The Geometry of Irregular Forms: Part 3

(Introduction to this series is at this link.)

Let's return to my experiment with the calculator's square root key. Although I didn't know it at the time, what I had stumbled upon was an iterative algorithm. That's fancy words for a process that's kind of like hitting a nail with a hammer. Every time you hit the nail, you drive it into the two-by-four a little bit. That's one iteration of the process. Then you take the output of that iteration (a nail that's part-way into the board), and you do the same thing to it. You do this over and over, until a terminating condition is met -- in this case, when the nail is all the way into the board.

When you start an iterative algorithm, you need an initial condition. From my square root experiment, 0.0000001 and 99999999 are examples of initial conditions. If we take the variable x to be the number on the calculator's display, then the initial condition is x₀.

For the first iteration, we plug x₀ into the process, and it yields the next x, or x₁. And the second iteration of the process yields x₂, and so on. In general:

the next x = do-something-to( the current x ), or
x_n+1 = do-something-to( x_n)

In my square root experiment, the "do-something-to" is the square root function. In the hammer-and-nail example, the "do-something-to" is striking the nail with the hammer. Using the x_n notation, the sequence from Part 1 looks like:

x₀ = 0.0000001
x₁ = 0.0003162
x₂ = 0.0177828
x₃ = 0.1333521
x₄ = 0.3651741
x₅ = 0.6042964
x₆ = 0.7773650
x₇ = 0.8816831
x₈ = 0.9389798
... 15 more ...
x₂₄ = 0.9999990
x₂₅ = 0.9999995
x₂₆ = 0.9999998
x₂₇ = 0.9999999
x₂₈ = 0.9999999
x₂₉ = 1

After hitting the square root key 29 times, I arrived at a terminating condition -- the calculator read 1, and the value was no longer changing -- and so I stopped.

The Geometry of Irregular Forms: Part 2

Suppose I lived on a country estate on which there lived a predator population of foxes and a prey population of grouse (the example given in Stoppard's Arcadia). On this idealized estate, this is the extent of biological diversity. If the foxes were numerous, they would eat most of the grouse, which would become scarce in the short run. But then the foxes would have nothing to eat, and their numbers would decline. With fewer foxes around, the grouse population would recover. In the case of a marginally stable ecological system, the numbers of foxes and grouse would be cyclical and experience population explosions and crashes. An unstable system would have the foxes eat all the grouse to extinction, followed by their own extinction from starvation. But if the system is stable, the fox and grouse populations would eventually settle to an equilibrium state.

I first learned about this interaction between predator and prey populations when I visited Epcot Center (now just "Epcot") in Orlando at the age of eleven. One of the Future World buildings had an interactive computer exhibit, which was pretty futuristic at a time not far removed from when a Pong game console represented the pinnacle of home computing. The exhibit had a bank of computers each running a simulation of a predator/prey system. It let you enter initial populations, and possibly other parameters such as reproductive rate, and then it would display a graph of how the populations would change over many years. Usually, the graphs would have either an exponential-decay shape or a sine-wave shape. I thought this was very cool.

The Geometry of Irregular Forms: Part 1

(Introduction to this series is at this link.)

I was about seven years old when I discovered the square root key on my father's calculator. This was back when electronic calculators plugged into the wall outlet, and a square root key was a real novelty. I'd pick a number, hit the square root key, and then multiply the result by itself -- I'd get the original number back. But the most interesting thing about that square root key was that, unlike the basic arithmetic operations ('+', '-', etc.), the square root is a unary operation. That meant I could hit that square root key over and over again without having to enter other numbers or the '=' key.

In my experimenting, I identified four cases: (a) I enter '0'. No matter how many times I hit the square root key, it stays at 0. (b) I enter '1'. No matter how many times I hit the square root key, it stays at 1. (c) I enter a number between 0 and 1. Every time I hit the square root key, the result gets larger and approaches 1, until the calculator reads precisely 1. (d) I enter a number greater than 1. Every time I hit the square root key, the result gets smaller and approaches 1, until the calculator reads precisely 1.

I would try extreme cases. What if I put in the smallest positive number the calculator would allow (0.0000001)? The largest (99999999)? I would count the number of times I had to press the square root key before the calculator read 1. Yes, I was a strange child, but in my defense, they didn't have Nintendo back then. Here is the sequence when I started at 0.0000001:

0.0000001
0.0003162
0.0177828
0.1333521
0.3651741
0.6042964
0.7773650
0.8816831
0.9389798
... 15 more ...
0.9999990
0.9999995
0.9999998
0.9999999
0.9999999
1

Twenty-nine times. Same as if you start with 99999999.

So far, the most difficult mathematical concept we've encountered is the square root. And everything described above can be done by a seven-year-old with a simple calculator.

The Geometry of Irregular Forms: Introduction

It's been two weeks since I posted my last update. I'm still here.

Last week I saw the play Arcadia by Tom Stoppard. Stoppard, with Terry Gilliam of Monty Python, co-wrote the screenplay for the movie Brazil (one of my favorites). That alone sets an expectation for genius. And just last night, the Broadway production of his trilogy The Coast of Utopia won the Tony award for best play.

I enjoyed Arcadia, even if it is not perfect. I'm still trying to figure out why some of the characters were even necessary (the mute child, for instance). But I was interested in how Stoppard works in some mathematics and science and uses some of the same themes as The Coast of Utopia dealing with the conflict between the Enlightenment of the 18th century and the Romanticism of the 19th century. Enlightenment philosophers, building on the discoveries of Galileo and Newton, thought that all problems, and all questions about how the world worked, could be solved by the application of reason. This was a great step forward from earlier periods dominated by superstitious thinking -- when women were being burned as witches, and lightning was seen as a sign of God's wrath. However, it became clear that not everything that happens in the world is as deterministic as a simple equation. Arcadia contains a tale of modern chaos theory being developed -- and then forgotten -- in the early 19th century as a synthesis of both Romanticism and the Enlightenment, even before such pioneers in the field as Sir Arthur Cayley were born.

One of the characters in Arcadia wrote a book titled The Geometry of Irregular Forms (which exists only in the play). I can't make any promises -- this is merely a blog, after all -- but I'll try to do a series of postings on this subject: the geometry of irregular forms.