunted by the day that follows. And so no storied valley or hillside is richer in humanity than this coombe. It is one of the countless Edens where we are in contact not with the soldier and ploughman and mason that change the surface of the earth, but with prophet and poet who have ever lived to trace to Nature and to the early ages the health and vigour of men. There is the greatest antiquity of all, peace and purity and simplicity, and in the midst is the mother Earth, the young mother of the world, with a face like Ceres before she had lost Persephone in the underworld. In fact, so blessed is this solitary hall that after climbing out it is mournful to see the rabbit-worn tunnels and the Roman camp on the ridge.
- 53. CORNWALL. In Cornwall, where the wrinkles and angles of the earth’s age are left to show, antiquity plays a giant’s part on every hand. What a curious effect have those ruins, all but invisible among the sands, the sea-blue scabious, the tamarisk and rush, though at night they seem not inaudible when the wild air is full of crying! Some that are not nearly as old are almost as magical. One there is that stands near a great water, cut off from a little town and from the world by a round green hill and touched by no road but only by a wandering path. At the foot of this hill, among yellow mounds of sand, under blue sky, the church is dark and alone. It is not very old—not five centuries—and is of plainest masonry: its blunt short spire of slate slabs that leans slightly to one side, with the smallest of perforated slate windows at the base, has a look of age and rusticity. In the churchyard is a rough grey cross of stone—a disc supported by a pillar. It is surrounded by the waving noiseless tamarisk. It looks northward over the sandhills at a blue bay, guarded on the west by tall grey cliffs which a white column surmounts. For a time the nearer sandhills have rested and clothed themselves in bird’s-foot trefoil, thyme, eyebright and short turf: but once the church was buried beneath them. Between the round hill and the church a tiny stream sidles along through a level hiding- place of flags and yellow flag flowers, of purple figwort and purple orchis and green grass. A cormorant flies low across the sky—that sable bird which seems to belong to the old time, the time of badger and beaver, of ancient men who rose up out of the crags of this coast. To them, when the cuckoo first called one April, came over the blue sea a small brown ship, followed by three seals, and out of it descended a Christian from Ireland, black-haired, blue-eyed, with ready red lips and deep sweet voice and spoke to them, all alone. He told them of a power that ruled the blue waters and shifting sands, who could move the round green hill to the rock of the white gulls; taller and
- 54. grimmer than the cloven headland yet sweet and gentle as the fennel above; deep-voiced as the Atlantic storm, tender also as the sedgewarbler in the flags below the hill; whose palace was loftier than the blue to which the lark was now soaring, milder and richer than the meadows in May and everlasting; and his attendants were more numerous and bright than the herring under a moon of frost. The milkpails should be fuller and the grass deeper and the corn heavier in the car if they believed in this; the pilchards should be as water boiling in the bay; and they should have wings as of the white birds that lounged about the precipices of the coast. And all the time the three seals lay with their heads and backs above the shallows and watched. Perhaps the men believed his word; perhaps they dropped him over the precipice to see whether he also flew like a gull: but here is the church named after him. All along the coast (and especially where it is lofty and houseless, and on the ledges of the crags the young grey gulls unable to fly bob their heads seaward and try to scream like their parents who wheel far and near with double yodeling cry), there are many rounded barrows looking out to sea. And there are some amidst the sandhills, bare and corrugated by the wind and heaved up like a feather-bed, their edges golden against the blue sky or mangily covered by drab marram grass that whistles wintrily; and near by the blue sea, slightly roughened as by a barrow, sleeps calm but foamy among cinder-coloured isles; donkeys graze on the brown turf, larks rise and fall and curlews go by; a cuckoo sings among the deserted mines. But the barrows are most noble on the high heather and grass. The lonely turf is full of lilac scabious flowers and crimson knapweed among the solid mounds of gorse. The brown-green-grey of the dry summer grass reveals myriads of the flowers of thyme, of stonecrop yellow and white, of pearly eyebright, of golden lady’s fingers, and the white or grey clover with its purest and earthiest of all fragrances. Here and there steep tracks descend slantwise among the thrift-grown crags to the sea, or promise to descend but end abruptly in precipices. On the barrows themselves, which are either isolated or in a group of two or three, grow thistle and gorse. They
- 55. command mile upon mile of cliff and sea. In their sight the great headlands run out to sea and sinking seem to rise again a few miles out in a sheer island, so that they resemble couchant beasts with backs under water but heads and haunches upreared. The cliffs are cleft many times by steep-sided coves, some with broad sand and shallow water among purple rocks, the outlet of a rivulet; others ending precipitously so that the stream suddenly plunges into the black sea among a huddle of sunless boulders. Near such a stream there will be a grey farm amid grey outbuildings—with a carved wooden eagle from the wreckage of the cove, or a mermaid, once a figure-head with fair long hair and round bosom, built into the wall of a barn. Or there is a briny hamlet grouped steeply on either side of the stream which gurgles among the pebbles down to the feet of the bearded fisherman and the ships a-gleam. Or perhaps there is no stream at all, and bramble and gorse come down dry and hot to the lips of the emerald and purple pools. Deep roads from the sea to the cliff-top have been worn by smuggler and fisherman and miner, climbing and descending. Inland shows a solitary pinnacled church tower, rosy in the warm evening—a thin line of trees, long bare stems and dark foliage matted—and farther still the ridges of misty granite, rough as the back of a perch. Of all the rocky land, of the sapphire sea white with quiet foam, the barrows are masters. The breaking away of the rock has brought them nearer to the sea as it has annihilated some and cut off the cliff-ways in mid-career. They stand in the unenclosed waste and are removed from all human uses and from most wayfaring. Thus they share the sublimity of beacons and are about to show that tombs also have their deaths. Linnet and stonechat and pipit seem to attend upon them, with pretty voices and motions and a certain ghastliness, as of shadows, given to their cheerful and sudden flittings by the solemn neighbourhood. But most of their hold upon the spirit they owe to their powerful suggestion that here upon the high sea border was once lived a bold proud life, like that of Beowulf, whose words, when he was dying from the wounds of his last victory, were: “Bid the warriors raise a funeral mound to flash
