November 28, 2024 Spiral of Theodorus

 

Figure 1

Figure 2

Today, we made Spirals of Theodorus in class over Zoom. One small piece of feedback I have is that the cardboard didn't really have a chance to play a part in the creative process, so I was relieved because I couldn't find any.

Anyways, this blog post will probably be shorter; basically, when I made the paper version of the spiral, I decided to fold along the hypotenuses. At first I wore the Cone of Theodorus on my head as a hat, but then I took it off. Then I thought: what if I folded the spiral in an accordion style? The result of that can be seen in the first image, which looks like a bunch of stacked ice cream cones where the gap gets narrower and narrower.

I owe an apology because I stated that the "right angles" (angle between the inner and between lines) I pointed out on the shaded triangles in Figure 2 are not actually right angles. What a tragic result! However, it should be noted that they tend towards 90 degrees as the spiral goes outwards. Figure 2 also looks like a cool art piece on its own -- maybe there's some rearranging that could also be done with the shaded triangles?

NOTE:

• "inner side" = the side of the shaded triangles facing the centre of the spiral
• "outer side" = the side of the shaded triangles facing the outside of the spiral; all length 1
• "between side" = the last side between the inner and outer sides

Some trends I've noticed with the accordion fold shaded triangles; I have not entertained deeper on things here, but I hope it raises some neat questions:

• The inner side's length tends towards 1.
• The between side's length gets smaller and smaller. I can't say in what pattern, but it appears to be asymptotic, where the length is longer at first but approaches 0 (but never reaches 0).
• The outer side lengths stay as 1 the entire time.
• The shaded triangle areas approach 0.
• The between line's angles both approach 90 degrees.
• We can say that the dimensions of the triangle approach a flat line, so the areas approach 0. My hypothesis is that the sum of areas converges if looking at Figure 1, but I'm not sure what it would converge to because I don't have an exact formula of the areas of the shaded triangles (by virtue of the right angles not being actual right angles...)
• Looking at Figure 1, even though I can't point out why, there is a very nice uniformity to it that just makes you want to figure out the pattern of things (like the spacing between the lines, or areas, or angles, etc.).
• A comment could be made on the horizontal position of the lines in Figure 1. What do they approach? Is there a curve or mathematical function that represents what we see here?

All in all, this was a surprisingly flow-inducing activity to work on today. Very nice stuff, and I would love to just hand my students these spirals and ask them to think about the trends they observe, and to manipulate the spiral in interesting ways! It certainly sparked my mathematical gears, just tracing borderline conspiracy-level patterns.