Turing fibonacci numbers

Alan
Turing

Fibonacci
Numbers
in
Nature

Melissa
Davis:

Alan
Turing
was
a
mathematical
genius.
He
speculated
that
there
was
a
relationship

between
math
and
nature
by
the
presence
of
Fibonacci
numbers
that
naturally
occur
in

plants.

Fibonacci
numbers
are
a
sequence
of
numbers,
where
you
can
add
one
of
the

numbers
with
the
number
to
the
right
of
it,
to
get
the
next
number.

For
example,
the
first

few
numbers
of
the
sequence
begin
as
follows:
0,
1,
1,
2,
3,
5,
8,
13,
21,
etc.

To
get
the

number
after
21,
simply
add
13
to
21,
which
gives
you
34.

These
numbers
may
correlate
to

the
number
of
petals,
leaves,
or
spirals
of

seeds
a
plant
has.

This
is
also
the
reason
why

four-‐leaf
clovers
are
so
rare,
since
four
is
not

a
number
k
appears
in
the
Fibonacci

sequence.

Turing
specifically
looked
at
sunflowers
to

study
this
phenomenon.
Turing
examined

how
the
number
of
spirals
in
the
seed

patterns
of
sunflowers
typically
resulted
in
a

Fibonacci
sequence.

This
finding
was

significant,
as
it
provided
much
information

for
phyllotaxis,
the
study
of
the
way
plants

grow.

“The
appearance
of
patterns
in
the

phyllotaxis
-‐
the
arrangement
of
leaves,

stems,
seeds
or
similar
-‐
has
been

studied
by
many
well-‐known
scientists,

including
Leonardo
Da
Vinci.”
–
BBC

News

Here
is
a
demonstration
of
one
way
to

count
the
spirals
of
seeds.
The
total

number
of
rows
is
34,
which
is
a

Fibonacci
number.

Because
of
Alan
Turing’s
abbreviated
life
due
to
his
mistreatment
in
society,
Turing
was

never
able
to
confirm
his
findings.

However,
the
Manchester
Science
Festival,
the
Museum

of
Science
and
Industry,
and
the
University
of
Manchester
are
asking
for
help
from
the

public
to
confirm
Turing’s
work
on
Fibonacci
numbers
in
sunflowers.

The
project
entails

people
planting
and
growing
their
own
sunflowers,
and
then
counting
the
rows
of
seeds
as

Turing
did.

This
project
also
aims
to
honor
Turing
during
the
one-‐hundred-‐year

anniversary
of
his
death.

In
addition
to
sunflowers,
the

Fibonacci
sequence
is
found

in
the
number
of
petals
in

many
flowers.

For
example,

buttercups,
wild
roses,
and

larkspurs
have
five
petals;

delphiniums
and
coreopsis

have
eight
petals;
ragworts

and
marigolds
have
thirteen

petals;
and
daisies
can
have

eighty-‐nine
petals.

References:

Couder,
Yves.
"Sunflower."
Photo.
Flower
Patterns
and
Fibonacci
Numbers.
2002.
22
May

2012.
<http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html>.

"The
Fibonacci
Series."
ThinkQuest.
Oracle
Foundation.
Web.
22
May
2012.

<http://library.thinkquest.org/27890/applications5.html>.

"Greater
Manchester
Sunflowers
to
Test
Alan
Turing
Theory."
BBC
News.
BBC,
22
Mar.

2012.
Web.
06
June
2012.
<http://www.bbc.co.uk/news/uk-‐england-‐manchester-‐
17469241>.

GrrlScientist.
"Sunflowers
and
Fibonacci."
The
Guardian.
Guardian
News
and
Media,
29
Mar.

0016.
Web.
22
May
2012.

<http://www.guardian.co.uk/science/grrlscientist/2012/apr/16/1>.

"How
To
Count
the
Spirals."
Photo.
Museum
of
Mathematics.
22
May
2012.

<http://momath.org/home/fibonacci-‐numbers-‐of-‐sunflower-‐seed-‐spirals>.

McKay,
Dennis.
"Larkspur."
Photo.
Drug
Discovery.
2009.
22
May
2012.

<http://digitalunion.osu.edu/r2/summer09/jaeger/MLA.html>.

