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.