- 56. with fire on a promontory above the sea, that it may stand high and be a memorial by which my people shall remember me, and seafarers driving their tall ships through the mist of the sea shall say: ‘Beowulf’s Mound.’” In Cornwall as in Wales, these monuments are the more impressive, because the earth, wasting with them and showing her bones, takes their part. There are days when the age of the Downs, strewn with tumuli and the remnants of camp and village, is incredible; or rather they seem in the course of long time to have grown smooth and soft and kind, and to be, like a rounded languid cloud, an expression of Earth’s summer bliss of afternoon. But granite and slate and sandstone jut out, and in whatsoever weather speak rather of the cold, drear, hard, windy dawn. Nothing can soften the lines of Trendreen or Brown Willy or Carn Galver against the sky. The small stone-hedged ploughlands amidst brake and gorse do but accentuate the wildness of the land from which they have been won. The deserted mines are frozen cries of despair as if they had perished in conflict with the waste; and in a few years their chimneys standing amidst rotted woodwork, the falling masonry, the engine rusty, huge and still (the abode of rabbits, and all overgrown with bedstraw, the stern thistle and wizard henbane) are in keeping with the miles of barren land, littered with rough silvered stones among heather and furze, whose many barrows are deep in fern and bramble and foxglove. The cotton grass raises its pure nodding white. The old roads dive among still more furze and bracken and bramble and foxglove, and on every side the land grows no such crop as that of grey stones. Even in the midst of occasional cornfield or weedless pasture a long grey upright stone speaks of the past. In many places men have set up these stones, roughly squaring some of them, in the form of a circle or in groups of circles—and over them beats the buzzard in slow hesitating and swerving flight. In one place the work of Nature might be mistaken for that of man. On a natural hillock stands what appears to be the ruin of an irregularly heaped wall of grey rock, roughened by dark-grey lichen, built of enormous angular fragments like the masonry of a giant’s child.
- 57. Near at hand, bracken, pink stonecrop, heather and bright gold tormentil soften it; but at a distance it stands black against the summer sky, touched with the pathos of man’s handiwork overthrown, yet certainly an accident of Nature. It commands Cape Cornwall and the harsh sea, and St. Just with its horned church tower. On every hand lie cromlech, camp, circle, hut and tumulus of the unwritten years. They are confused and mingled with the natural litter of a barren land. It is a silent Bedlam of history, a senseless cemetery or museum, amidst which we walk as animals must do when they see those valleys full of skeletons where their kind are said to go punctually to die. There are enough of the dead; they outnumber the living; and there those trite truths burst with life and drum upon the tympanum with ambiguous fatal voices. At the end of this many-barrowed moor, yet not in it, there is a solitary circle of grey stones, where the cry of the past is less vociferous, less bewildering, than on the moor itself, but more intense. Nineteen tall, grey stones stand round a taller, pointed one that is heavily bowed, amidst long grass and bracken and furze. A track passes close by, but does not enter the circle; the grass is unbent except by the weight of its bloom. It bears a name that connects it with the assembling and rivalry of the bards of Britain. Here, under the sky, they met, leaning upon the stones, tall, fair men of peace, but half- warriors, whose songs could change ploughshare into sword. Here they met, and the growth of the grass, the perfection of the stones (except that one stoops as with age), and the silence, suggest that since the last bard left it, in robe of blue or white or green—the colours of sky and cloud and grass upon this fair day—the circle has been unmolested, and the law obeyed which forbade any but a bard to enter it. Sky-blue was the colour of a chief bard’s robe, emblematic of peace and heavenly calm, and of unchangeableness. White, the colour of the Druid’s dress, was the emblem of light, and of its correlatives, purity of conduct, wisdom, and piety. Green was the colour of the youthful ovate’s robe, for it was the emblem of growth. Their uniformity of colour signified perfect truth. And the inscription upon the chair of the bards of Beisgawen was, “Nothing is that is not for ever and ever.” Blue and white and green, peace and
- 58. light and growth—“Nothing is that is not for ever and ever”—these things and the blue sky, the white, cloudy hall of the sun, and the green bough and grass, hallowed the ancient stones, and clearer than any vision of tall bards in the morning of the world was the tranquil delight of being thus “teased out of time” in the presence of this ancientness. It is strange to pass from these monumental moors straight to the sea which records the moments, not the years or the centuries. In fine weather especially its colour—when, for example, it is faintly corrugated and of a blue that melts towards the horizon into such a hue that it is indistinguishable from the violet wall of dawn—is a perpetual astonishment on account of its unearthliness and evanescence. The mind does not at once accept the fact that here underneath our eyes is, as it were, another sky. The physical act of looking up induces a special mood of solemnity and veneration, and during the act the eyes meet with a fitting object in the stainless heavens. Looking down we are used to seeing the earth, the road, the footpath, the floor, the hearth; but when, instead, it is the sea and not any of these things, although our feet are on firm land, the solemnity is of another kind. In its anger the sea becomes humanized or animalized: we see resemblances to familiar things. There is, for instance, an hour sometimes after sunset, when the grey sky coldly lights the lines of white plumes on a steely sea, and they have an inevitable likeness to a trampling chivalry that charges upon a foe. But a calm sea is incomparable except to moods of the mind. It is then as remote from the earth and earthly things as the sky, and the remoteness is the more astonishing because it is almost within our grasp. It is no wonder that a great idea was expressed by the fortunate islands in the sea. The youthfulness, the incorruptibility of the sea, continually renewing itself, the same from generation to generation, prepares it as a fit sanctuary of the immortal dead. So at least we are apt to think at certain times, coming from the heavy, scarred, tormented earth to that immense aëry plain of peacock blue. And yet at other times that same unearthliness will suggest quite other thoughts. It has not changed and shrunken and grown