“Sunflowers
and
Fibonacci
–
Numberphile.”
Video.
(2012).
Retrieved
May
22,
2012
from

http://www.youtube.com/watchfeature=player_embedded&v=DRjFV_DETKQ.

Wainwright,
Martin.
"Grow
a
Sunflower
to
Solve
Unfinished
Alan
Turing
Experiment."
The

Guardian.
Guardian
News
and
Media,
24
Nov.
0048.
Web.
22
May
2012.

<http://www.guardian.co.uk/uk/the-‐northerner/2012/mar/26/alan-‐turing-‐sunf....>

Jing
(Sophie)
Xia:

However,
flowers
are
not
the
only
organisms
in
which
Fibonacci
numbers
are
present;

Fibonacci
numbers
are
also
found
in
pine
cones
and
plant
leaves.
Pine
cones
display
the

Fibonacci
Spirals
clearly.
The
best
way
to
examine
these
patterns
is
to
observe
pine
cones

from
the
base
where
the
stalk
connects
it
to
the
tree.
For
instance,
one
set
of
spirals
goes
in

one
uniform
direction
whereas
another
set
of
spirals
goes
in
the
opposite
direction
(see

images
below).
For
example,
in
one
direction,
there
are
8
whirls
whereas
in
the
other

direction,
there
are
13
whirls.
It
is
not
coincidence
that
both
8
and
13
are
Fibonacci

numbers.

Pine
cones
contain
evidence
of
Fibonacci
spirals
since
their
patterns
are
arranged
in
two

different
directions
of
spirals.

In
addition,
many
plants
show
Fibonacci
numbers
in
the
arrangements
of
the
leaves
around

their
stems.
When
looking
down
on
a
plant,
one

can
notice
that
its
leaves
are
arranged
so
that

the
leaves
higher
up
on
the
stem
do
not
hide

leaves
below.
This
ensures
that
no
matter

where
the
leaves
are
located
on
a
stem,
they

are
able
to
receive
sunlight.
Fibonacci
numbers

are
evident
in
two
ways
in
terms
of
leaves
per

turn.
First,
they
occur
when
counting
the

number
of
times
they
go
around
the
stem.

Secondly,
it
occurs
when
counting
leaves
until

finding
a
leaf
directly
above
the
leaf
in
which

one
started
counting.
If
one
counts
in
the

opposite
direction,
there
is
a
different
number

of
turns
with
the
same
number
of
leaves.
The

number
of
turns
in
each
direction
and
the

number
of
leaves
met
are
three
consecutive

Fibonacci
numbers.
For
example,
one
must

rotate
three
turns
clockwise
to
meet
a
leaf
that

is
directly
above
the
first
leaf
counted.
On
the

way,
one
passes
by
five
leaves.
But
when
one

counts
counter-‐clockwise,
they
only
turn
two

times.

Because
2,
3,
and
5
are
consecutive

Fibonacci
numbers,
this
example
demonstrates

the
existence
of
Fibonacci
numbers
in
plant

leaves.

References:

"Evolution."
How
Stuff
Works.
N.p.,
n.
d.
Web.
20
May.
2012.

<http://science.howstuffworks.com/environmental/life/evolution>.

"Fibonacci
Numbers
and
Nature."
Rabbits,
Cows
and
Bees
Family
Trees
.
N.p.,
n.
d.
Web.
20

May.
2012.
<http://www.maths.surrey.ac.uk/hosted-‐sites/R.Knott/Fibonacci/fibnat.html>.

"Fibonacci
Numbers
and
the
Golden
Section."
N.p.,
n.
d.
Web.
20
May.
2012.

<http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm>.

"Fibonacci
numbers
and
Golden
ratio."
Natural
occurrence
of
Fibonacci
numbers.
N.p.,
n.
d.

Web.
20
May.
2012.
<http://gwydir.demon.co.uk/jo/numbers/interest/golden.htm>.

Parveen,
Nikhat.
"Fibonacci
in
Nature."
N.p.,
n.
d.
Web.
20
May.
2012.

<http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm>.

