Tag Archives: Mistakes

Trees are Air and Water

Trees are made from air and water. How much of the wood comes from air and how much comes from water? Let’s do the calculations.

Dry wood is mostly cellulose, chains of C6.H10.O5, molecular weight 6*12+10*1+5*16 = 162.

The production of cellulose, in overview, is 12*C.O2 (from air) + 10*H2.O (water) –> 2*C6.H10.O5 + 12*O2 (back to air). Each C.O2 has weight 44, each H2.O has weight 18, and each O2 has weight 32. Thus the formula, in terms of molecular weights, is 12*44 (carbon dioxide from air) + 10*18 (water) –> 2*162 (cellulose) + 12*32 (oxygen back to air).

Multiplied out, 528 grams of carbon dioxide from the air, plus 180 grams of water, makes 324 grams of cellulose and releases 384 grams of oxygen back to the air. The process uses 75 percent air and 25 percent water, by weight.

Incidentally, notice that the number of C.O2 gas molecules consumed in the formation of cellulose, equals the number of O2 gas molecules released. There is no change in air pressure as a result of tree growth. If there were an excess or deficit of gas molecules resulting from plant growth, we might expect some interesting breezes and weather effects near forests and fields. Every aspect this world fits together! I have a naturalist friend who says he can tell there is lots of oxygen in the area behind my house — which is surrounded by lush woods and growing plants. Maybe so. I used to be able to detect low oxygen in the stacks of a poorly ventilated library, where the many book pages were slowly oxidizing.

The above is not, however, the full story about the constituents of wood. Living trees are not dry wood. They contain about 50 percent by weight of water. So let’s revisit the calculation. Consider an amount of living tree, let’s say 200 grams. Half of it (100 grams) is 75 percent from air and 25 percent from water. The other half (100 grams) is 100 percent water. The total amount of living tree comes 37 percent from air and 63 percent from water.

There’s even more. Dry wood is not free of water. It contains about 12 percent water. So again consider an amount of dry wood, let’s say 200 grams. The majority of it, say 88 percent or 176 grams, comes 75 percent from air and 25 percent from water. The remaining 12 percent, 24 grams, is moisture trapped within the “dry” wood. Total water content of the 200 grams, as water or as cellulose partly derived from water, is 34 percent. The dry wood is two-thirds (66 percent) derived from air, and one-third water or derived from water.

Conclusion: Dry wood is mostly derived from air. Green wood is mostly either water or derived from water.

Here’s a nice tree experiment from about 350 years ago. It exemplifies scientific patience and careful observation. It also indicates how, despite careful and accurate observation, one can make a mistake by using the wrong model.

This experiment was performed by Jan (Johannes) Baptista van Helmont, and reported in 1648. Van Helmont wished to determine which of the four elements (earth, water, air, fire) was the predominant contributor to plants. More exactly, he believed that plants were mostly made from water, and wished to validate his hypothesis. Here is his report of his experiment, which took five years:

Van Helmont: “That all plants immediately and substantially stem from the element water alone I have learnt from the following experiment. I took an earthern vessel in which I placed two hundred pounds of earth dried in an oven, and watered with rain water. I planted in it the stem of a willow tree weighing five pounds. Five years later it had developed a tree weighing one hundred and sixty-nine pounds and and three ounces. Nothing but rain (or distilled water) had been added. The large vesel was placed in earth and covered by an iron lid with a tin-surface that was pierced with many holes. I have not weighed the leaves that came off in the four autumn seasons. Finally I dried the earth in the vessel again and found the same two hundred pounds of it diminished by about two ounces. Hence one hundred and sixty-four pounds of wood, bark and roots had come up from water alone.”

That quote is from William H. Brock’s book “The Chemical Tree: A History of Chemistry”, which has also been published under the alternative titles “The Fontana History of Chemistry” or “The Norton History of Chemistry”. It begins the introduction of Brock’s book and justifies his title “The Chemical Tree”. A well-written book, enjoyable reading.