- 59. like the earth; it is not sun-warmed: it is a monster that has lain unmoved by time, sleeping and moaning outside the gates within which men and animals have become what they are. Actually that cold fatal element and its myriad population without a sound brings a wistfulness into the mind as if it could feel back and dimly recall the dawn of time when the sea was incomprehensible and impassable, when the earth had but lately risen out of the waters and was yet again to descend beneath: it becomes a type of the waste where everything is unknown or uncertain except death, pouring into the brain the thoughts that men have had on looking out over untrodden mountain, forest, swamp, in the drizzling dawn of the world. The sea is exactly what it was when mountain, forest, swamp were imperturbable enemies, and the sight of it restores the ancient fear. I remember one dawn above all others when this restoration was complete. When it was yet dark the wind rose gustily under a low grey sky and a lark sang amidst the moan of gorse and the creak of gates and the deeply-taken breath of the tide at the full. Nor was it yet light when the gulls began to wheel and wind and float with a motion like foam on a whirlpool or interwoven snow. They wheeled about the masts of fishing-boats that nodded and kissed and crossed in a steep cove of crags whose black edges were slavered by the foam of the dark sea; and there were no men among the boats or about the grey houses that looked past the walls of the cove to the grim staircase and sea-doors of a black headland, whose perpendicular rocks stood up far out of the reach of the wings fashioned in the likeness of gigantic idols. The higher crags were bushy and scaly with lichen, and they were cushioned upon thrift and bird’s-foot trefoil and white bladder campion. It was a bristling sea, not in the least stormy, but bristling, dark and cold through the slow colourless dawn, dark and cold and immense; and at the edge of it the earth knelt, offering up the music of a small flitting bird and the beauty of small flowers, white and gold, to those idols. They were terrible enough. But the sea was more terrible; for it was the god of whom those rocks were the poor childish images, and it seemed that the god had just then disclosed his true nature and
- 60. hence the pitiful loveliness of the flowers, the pitiful sweetness of the bird that sang among the rocks at the margin of the kind earth. Now and then the sea will startle by some resemblance to the earth. Thus I have come unexpectedly in sight of it on a strange coast and have not known that it was the sea. A gale from the north-east was blowing, and it was late afternoon in mid-winter. The land was sandy moorland, treeless and dark with iron-coloured heather. A mile away I saw rising up into the sky what seemed a peaty mountain in Cardiganshire, as it would be in a tempest of rain, and it was only when I was near the cliff and could see the three long walls of white waves towards the shore that I knew it was the sea. More common is the calm dark-blue sea in mid-summer, over which go criss-cross bands of lighter hue, like pale moorland paths winding about a moor. In a stern land like Cornwall that so often refuses the consolations of grass and herb and tree, the relentings are the more gracious. These are to be found in a whole valley where there are sloping fields of corn and grass divided by green hedges, and woods rich and misty and warm, and the bones of the land are buried away until it ends in a bay where high and cavernous dark rocks stand on either side of blue water and level sand. Often all the sweetness of the country round seems to have run into one great roadside hedge as dewdrops collect in the bosom of a leaf. The stones of the original wall are themselves deeply hidden in turf, or from the crevices ferns descend and the pale blooms of pennywort rise up; the lichen is furry and the yellow or pink stonecrop is neat and dense; ivy climbs closely up and hangs down in loose array. Up from the top of the wall or mound rise bramble and gorse and woodbine over them, or brier and thorn and woodbine again; and the tallest and massiest of foxgloves cleave through these with their bells, half a hundred of them in rows five deep already open and as many more yet in bud, dense as grapes, dewy, murmurous; and below the foxgloves are slender parsleys, rough wood sage and poppies. At the foot of the wall, between it and the road, is a grassy strip, where the yarrow grows feathery with gilded cinquefoil and tormentil—or above nettles
- 61. as dense as corn rise large discs of white hog parsnip flower, a coarse and often dirty flower that has a dry smell of summer—or bramble and brier arch this way and that their green and rosy and purple stems, bright leaves, flowers pink and white. Only the shin- breaking Cornish stiles of stone, interrupting the hedge and giving a view of barren hills or craggy-sided sea, destroy the illusion created by this exuberance of herb and bush and the perfume of woodbine and rose. Nowhere is the stateliness or grace or privacy of trees more conspicuous than about the Cornish towns and farms. The tall round-topped elms above Padstow, for example, would be natural and acceptable unconsciously elsewhere; but above those crossing lines of roof they have an indescribable benevolence. The farmhouses are usually square, dry and grey, being built of slate with grey-slated roofs painted by lichen; some are whitewashed; in some, indeed, the stones are of many greys and blues, with yellowish and reddish tinges, hard, but warm in the sun and comforting to look at when close to the sea and some ruinous promontory; few are screened by ivy or climbing rose. The farm buildings are of the same kind, relieved by yellow straw, the many hues of hay, the purple bracken stacks, the dark peat. The gates are coarse and mean, of iron or of cheap or rough wood, lightly made, patched, held together by string, and owing their only charm to the chance use of the curved ribs of ships as gate-posts. But to many of the buildings sycamore and ash and apple trees bent above tall grass lend their beauty of line, of mass, of colour, of shade, of sound and of many motions. I can never forget the rows of ash trees, the breezy sycamores and the tamarisks by ancient Harlyn, with its barrows on the hill, its ruins of chapel and church among rushes and poppies, its little oak wood by the sandy river mouth where the men of old time buried their dead, the poppied corn, the white gulls and their black shadows wheeling over sunny turf. The file of lean woods seen between Perranporth and St. Agnes inland. The sycamores above the farm near Towan cross where the road dips and the deep furrow of a little valley winds, with hay upon its slopes, out to sea.