Shiwei
Huang

Alan
Turing’s
interest
in
Fibonacci
series
was
inspired
by
zoologist
D’arcy
Wentworth

Thompson’s
book,
On
Growth
and
Form.
Thompson
wanted
to
explain
how
physical
and

mathematical
laws
could
explain
the
forms
and
patterns
of
living
things.
After
examining

the
patterns
of
fir
cones
and
sunflowers,
Thompson
observed
that
the
scales
of
a
fir
cone

and
the
florets
of
a
sunflower
are
grouped
in
the
numbers
in
Fibonacci
series.
However,

Thompson
claimed
that
the
appearance
of
Fibonacci
series
in
these
plants
were
purely
for

mathematical
reasons,
and
the
purpose
of
introducing
of
this
series
into
plants
throughout

the
course
of
natural
selection
was
not
worth
studying.

Alan
Turing
was
the
first
scientist
to
study
the
mechanisms
behind
the
development
of

pattern
in
living
organisms
using
computer
simulation.
When
Manchester
Electronic

Computer,
also
called
Ferranti
Mark
I,
was
installed
in
Manchester
University,
Turing
wrote

to
his
colleague,
“I
am
hoping
as
one
of
the
first
jobs
to
do
something
about
‘chemical

embryology.’
In
particular
I
think
one
can
account
for
the
appearance
of
Fibonacci
numbers

in
connection
with
fir
cones.”
Turing
formulated
his
reaction-‐diffusion
model
from
the

observation
of
fir
cones
and
sunflowers.
He
believed
that
diffusing
chemicals
reacted
with

each
other
and
caused
the
development
of
forms
of
living
organisms.
He
postulated
that
the

model
could
be
applied
to
gastrulation
of
an
embryo,
which
is
the

rearranging
of
the
cells
in
an
embryo,
and
the
formation
of
leaf
pattern.
Modern
computer

has
simulated
Turing’s
mechanism,
and
it
has
successfully
produced

leopard-‐like,
cheetah-‐like,
and
giraffe-‐like
stripes.
In
his
paper
“Chemical
basis
of

morphogenesis,”
he
called
these
interacting
chemicals
“morphogens.”
Unfortunately,

Turing’s
work
on
morphogenesis
was
considered
ahead
of
the
time,
and
he
died
and
left
a

large
number
of
research
materials
and
notes
that
could
not
be
understood
today.

Turing’s
system
(the
simplified
version):

∂c/∂t=f(c)+D∇2c

f(c)
represents
the
local
chemical
reaction
that
different
chemicals
are
reacted
and

formed.
D
is
the
diffusion
constant,
which
describes
the
flow
of
a
chemical
due
to
its

concentration
gradient
and
its
diffusion.
Simply
put,
the
equation
states
that
the

distribution
of
a
chemical
is
determined
by
the
chemical
reaction
that
generates
this

chemical
and
the
diffusion
of
this
chemical.
Under
certain
conditions,
two
or
more

chemicals
will
diffuse
and
react
with
each
other
in
the
embryo,
and
they
will
reach
a
stable

pattern
of
concentration.
For
instance,
if
there
was
a
ring
of
cells,
model

could
give
us
a
pattern
of
chemical
gradient
surrounding
the
ring
of
cells.
The
same

concentrations
would
occur
at
places
with
the
same
distances
to
each
other,
and
Turing

called
this
chemical
wave
if
it
was
stationary.
Conversely,
if
the
gradient
was
changing,
it

would
be
called
traveling
waves.
Turing
pointed
out
the
structure
of
the
embryo
could

break
this
pattern
and
cause
asymmetrical
chemical
waves.
Turing
believed
that
genes

catalyzed
the
production
of
morphogens
and
might
influence
the
rate
of
the
reaction
to

determine
the
pattern
in
animals.

Program
sheet
written
by
Alan
Turing
during
his

study
of
fir
cone
patterning

’

Computer
output.
Turing
wrote
“How
did

this
happen?”
on
the
sheet.

Turing’s
numbering
on
the
sunflower

References:

Charvoin,
J.
and
Sadoc,
J-‐F.
(2011)
“A
Phyllotactic
Approach
to
The
Structure
of
Collagen

Fibrils.”
<http://arxiv.org/pdf/1102.2359v2.pdf>

Copeland,
B.J.(2004)
“The
Essential
Turing,
Seminal
Writings
in
Computing,
Logic,

Philosophy,
Artificial
Intelligence,
and
Artificial
Life
plus
The
Secrets
of
Enigma.”
Oxford:

Clarendon
Press.

