Thursday, April 3, 2014

Math is Beautiful

Like pretty pictures? I got these from Cye Waldman, a retired engineer who's been doing mathematical modeling for over 40 years. The images below weren't created in SketchUp (he uses Matlab), but he just got one of my books to learn how to make them in SketchUp.

Here are some of his Voderberg patterns:

And here's the book that shows how to make these, and others:

If you're a math geek (and maybe even if not), you'll love what Cye has to say:
By way of introduction, I’m a retired engineer/physicist/mathematician with 40-odd years of experience in mathematical modeling. Upon retiring, I took up recreational math and have had a number of papers accepted at the National Curve Bank (specific links below). A few months ago I turned my attention to tiling. I do all my work in the complex plane, it really simplifies things. This morning I undertook to model Voderberg tiling. I parameterized it in terms of two angles and calculated all nine arm lengths. (Actually, I start out by taking one of them to be unity.) Well, one of the angles must be 12 degrees and the other is limited to ~111-153 degrees without lines crossing. Thus, with 20 lines of code I have a generalized Voderberg tile. The material in your book helped me proceed further very quickly. Especially the tile sets required for the outer rings. I also found that I could use the conjugate tiles to good advantage and that I could construct an ordinary rectangular tiling as well. This tiling gives me access to any of the wild transformations you see at the Web site. The computation of tiling takes about 0.02 seconds and then another 0.2 seconds to render.

Cye also can tile those Voderbergs linearly:

And like this:

Lastly, here's his Hirschhorn 72-degree tile.

Cye's five interior angles, which would be a bit cumbersome to enter in SketchUp are:


If you like this stuff, here are some links to some of Cye's animations on the Curve Bank pages. Warning - hypnotic!

Sinusoidal Curves

Fibonacci Spiral and more Fibonacci Spiral

Polynomial Spiral

Gamma Pulse

 and my favorite, Valentine Heart Tesselations

Anyone can design anything in 3D!

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