Friday, January 5, 2007
Geometry and Mathemagic
I recently took up drawing and painting, so I'm learning to see the world around me as shapes and forms, colors and lines. This reminds me of a book I once read: A Beginners Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science, by Michael Schneider. In it, the author shows you how to construct perfect geometrical shapes, from a triangle to a decagon, using only a compass and a straightedge. He also discusses (and illustrates!) the roles these shapes, and the numbers 1-10 play in art, architecture, and nature.
I don't have the book in front of me, but I recall that you can make each shape, from 3 sides to 12 sides, except 7 and 11. I'm going to give it a try (using Illustrator instead of paper) and show you here if I can figure it out.
Triangle is the easiest of course. Three circles of the same size, with the edge of each one aligned with the centers of the others, gives you an equilateral triangle. If you want to try this, or other shapes, with a pencil and paper, here are the rules:
1. You can make circles and straight lines.
2. You can mark the centers of circles, the points where lines cross, and the point where a line is tangent to a circle.
3. You can use the compass to mark off distances.
4. You may not measure distances with a ruler, or angles with anything.
Go to it! I'll post my attempts as soon I complete them.
Also, if there are any mathematical minds out there, I've been wondering if there is a way to test which shapes can and cannot be drawn by this method. For example, why not seven sides? Is there a formula to tell you which would work?