- 62. The green wood, long and beautiful, below the gentle brown slopes of Hudder Down. The several companies of trees in the valley by the Red River, and the white farm of Reskajeage near by, under ash and elm, sycamore and wych-elm and lime, a rough orchard of apples and a gnarled squat medlar to one side—the trees grouped as human figures are when they begin to move after some tense episode. The wych-elm, sycamore and ash round the tower of Gwithian church and in amongst the few thatched cottages alongside the yellow towans and violet sea. In a land of deserted roofless houses with solid chimneys that no man wants, the narrow copse of small spindly oaks upholding with bare crooked stems as of stone a screen of leaves, above a brooklet that runs to the sea through dense rush and foxglove and thistle where the sedge- warbler sings. The long low mound of green wood nearest to Land’s End. Between Tregothal and Bosfranken, the wet copse in a narrow valley, where red campion and bracken and bramble are unpenetrated among flowery elders, sallows, thorns and sycamores. A farm that has a water-mill and water gloomy and crystal under sycamore and ash. The thin halting procession of almost branchless trees on the ridge of the Beacon above Sancreed—a procession that seems even at mid-day to move in another world, in the world and in the age of the stone circles and cairns and cromlechs of the moor beyond. The sycamore and elder that surround and tower above Tregonebris near Boscawen Un. The avenue of ash and elm and wych-elm and sycamore, very close together, leading from grey Nancothan mill, where the dark-brown water mingles its noise with the rustling trees. The wych-elms and golden-fruited sycamores about the roads near St. Hilary, and the long avenue of ash up to the church itself, and the elms through which the evening music floats, amidst the smell of hay, in a misty mountained sunset. Under the flaming fleeces of a precipitous sky, in a windless hush and at low tide, I descended to a narrow distinct valley just where a stream ran clear and slow through level sands to a bay, between headlands of rocks and of caves among the rocks. The sides of the valley near the sea were high and steep and of grass
- 63. until their abrupt end in a low but perpendicular wall of rock just above the river sands. Inland the valley began to wind and at the bend trees came darkly trooping down the slopes to the water. Immediately opposite the ford—the wet sands being unscathed by any foot or hoof or wheel—a tributary ran into the river through a gorge of its own. It was a gorge not above a hundred feet across, and its floor was of sand save where the brook was running down, and this floor was all in shadow because the banks were clothed in thick underwood and in ash, sycamore, wych-elm and oak meeting overhead. And in these sands also there was no footprint save of the retreated sea. There was no house, nor wall, nor road. And there was no sound in the caverns of foliage except one call of a cuckoo as I entered and the warbling of a blackbird that mused in the oaks and then laughed and was silent and mused again and filled the mind with the fairest images of solitude—solitude where a maid, thinking of naught, unthought of, unseen, combs out her yellow hair and lets her spirit slip down into the tresses—where a man fearful of his kind ascends out of the deeps of himself so that his eyes look bravely and his face unstiffens and unwrinkles and his motion and gesture is fast and free—where a child walks and stops and runs and sings in careless joy that takes him winding far out into abysses of eternity and makes him free of them, so that years afterward the hour and place and sky return, and the eternity on which they opened as a casement, but not the child, not the joy. I like trees for the cool evening voices of their many leaves, for their cloudy forms linked to earth by stately stems—for the pale lifting of the sycamore leaves in breezes and also their drooping, hushed and massed repose, for the myriad division of the light ash leaves—for their straight pillars and for the twisted branch work, for their still shade and their rippling or calm shimmering or dimly glowing light, for the quicksilver drip of dawn, for their solemnity and their dancing, for all their sounds and motions—their slow- heaved sighs, their nocturnal murmurs, their fitful fingerings at thunder time, their swishing and tossing and hissing in violent rain, the roar of their congregations before the south-west wind when it
- 64. seems that they must lift up the land and fly away with it, for their rustlings of welcome in harvest heat—for their kindliness and their serene remoteness and inhumanity, and especially the massiest of the trees that have also the glory of motion, the sycamores, which are the chief tree of Cornwall, as the beeches and yews are of the Downs, the oaks of the Weald, the elms of the Wiltshire vales. Before I part from trees I should like to mention those of mid- Somerset—and above all, the elms. I am thinking of them as they are at noon on the hottest days of haymaking at the end of June. The sky is hot, its pale blue without pity and changing to a yellow of mist near the horizon. The land is level and all of grass, and where the hay is not spread in swathes the grass is almost invisible for the daisies on its motionless surface. Here and there the mower whirrs and seems natural music, like the grasshopper’s, of the burning earth. Through the levels wind the heavy-topped grey willows of a hidden stream. In the hedges and in the wide fields and about the still, silent farmhouses of stone there are many elms. They are tall and slender despite their full mounded summits. They cast no shade. In the great heat their green is all but grey, and their leaves are lost in the mist which their mingling creates. Grey-hooded, grey-mantled, they seem to be stealing away over the fields to the sanctuary of the dark-wooded hills, low and round and lapped entirely in leaves, which stand in the mist at the edge of the plain—to be leaving that plain to the possession of the whirring mower and the sun of almighty summer. Sycamores solemnized the Cornish farm in the twilight, where I asked the farmer’s wife if she could let us have two beds for the night. She stood in the doorway, hands on hips, watching her grandchildren’s last excited minutes of play in the rickyard. “He’s the master,” she replied, pointing to the farmer who was talking to his carter, between the rickyard and the door, under the sycamores. “Two beds?”