Van Helmont’s mistake, is that his model did not include the invisible air, which it turns out contributes about three-quarters of the mass of the tree material. He also excludes sunlight (= energy = fire) from his contributing elements. And of course the earth is also necessary, for nutrients and trace materials, and for support and context for the tree’s life (growth medium). One might say, in the four-element categorization, that trees depend upon all four of the basic elements, and could not flourish were availability of any one of those elements blocked.

We make advances by relying upon our predecessors. Van Helmont was a creative and careful experimenter. But all experiments benefit from prior experiments and also from sincere critiques from others.

An article in Plant Science Bulletin by David Hershey has a very interesting discussion of Van Helmont’s experiment. There are predecessors, contemporaries and later scientists who made related experiements. Hershey draws some conclusions about the scientific process, and also about the teaching process. The best formatting of the article is at this URL:
http://helmont1.tripod.com/hersheypsb49-3.htm
but it is also available at Plant Science Bulletin, vol 49 no 3 (2003), at this URL:
http://botany.org/PlantScienceBulletin/psb-2003-49-3.php
The Hershey article also quotes the above paragraph of Van Helmont’s in a 1662 translation, rather archaic language but useful as there are subtle differences between the 1662 translation and the more recent translation. If I could read Latin, perhaps there would be other subtleties to discover. However, there are other avenues of investation to pursue, and Latin does not make my priority list. German though … very important for science, of the past century and contemporary work.

Let’s have a photo of Jan Baptista van Helmont:

There’s a story behind this portrait. It has been mis-identified as Robert Hooke (of whom no portrait exists). However, one can see by comparing with other images of Van Helmont, for example in one of his books, that this portrait is Van Helmont.

Articles from Wikipedia about wood drying (re moisture content), about cellulose, about Van Helmont:
https://en.wikipedia.org/wiki/Wood_drying
https://en.wikipedia.org/wiki/Cellulose
https://en.wikipedia.org/wiki/Jan_Baptist_van_Helmont

Best wishes,
Ken Roberts
22-Jun-2015

Vepstas Regions 2

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Continuing with the topic of Vepstas regions … Did you spot my mistake in the prior post? To recap, f(z) = z^2/(z-1), where z is complex. Given positive R, the set of z for which abs(f(z) < R is called the Vepstas region V(R), and the boundary set B(R) is the set of z for which abs(f(z)) = R. I'm interested in the shape of Vepstas regions. As a preliminary step, I want to calculate the real values x = x(R) for which B(R) intersects the real axis in the z-plane. That means solving the equation f(x) = x^2/(x-1) = +R or -R, since we are restricting to the case of x being real. But I missed one of the solutions. Too many plus/minus alternatives on the go, I guess. So let's try again at solving abs(f(x)) = abs(x^2/(x-1)) = R, restricting x to be real.

To be systematic, let's use p to represent either of +1 or -1, so that we are solving, to find real x for which x^2/(x-1) = pR. That takes in both alternatives for +R and -R. We can think of p as being one of the two square roots of 1. Now write that equation as a quadratic in the form x^2 = pRx – pR, or x^2 – pRx + pR = 0. Using the quadratic formula, the solutions can be written as (1/2) times (pR plus or minus the square root of a quantity D). The quantity D is called the discriminant), and in this case the discriminant is D = p^2R^2 – 4pR. We can use a symbol q to represent either of +1 or -1, so that taking the square root of D will produce a result that can be written q*sqrt(D), and we do not have to say "plus or minus" when reading the equation. The solution (for real x) is x = (1/2)*(pR + q*sqrt(D)). Moreover, p^2 in the discriminant D is equal to 1 (because p is a square root of 1), and so D equals R^2 – 4pR. So the resulting solution simplifies to x = (1/2)*(pR + q*sqrt(R^2 – 4pR)). In this solution, each of p and q represents either of +1 or -1. There are four alternatives; p might be +1 whereas q is -1, or vice versa, or both p and q could be +1, or both could be -1. On the other hand, the two appearances of p in the solution are the same value. It seems a bit cumbersome to write symbols p and q, but one does not have to consider separate cases quite so much; some of the work, manipulating the quadratic equations, is combined into a common portion of the reasoning. Something like computer programming.