Engelhardt,
R.
(1994)
“Modeling
Pattern
Formation
in
Reaction-‐Diffusion
System.”

<http://www.robinengelhardt.info/speciale/main.pdf>

Maini,
P.
K.
(2007)
“The
impact
of
Turing's
work
on
pattern
formation
in
biology.”

<people.maths.ox.ac.uk/maini/PKM%20publications/172.pdf>

Swinton,
J.
(2003)
“Watching
the
Daisies
Grow:
Turing
and
Fibonacci
Phyllotaxis.”

<user29459.vs.easily.co.uk/wp-‐content/uploads/2011/05/swinton.pdf>

Thompson,
D.
W.
(1966)
“On
Growth
and
Form.”
Cambridge:
University
Press.

Turing,
A.
M.
(1952)
“The
Chemical
Basis
of
Morphogensis.”

<http://links.jstor.org/sici?sici=0080-‐
4622%2819520814%29237%3A641%3C37%3ATCBOM%3E2.0.CO%3B2-‐I

Jen-‐Ling
Nieh:

Morphology,
at
the
most
basic
sense,
consists
of
two
aspects
–
shape
and
pattern.

The

changes
of
morphology
that
occur
during
biological
development
of
an
organism
are
called

“morphogenesis.”

Turing
proposed
that
both
shape
and
pattern
seem
to
be
set
up
in

embryos
by
the
same
mechanism,
a
pre-‐pattern
of
chemical
changes
that
waits
for
the

appropriate
stage
of
development,
and
then
triggers
either
pigments,
to
create
pattern,
or

cellular
changes,
to
create
shape.

He
showed
that
this
kind
of
system
may
have
a

homogeneous
stationary
state
which
is
unstable
against
perturbations,
such
that
any

random
deviation
from
the
stationary
state
leads
through
diffusion
to
a
symmetry
break.

This
process
is
called
diffusion-‐driven
instability.

Since
complex
spatial
patterns
are

commonly
found
in
nature,
for
example,
in
animal
skins
and
also
in
some
polymer
systems,

it
is
quite
natural
to
think
that
such
pattern
formations
could
be
caused
by
some
general

physicochemical
process.

Many
animals
develop
their
coat
patterns
in
stages.
Typically,
a
secondary
pattern
will

emerge
as
the
animal
transitions
to
adulthood.
The
following
examples
all
use
multiple

stages:

Turing's
mathematical
model
of
chemical
morphogenesis
helps
us
understand
why

tigers
and
zebras
have
stripes.
Turing's
model
from
1952
consists
on
a

set
of
equations
that
iteratively
simulate
the
distribution
of
a
chemical
agent
(activator)

modulated
by
the
presence
of
another
agent
called
inhibitor.
In
his
seminal
1952
paper,

Alan
Turing
predicted
that
diffusion
could
spontaneously
drive
an
initially
uniform
solution

of
reacting
chemicals
to
develop
stable
spatially
periodic
concentration
patterns.
It
is

believed
that
such
interactions
take
place
in
nature
to
form
patterns
that
can
be
found
in

mammals
and
fish,
and
the
first
model,
generating
spots.

According
to
the

Model,
the
diffusion
of
an
activator
and

inhibitor
through
an
evolving
cellular

system
over
a
period
of
time,
the

concentration
gradients
dictating
cell

differentiation,
i.e.
a
zebra
skin
cell
can
be

black
or
white
according
to
the

concentration
of
a
white-‐cell
activator
at

the
point
when
it
forms.
Biologists
would

call
these
activators
morphogens,
as
these

are
the
proteins
that
regulate
gene

expression.

Currently,
pigmentation
patterns
in

animal
skins,
feathers
of
birds,
and

shells
of
snails
are
the
only
examples

in
which
we
can
detect
the
dynamic

nature
of
Turing
waves
as
a
time

course
of
the
pattern
change.

Especially,
the
two-‐dimensional
(2-‐D)

skin
pattern
of
fish
is
quite
convenient

to
study
because
their
waves
are

sometimes
alive
even
when
the
fish

has
grown
up
into
an
adult
(Shigeru

Kondo).