- 65. “That is what we should like,” said my friend and I. “What do you want with two beds?” he asked with a tinge of scorn as well as of pity in his frank amusement. “My missus and I have only had one bed these forty years.” Here he laughed so gaily that he could not have embarrassed the very devil of puritanism, and turning to his man he called forth a deep bass laughter and from his wife a peal that shook her arms so that she raised them to the sides of the porch for better support; the children also turned their laughter our way. “But perhaps one of you kicks in his sleep?... We don’t.... Come inside. I dare say you are tired.... Good-night, John. Now, children, up with you.” I think they were the most excellent pair of man and woman I ever saw. Both were of a splendid physical type, she the more energetic, black-haired, black-eyed, plump and tall and straight; he the more enduring, fair-haired and bearded, blue-eyed, hardly her equal in height, certainly not in words. In forty years neither had overpowered the other. They had not even agreed to take separate paths, but like two school-boys, new friends, they could afford to contend together in opinions without fear of damage or of lazy truce. He had ploughed and sowed and reaped: she had borne him seven children, had baked and churned and stitched. They had loved sweet things together, and, with curses at times, their children and the land. Physical strength and purity—that were in them the whole of morality—seemed to have given them that equality with the conditions of life which philosophy has done nothing but talk about. They of all men and women had perhaps jarred least upon the music of the spheres. They had the right and power to live, and the end was laughter. In all those years they had been separated but once. Until four years ago she had not been out of Cornwall except to bury her mother, who had suddenly died in London. Two hundred pounds fell to her share on that death and the money arrived one morning after
- 66. the harvest thanksgiving. For a week she continued to go about her work in the old way save that she sent rather hurriedly for a daughter who had just left her place as cook in Exeter. At the end of the week, having stored the apples and shown her daughter how to use the separator, she walked in to Penzance in her best clothes but without even a handbag; her husband was out with his gun. By the next day she was at Liverpool. She sent off a picture postcard, with a little note written by the shopkeeper, saying that she would be back by Christmas, and telling her husband to sell the old bull. Then she sailed for New York. She saw Niagara; she visited her nephew, John Davy, at Cincinnati; she spent two weeks in railway travelling west and south, and saw the Indians. Four days before Christmas she was back in the rickyard, driving before her a young bull and carrying in her hand a bunch of maize. “Well, Ann, you’re back before your time,” said her husband, after praising the beast. “Yes, Samuel, and I feel as if I could whitewash the dairy, that I do,” said she. “Suppose you wait till to-morrow,” proposed Sam Davy. “I think I will, for I can hear that Mary is behind with the separator.” “She’s a good girl, but she hasn’t got your patience, my dear.” “Oh, here, Sam, here’s the change,” she said, giving him the bunch of maize. In Cornwall many of the women looked less English than the men. The noticeable men were fair-haired and, of fair complexion, blue-eyed and rather small-headed, upright and of good bearing. The noticeable women had black hair, pale, seldom swarthy, faces, very dark eyes. Perhaps the eyes were more foreign than anything
- 67. else in them: they were singularly immobile and seldom changed in expression with their voices. Several of the dark-eyed, black-haired women had a beauty of a fearless character like gypsy women, in their movement and expression. But the wives of small farmers and miners on piecework look old very soon and are puckered and shadowy in the face. Some of these middle-aged and old women suggested an early and barbarous generation. The eyes were small and deep-set, and the face narrowed forward like an animal’s; which gave the whole a peering expression of suspicion and even alarm. The eyes of most human beings are causes of bewilderment and dismay if curiously looked at; but the strangest I ever saw were in an old Cornish woman. They were black and round as a child’s, with a cold brightness that made them seem not of the substance of other eyes, but like a stone. They were set in a narrow, bony face of parchment among grey hair crisp and disarrayed. I saw them only for a few minutes while I asked a few questions about the way, and it was as much as I could do to keep up the conversation, so much did those motionless eyes invite me to plunge into an abyss of human personality—such intense loneliness and strangeness did they create, since they proclaimed shrilly and clearly that beyond a desire to be fed and clothed we had nothing in common. Had they peered up at me out of a cromlech or hut at Bosporthennis I could not have been more puzzled and surprised. Men and women were hospitable and ready to smile as the Welsh are; and they have an alluring naïveté as well as some righteousness. One family was excessively virtuous or had a wish to appear so: I do not know which alternative to like the less, since it was in a matter of game. They rented land on a large estate and had a right to the rabbits: the hares were sacred to the great landowner. The farmer’s wife assured me that one of her sons had lately brought in a lame hare and proposed to put it out of its pain, but that she had said: “No, take it out and let it die outside anywhere. The best thing is to be afraid in things of this kind and then you won’t go wrong.” Doing much the same kind or quantity of manual work as their husbands and being much out of doors, the
- 68. women’s manners were confident and free. Their speech was as a rule fluent and grammatical and clearly delivered, with less accent than in any part of England. Coming into a mining village one day and wanting tea, I asked a woman who was drawing water from a farmyard well if she could make me some, thinking she was the farmer’s wife. She said she would, but took me to one of a small row of cottages over the way, where her husband was half-naked in the midst of his Saturday wash. Taking no notice of him she led me into the sitting-room and, with a huge loaf held like a violin, began buttering and cutting thin slices while she talked to me, to the little children and to her husband, from the adjacent kitchen. She was tall, straight as a pillar, black-haired, with clear untanned but slightly swarthy skin, black eyes, kindly gleaming cheeks and red lips smiling above her broad breast and hips. Her clothes were black but in rags that hardly clung to her shoulders and waist. She was barely five and twenty, but had six young children about her, one in a cradle by the hearth and another still crawling at her feet. Her only embarrassment came when I asked to pay for my tea—she began adding up the cost, a pennyworth of bread and butter, a halfpennyworth of tea, etc.! The kitchen consisted simply of a large grate and baking oven, plain tables and chairs on a flagged floor. But the sitting-room was a museum—with photographs of a volunteer corps, of friends and relations on the wall over the fire; foxgloves in jam-pots surrounded by green crinkled paper in the fireplace; on the mantelpiece, cheap little vases and scraps of ore and more photographs. On the walls were three pictures: one of two well- dressed children being timidly inspected by fallow deer; another of a grandmother showing a book to a child whose attention is diverted by the frolics of two kittens at her side; and a third of Jesus, bleeding and crowned with thorns, high on a cross over a marble city beneath a romantic forest ridge, behind which was the conflagration of a crimson sunset. Other sitting-rooms were similarly adorned, with the addition of a picture of John Wesley as a child escaping from the window of a burning house, with many anxious men holding up their hands from