Here are the graphs of the x-intercepts of the boundary curves B(R) with the x-axis, for various R values.

If you compare with the prior post, you will see that there is an additional curve at the top. For R greater than 4, the boundary B(R) of a Vepstas region will intersect the x-axis four times. For example, if R is 5, the boundary curve B(5) will intersect the x-axis at x equal to any of four values, approximately x = -6, +3.7, 0.8 and 1.4 — just approximate, reading off the graph. You can calculate precise values from the formula for x, taking the various alternatives for p and q.

Now, about mistakes … how did I know I made a mistake in my prior post? It just didn’t feel right, but why? It seemed strange to have three solutions (for general R larger than 4), not four solutions. Walking along the x-axis, from very large negative x to very large positive x, one expects to be outside V(R), then inside by the time you reach the origin. Then outside as you approach x=1, where f(x) is undefined (plus or minus infinity). Then (for R bigger than 4) inside again as you reach x = 2, then outside once again for very large positive x. Outside, inside, …(some intermediate wobbles perhaps) … outside — that should involve an even number of transitions through the boundary B(R). So it was the kinethestics of the solution that bothered me. I don’t know how you feel about complex variables and complex functions, but for me they are genuinely FELT — movements of points and surfaces. When one considers the map z –> w = z^2 between two complex planes, it is a wrapping (a physical wrapping) of a surface (the z-plane) around the origin of the w-plane, and it must wrap around twice. You could do it with bedsheets, in principle, except for certain complexities as the sheet has to connect back to itself after two trips around the origin. Anyway, all I can recommend is to cultivate a geometric or physical “feeling” for complex variables and complex functions. For me, it is kinesthetic, and probably involves movement of my muscles (at least the hands and arms) when I am visualizing something.

More later about Vepstas regions.

Best wishes,
Ken Roberts
12-Apr-2014

Vepstas Regions

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The term Vepstas Region is a phrase I’m using for a region of the complex plane which appears in a paper by Linus Vepstas. It’s my term, not his! I just need something to use when speaking about regions of this type. A Vepstas region V(R) is the set of complex points z such that abs(z^2/(z-1)) is less than some constant R, let’s say R = 4 typically. The points of V(4) are the ones for which a computational algorithm devised by Vepstas will converge. The points of V(R) for R meaningfully less than 4, let’s say V(3) or V(2), are the ones for which his algorithm will converge raapidly. In this cluster of posts, I just want to play around a bit with the regions themselves, to look at their shapes. To have some fun with the underlying geometric objects, not worry about convergence of an algorithm.

Let f(z) = z^2/(z-1). The boundary of V(R) is the set B(R) for which abs(f(z)) = R. To get an initial handle on the regions, note that as z tends to 1 (in complex plane), f(z) behaves like 1/(z-1) so has a simple pole at z = 1. The point z = 1 is not going to be in any V(R) region. On the other hand, f(0) = 0 so the point z = 0 is going to be in V(R) for any positive R. The regions V(R) are not empty, and they have boundary points (in the set B(R)) along any path from z = 0 to z = 1. I guess I should mention that f(z) is analytic (except at z = 1), if you haven’t remembered that already, so it is continuous, has derivatives of all orders, is a conformal map, has a power series representation about any point in its domain (that is, except at z = 1 which is not in the domain of f(z)), and so on. The usual properties of analytic functions.

Given fixed positive R, there is a point (or points) z = x along the real axis, between 0 and 1, for which the absolute value of f(x) is R. Let’s find those points. Since we’re using only real values of z = x for the moment, the value of f(x) will also be real, so there are only two possibilities: f(x) is either +R or -R.