For
example,
when
a
striped

angelfish
(Pomacanthus
imperator)

grows,
the
branching
points
of
the

stripes
slide
horizontally
as
the
zip

opens
and
add
a
number
of
stripes;

eventually
the
spacing
between
the

stripes
remains
stable.
In
the
case
of
the
spotted
catfish
(Plecostoms),
both
division
of
the

spots
and
insertion
of
the
new
spots
occur
to
retain
the
density
and
size
of
the
spots.
Both

stripes
and
the
spots
are
the
most
typical
2-‐D
patterns
generated
by
the
RD
mechanism,

and
the
time
course
of
the
pattern
change
possesses
the
characteristics
of
the
dynamics
of

RD
waves,
strongly
suggesting
that
the
RD
mechanism
underlies
the
process
of
pigment

pattern
formation
of
fish.

References:

www.scholarpedia.org/w/images/8/8d/TROPH.jpg">http://www.scholarpedia.org/w/im
ages/8/8d/TROPH.jpg

http://cgjennings.ca/toybox/turingmorph/texture1.png

http://cgjennings.ca/toybox/turingmorph/texture2.png

http://cgjennings.ca/toybox/turingmorph/texture3.jpg

www.urbagram.net/images/turing.jpg">http://www.urbagram.net/images/turing.jpg

www.urbagram.net/v1/revision/Morphogenesis?rev=1">http://www.urbagram.net/v1/r
evision/Morphogenesis?rev=1

http://27.media.tumblr.com/tumblr_lzeh1qUhRe1r3lyy3o1_500.jpg

Alexandra
Pourzia:

Alan
Turing
proposed,
based
purely
on
logical
reasoning,
that
pattern
formation
in
nature

involved
an
‘activating’
substance
and
an
‘inhibiting’
substance.
The
repetition
of
activator

and
inhibitor
could
create
patterns
such
as
stripes.1
Previously,
developmental
biologists

were
puzzled
by
pattern
formation
because
they
could
not
explain
it
using
the
linear

models
that
were
the
extent
of
their
knowledge
at
the
time.
Turing
proposed
a
nonlinear

model
by
introducing
diffusion
as
the
generator
of
instability
in
the
model,
instead
of
being

a
byproduct
of
the
model.
2
The
implications
of
Turing’s
mechanism
were
astounding:
he

predicted
the
mode
of
action
of
the
Hox
genes
in
Drosophila,
which
result
in
the
patterning

of
the
embryo’s
body
segments.
3

Hox
gene
patterning
by
body
segment
in
Drosophila

The
Hox
genes
induce
patterning
by
activating
transcription
of
their
unique
set
of
genes

while
repressing
others
not
related
to
their
segment.
They
in
turn
are
regulated
by

patterning
genes
(gap,
pair-‐rule,
or
segment
polarity
genes),
which
follow
Turing’s

proposed
model
very
closely.
These
patterning
genes
are
induced
by
high
or
low

concentrations
of
maternal
proteins
in
the
embryo,
which
was
formed
from
the
maternal

egg
and
paternal
sperm.
For
example,
high
concentrations
of
maternal
protein
induce
the

expression
of
Bicoid
and
Hunchback,
while
inhibiting
Giant
and
Kruppel.
The
concentration

of
these
“morphogens”,
as
Turing
first
called
them,
lead
to
the
formation
of
a
pattern
–

segment
two
of
the
fly
embryo.3

Pair-‐rule
genes:
expressed
between
certain
segments

References:

1. Hughes,
Virginia.
“Alan
Turing’s
60-‐Year-‐Old
Prediction
About
Patterns
in
Nature

Proved
True.“
Smithsonian.com.
The
Smithsonian
Institution,
21
Feb
2012.
Web.
20

May
2012.
<http://blogs.smithsonianmag.com/science/2012/02/alan-‐turing-‐
predicted-‐natures-‐stripes-‐and-‐patterns/>

2. Reinitz,
John.
“Pattern
formation.”
Nature.
Feb
2012.

3. “Hox
gene.”
Wikipedia.
Wikimedia
Foundation,
Inc,
n.d.
20
May
2012.

<http://en.wikipedia.org/wiki/Hox_gene>

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