- 69. below. The smell of flowers and of sun-warmed furniture and old upholstery mingles in such rooms. But the kitchens are often as charming as in Wales. I remember one especially near Carn Galver. The farmhouse was of whitened stone under a steep thatch. In front were fuchsia trees in the corner of a stony yard; to one side, the haystacks and piles of furze and bracken and peat. The farmer’s wife was carrying peat on an iron hook into the kitchen and I followed her. A pan of yellow scalded cream stood inside. The fireplace was a little room in itself, with seats at each side and a little fire of wood and three upright turves in one corner of the great stone hearth: over the fire the kettle boiled. Horse ornaments of polished brass surmounted the fireplace. The wallpaper had given up its pattern long since to a smoky uneven gold; nailed to it were calendars and lists of fairs and sales; against it were two small tables, one to support a Bible and an almanac, the other spread with a white cloth on which was a plate and a bowl of cream. Behind the door and between it and the fire was a high- backed settle of dark wood, with elbow-rests. The floor was flagged and sanded. The light came in through a little square window on to the Bible by the opposite wall, and through the open door on to the figure of the housewife, a woman of forty. A delicate white face shone beneath a broad untrimmed straw hat that was tied tightly under her chin so as to hide her ears and most of her black hair. Her black skirt was kilted up behind; a white apron contrasted with black shoes, black stockings and black clothes. At first her face was hardly seen, not only because but a part of it emerged from the shell of her hat, but because the spirit that emanated from it was more than the colour and features and so much in harmony with the sea and crag and moor and dolmen of her land. It is evading an insuperable difficulty to say that this spirit was not so much human as fay. It was the spirit of which her milky complexion, the bright black eyes, white teeth and fine red lips of her readily smiling and naïvely watching fearless face, her slender form, her light and rapid movements upon small feet, were only the more obvious expressions. Her spirit danced before her—not quite visibly, not quite audibly—as she
- 70. moved or spoke or merely smiled; if it could have been seen it would have been a little singing white flame changing to blue and crimson in its perpetual flickering. It was a spirit of laughter, of laughter unquenchable since the beginning of time, of laughter in spite of and because of all things, the laughter of life like a jewel in desolate places. It was a spirit most ancient and yet childlike, birdlike: it belonged to a world outside any which other human beings ever seemed to touch, but the laughter in it made it friendly, for it was far deeper than humour, it was gaiety of heart. Her goings to and fro on those light feet had the grace, quickness, suddenness of a bird, of a wren that slips from twig to twig and jets out its needle of song, of a moorhen flicking its tail and hooting sharply. Her laugh startled and delighted like the laugh of the woodpecker as it leaps across the glades—like the whistling of birds up amongst the dark clouds and the moon. But most of all she called to mind the meadow pipit of her own crags, that rises from green ledges out over the sea and then, falling slantwise with body curved like a crescent, utters his passionate pulsating song, so rapid and passionate that it seems impossible and unfit that it should end except in death, yet suddenly ceasing as it lands again upon the samphire or the thrift. The spirit was as quicksilver in the corners of her eyes, as quicksilver in the heart. Such a maid she must have been as the bard would have thought to send out the thrush to woo for him, when he heard the bird of ermine breast singing from the new-leaved hazel at dawn, on the edge of a brook among the steep woods—singing artfully with a voice like a silver bell—solemnly, too, so as to seem to be performing a sacrifice—and amorously, bringing balm to lovers’ hearts and inspiring the bard to send by him a message to the sun of all maidens that she, white as the snow of the first winter night, should come out to the green woods to him. She had lived for generations on the moor, for generations upon generations, and this was what she had gained from heather and furze and crag and seawind and sunshine tempered by no trees—inextinguishable laughter. But she was inarticulate. She milked the cows, made butter, baked bread, kept the peat fire burning and tended her children. When she talked, I asked for more cream. Perhaps after several more generations
- 71. have passed she will be a poet and astonish the world with a moorland laughter of words that endure. Everything in that house was old or smooth and bright with use, and the hollowed threshold of the doorway in the sun put me in mind of a hundred old things and of their goodliness to mortal eyes —the wrecked ship’s ribs, their bolt-holes rusty, that stand among nettles as gate-posts—the worn dark stones that rock to the tread among the ripples of an umbrageous ford—many a polished stile and gate—the group of rigid but still gracious bowery thorns dotted with crimson haws in the middle of a meadow, their holes and lower branches rubbed hard and smooth and ruddy like iron by the cows— the ash staff beginning to bend like its master, the old man upon the roads who once wore scarlet and wound the horn for Mr. ——’s hounds. Odd it is how old use sanctifies a little thing. There was once a hut where a good man, but a poor and a weak and unwise, stayed all one fair summer and talked of English roads—he was a lord of the roads, at least of South Country roads—and of ships, which he knew. Now on the first night of his stay, needing a candlestick he kicked off the top of a pointed wooden paling, so as to make a five-angled piece on which he stuck the candle in its own grease. All through his stay he used the candlestick, when he read the Divina Commedia and Pantagruel and Henry Brocken and recollected airs of Italy and Spain, amidst the sound of nightjars and two leafy streams: the light flickered out as he mused about the open sea, calm but boundless and without known harbour, on which he was drifting cheerfully, regardless of Time, pied with nights and days. The hut was burnt and the man went—to drown a little afterwards with a hundred unlike himself in the sea—but among nettle and dock the candlestick was picked up safe. It had broken off straight and the simple shape was pleasant; it was dark with age; along with the mound and little pillar of wax remaining it had the shape of a natural thing; and it was his.