If f(x) = +R, we have to solve x^2/(x-1) = R, or x^2 = Rx – R, or x^2 – Rx + R = 0, which is quadratic, and has the solution x(R) = (1/2) * (R plus or minus square root of (R^2 – 4R)). In order to get a real result from that formula, we must have R^2 – 4R non-negative; that is there is a solution f(x) = +R only for R at least 4. Here’s a little graph of those solutions, showing x(R) as a function of R, for R at least 4:

Well, it appears that none of those points x(R) which solve f(x) = +R lies on the segment (0,1) of the real axis. They are all further out than 1. For large R, they get close to 1, never reaching 1. For R a little bit more than 4, they get close to 2, and do reach 2. So the boundary B(R) intersects the real axis somewhere between 1 and 2, provided R is at least 4. Let’s just check: f(2) is 2^2/(2-1) = 4, so that’s right; the point x = 2 is in V(R) for R equal to or greater than 4.

Now, what about the other case, f(x) = -R. Does that have solutions for x between 0 and 1? If f(x) = -R, we have to solve x^2/(x-1) = -R, or x^2 = -Rx + R, or x^2 + Rx – R = 0, which is quadratic, and has the solution x(R) = (1/2) * (-R plus or minus square root of (R^2 + 4R)). That formula will always give a real result, two of them. The square root of (R^2 + 4R) will be greater than R, so one of the results will be negative (outside the interval (0,1), and the other will be positive, and equal to (1/2)*(sqrt(R^2+4R) – R). Here’s a graph of those x(R) values. This time, I’m going to consider any positive R, not just R at least 4.

Actually, I’ve shown three x(R) curves on that graph. The top curve is the previous one, squished. The horizontal line is just to show the line with ordinate 1, so you can see the asymptotic behaviour. The middle curve is the position of the boundary point of B(R) on the x-axis between 0 and 1, and the lower curve is the position of the boundary point of B(R) for negative x.

That’s enough for now. I encourage you to play a bit with determining the shape of the V(R) regions for various R values. More later…

Best wishes,
Ken Roberts
10-Apr-2014

ps. Are there mistakes in the above? Perhaps. I make lots of mistakes; it is how I learn. See post some time ago about Mistakes being an essential aspect of the learning process. You will learn the most if you catch me in a mistake before I get around to correcting it (if ever).

Mistakes — How to Learn from Them

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Making mistakes is one of the best ways to learn. Yet, we are taught to believe that making mistakes is wrong, and students are penalized for mistakes. That is unfortunate, and in this post I wish to argue for the merits of making mistakes.

Watch a baby learn — moving about, grasping, throwing, speaking, and so forth. A parent would be justified in serious concern if their child did not make constant, though unsuccessful, attempts to move about, grasp, throw, or speak. The child learns to succeed through repeated attempts, most of which are unsuccessful. Thousands of attempts to move about, to grasp, to throw, or to speak strengthen the neural and motor structures which allow for later success.

Youth is the period of our lives when we learn the fastest. I think that is partly related to it being the time when we attempt the most, that is, make lots of mistakes. Perfection and predictability are in opposition to the ability to learn.

I heard a story of an engineer in a research institute, who, when he had an idea, would try it out on his colleagues, one by one. He would walk into someone’s office, set forth the idea, and it would be shot down, full of flaws. He would revise his idea, and go to the next colleague — same result. After he had visited many colleagues, his idea would have become refined by the various circumstances and considerations set forth by his colleagues. If his idea survived, it would be robust. Most ideas would not survive unscathed; in fact, most might not survive the chain of validations at all. But the good ideas — the few percent that were really worth pursuing — would become much better. That engineer must have been known to his colleagues as a person who made lots of mistakes. It takes guts, or humility, to appear “silly” in order to keep learning new material, since that invariably involves mistakes.

I was trained as a mathematician, a field which, more than most others, puts a high premium on error-free problem solving, the avoidance of mistakes. When I became a computer programmer, it took a while for me to discover that the most effective way to program is to develop partial solutions, and then refine them (debug them) against realistic requirements. Effective programming involves the art of making mistakes, repeatedly, and learning from and extending the software based upon the experience of those mistakes.

Having returned to the academic world, in retirement, as a student, I notice that academic life puts a high value on being relatively mistake-free. That is a trap, a social pressure that can lead to reduced productivity. Mistakes, lots and lots of mistakes, whether in academic or industrial settings, are important in order to make progress.

Here’s hoping that you are able to enjoy making mistakes, as much as I do!

Ken Roberts
28-Jan-2014