- 72. Animate as well as inanimate things are open to this sanctification by age or use. I am not here thinking of ceremonious use—for which I have small natural respect, so that I have been denied the power of appreciating either a great religious pomp or the dancing of Mademoiselle Genée. But some men, particularly sailors and field labourers, but also navvies and others who work heavily with their hands, have this glory of use. Their faces, their clothes, their natures all appear to act and speak harmoniously, so that they cause a strong impression of personality which is to be deeply enjoyed in a world of masks, especially of black clerical masks. One of the best examples of this kind was a gamekeeper who daily preceded me by twenty or thirty yards in a morning walk up through a steep wood of beeches. He was a short, stiffly-built and stoutish man who wore a cap, thick skirted coat and breeches, leather gaiters and heavy boots, all patched and stained, all of nearly the same colour as his lightish-brown hair and weathered skin, but not so dark as the gun over his shoulder. The shades of this colour were countless and made up like the colour of a field of ripe wheat, which they would have resembled had they not been liberally dusted all over, just as his brown beard was grizzled. He went slowly up, swinging slightly at the shoulders and always smoking a pipe of strong shag tobacco of which the fumes hovered in the moist air with inexpressible sweetness and a good brown savour: if I may say so, the fit emanation of the brown woodland man who, when he stood still, looked like the stump of a tree.
- 73. CHAPTER X SUMMER—SUSSEX Far up on the Downs the air of day and night is flavoured by honeysuckle and new hay. It is good to walk, it is good to lie still; the rain is good and so is the sun; and whether the windy or the quiet air be the better let us leave to a December judgment to decide. One day the rain falls and there is no wind, and all the movement is in the chaos of the dark sky; and thus is made the celestial fairness of an earth that is brighter than the heavens; for the green and lilac of the grasses and the yellow of the goat’s-beard flowers glow, and the ripening corn is airy light. But next day the sun is early hot. The wet hay steams and is sweet. The beams pour into a southward coombe of the hills and the dense yew is warm as a fruit-wall, so that the utmost of fragrance is extracted from the marjoram and thyme and fanned by the coming and going of butterflies; and in contrast with this gold and purple heat on flower and wing, through the blue sky and along the hill-top moist clouds are trooping, of the grey colour of melting snow. The great shadows of the clouds brood long over the hay, and in the darker hollows the wind rustles the dripping thickets until mid-day. On another morning after night rain the blue sky is rippled and crimped with high, thin white clouds by several opposing breezes. Vast forces seem but now to have ceased their feud. The battle is over, and there are all the signs of it plain to be seen; but they have laid down their arms, and peace is broad and white in the sky, but of many colours on the earth—for there is blue of harebell and purple of rose-bay among the bracken and popping gorse, and heather and foxglove are purple above the sand, and the mint is hoary lilac, the meadow-sweet is foam, there is rose of willow-herb and yellow of fleabane at the edge
- 74. of the water, and purple of gentian and cistus yellow on the Downs, and infinite greens in those little dense Edens which nettle and cow- parsnip and bramble and elder make every summer on the banks of the deep lanes. A thousand swifts wheel as if in a fierce wind over the highest places of the hills, over the great seaward-looking camp and its three graves and antique thorns, down to the chestnuts that stand about the rickyards in the cornland below. These are the hours that seem to entice and entrap the airy inhabitants of some land beyond the cloud mountains that rise farther than the farthest of downs. Legend has it that long ago strange children were caught upon the earth, and being asked how they had come there, they said that one day as they were herding their sheep in a far country they chanced on a cave; and within they heard music as of heavenly bells, which lured them on and on through the corridors of that cave until they reached our earth; and here their eyes, used only to a twilight between a sun that had set for ever and a night that had never fallen, were dazed by the August glow, and lying bemused they were caught before they could find the earthly entrance to their cave. Small wonder would this adventure be from a region no matter how blessed, when the earth is wearing the best white wild roses or when August is at its height. The last hay-waggon has hardly rolled between the elms before the reaper and the reaping-machines begin to work. The oats and wheat are in tents over all the land. Then, then it is hard not to walk over the brown in the green of August grass. There is a roving spirit everywhere. The very tents of the corn suggest a bivouac. The white clouds coming up out of the yellow corn and journeying over the blue have set their faces to some goal. The traveller’s-joy is tangled over the hazels and over the faces of the small chalk-pits. The white beam and the poplar and the sycamore fluttering show the silver sides of their leaves and rustle farewells. The perfect road that goes without hedges under elms and through the corn says, “Leave all and follow.” How the bridges overleap the streams at one leap, or at three, in arches like those of running hounds! The far-scattered,
- 75. placid sunsets pave the feet of the spirit with many a road to joy; the huge, vacant halls of dawn give a sense of godlike power. But it is hard to make anything like a truce between these two incompatible desires, the one for going on and on over the earth, the other that would settle for ever, in one place as in a grave and have nothing to do with change. Suppose a man to receive notice of death, it would be hard to decide whether to walk or sail until the end, seeing no man, or none but strangers; or to sit—alone—and by thinking or not thinking to make the change to come as little as is permitted. The two desires will often painfully alternate. Even on these harvest days there is a temptation to take root for ever in some corner of a field or on some hill from which the world and the clouds can be seen at a distance. For the wheat is as red as the most red sand, and up above it tower the elms, dark prophets persuading to silence and a stillness like their own. Away on the lesser Downs the fields of pale oats are liquid within their border of dark woods; they also propose deep draughts of oblivion and rest. Then, again, there is the field—the many fields—where a regiment of shocks of oats are ranked under the white moon between rows of elms on the level Sussex land not far from the sea. The contrast of the airy matter underfoot and the thin moon overhead, with the massy dark trees, as it were, suspended between; the numbers and the order of the sheaves; their inviolability, though protected but by the gateway through which they are seen—all satisfy the soul as they can never satisfy the frame. Then there are the mists before heat which make us think of autumn or not, according to our tempers. All night the aspens have been shivering and the owls exulting under a clear full moon and above the silver of a great dew. You climb the steep chalk slope, through the privet and dog-wood coppice; among the scattered junipers—in this thick haze as in darkness they group themselves so as to make fantastic likenesses of mounted men, animals, monsters; over the dead earth in the shade of the broad yews, and thence suddenly under lightsome sprays of guelder-rose and their cherry-coloured berries; over the tufted turf; and then through the massed beeches, cold and dark as
- 76. a church and silent; and so out to the level waste cornland at the top, to the flints and the clay. There a myriad oriflammes of ragwort are borne up on all stems of equal height, straight and motionless, and near at hand quite clear, but farther away forming a green mist until, farther yet, all but the flowery surface is invisible, and that is but a glow. The stillness of the green and golden multitudes under the grey mist, perfectly still though a wind flutters the high tops of the beech, has an immortal beauty, and that they should ever change does not enter the mind which is thus for the moment lured happily into a strange confidence and ease. But the sun gains power in the south-east. It changes the mist into a fleeting garment, not of cold or of warm grey, but of diaphanous gold. There is a sea-like moan of wind in the half-visible trees, a wavering of the mist to and fro until it is dispersed far and wide as part of the very light, of the blue shade, of the colour of cloud and wood and down. As the mist is unwoven the ghostly moon is disclosed, and a bank of dead white clouds where the Downs should be. Under the very eye of the veiled sun a golden light and warmth begins to nestle among the mounds of foliage at the surface of the low woods. The beeches close by have got a new voice in their crisp, cool leaves, of which every one is doing something—cool, though the air itself is warm. Wood- pigeons coo. The white cloud-bank gives way to an immeasurable half-moon of Downs, some bare, some saddle-backed with woods, and far away and below, out of the ocean of countless trees in the southern veil, a spire. It is a spire which at this hour is doubtless moving a thousand men with a thousand thoughts and hopes and memories of men and causes, but moves me with the thought alone that just a hundred years ago was buried underneath it a child, a little child whose mother’s mother was at the pains to inscribe a tablet saying to all who pass by that he was once “an amiable and most endearing child.” And what nights there are on the hills. The ash-sprays break up the low full moon into a flower of many sparks. The Downs are heaved up into the lighted sky—surely they heave in their tranquillity as with a slowly taken breath. The moon is half-way up the sky and
- 77. exactly over the centre of the long curve of Downs; just above them lies a long terrace of white cloud, and at their feet gleams a broad pond, the rest of the valley being utterly dark and indistinguishable, save a few scattered lamps and one near meadow that catches the moonlight so as to be transmuted to a lake. But every rainy leaf upon the hill is brighter than any of the few stars above, and from many leaves and blades hang drops as large and bright as the glowworms in their recesses. Larger by a little, but not brighter, are the threes and fours of lights at windows in the valley. The wind has fallen, but a mile of woods unlading the rain from their leaves make a sound of wind, while each separate drop can be heard from the nearest branches, a noise of rapt content, as if they were telling over again the kisses of the shower. The air itself is heavy as mead with the scent of yew and juniper and thyme.
- 78. CHAPTER XI HAMPSHIRE—AN UMBRELLA MAN A beggar is a rich man on some of these August days, especially one I know, whom first I met some Augusts ago now. A fine Sunday afternoon had sprinkled the quiet and thinly-peopled land with black- dressed men and white-dressed women, the older married couples and their trains of children keeping chiefly to the roads and most straightforward paths, the younger, with one child or none, choosing rather the green lanes, while the lovers and the boys found out tall hedge-sides and the footpaths across which more than one year’s growth of hazel had spread, so that the shortest of the maids must stoop. Many showers following a dry season made miles of the country as clean and fragrant as a garden. Honeysuckle and privet were in every hedge with flowers that bring a thrill of summer bridals on their scent. The brisk wind was thymy from the Downs. The ragwort was in its glory; it rose tall as a man in one straight leap of dark-foliaged stem, and then crowned itself in the boldest and most splendid yellows derived from a dark golden disc and almost lemon rays; it was as if Apollo had come down to keep the flocks of a farmer on these chalk hills and his pomp had followed him out of the sky. A few birds still sang; one lark now and then, a cirl-bunting among the topmost haws of a thorn, chiffchaffs in the bittersweet and hazel of the little copses. There was apparently comfort, abundance and quiet everywhere. They were seen in the rickyards where grand haystacks, newly thatched, stood around ancient walnut-trees. Even the beeches had a decorous look in their smooth boles and perfect lavish foliage. The little patches of flowery turf by the roadside and
- 79. at corners were brighter and warmer than ever, as the black bees and the tawny skipper butterflies flew from bloom to bloom of the crimson knapweed. Amplest and most unctuous of all in their expression of the ceremonious leisure of the day and the maturity of the season were the cart-horses. They leaned their large heads benignly over the rails or gates; their roan or chestnut flanks were firm and polished; manes, tails and fetlocks spotless; now and then they lifted up their feet and pressed their toes into the ground, showing their enormous shoes that shone and were of girth sufficient to make a girdle for the lightest of the maids passing by. Sunday with not too strict a rod of black and white ruled the land and made it all but tedious except in the longest of the green lanes, which dipped steeply under oaks to a brook muffled in leaves and rose steeply again, a track so wet in spring—and full of the modest golden green of saxifrage flowers—that only the hottest Sunday ever saw it disturbed except by carter and horses. In a hundred yards the oak-hidden windings gave the traveller a feeling of reclusion as if he were coiled in a spool; very soon a feeling of possession ripened into one of armed tyranny if another’s steps clattered on the stones above. Sometimes in a goodly garden a straight alley of shadows leads away from the bright frequented borders to—we know not quite whither, and perhaps, too much delighted with half-sad reverie, never learn, smother even the guesses of fancy, lest they should bring some old unpleasant truth in their train; but if the fancy will thread the alley and pass the last of the shadows it is into some such lane as this that it would gladly emerge, to come at last upon the pure wild. It seemed that I had come upon the pure wild in this lane, for in a bay of turf alongside the track, just large enough for a hut and thickly sheltered by an oak, though the south-west sun crept in, was a camp. Under the oak and at the edge of the tangled bramble and brier and bracken was a low purple light from those woodside flowers, self-heal and wood- betony. A perambulator with a cabbage in it stood at one corner; leaning against it was an ebony-handled umbrella and two or three umbrella-frames; underneath it an old postman’s bag containing a
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