Back in highschool, I had a fascination with geodesic domes and such, and built various balls out of paper with various number of triangles in them. The most complicated one I worked out the math for and built had 180 sides to it, with three slightly different sized and shaped types of triagngles. Its the rightmost one in the picture above.
After having built a lot out of paper, I decided to build one a little more solid. Making one out of wood was more challenging, with lots of compound miters, As such, I only used a 60 sided figure, which consists of 60 identical, but not equilateral triangles. The topology for this one is the same as the paper model in the middle in the photo above.
I left one section of 5 triangles only screwed in, and afterwards thought of entrapping the cat in it to see what it thought of it. Surprisingly, the cat seemed to enjoy it for a short while, but quickly got bored of it and meowed to be let out.
The most recent geodesic type ball I built was as a prop for the FASS 2006 show. What was fun about this one is that I did the geometry entirely using a compass, square and a ruler. No calculator or calculations. No protractor either. I worked out all the dimensions by geometrical construction. The one thing I did need that I couldn't construct was a 72 degree angle, but, I could check it by doing it five times. And, as it turned out I guessed exactly on the first time I tried. The design for this one is the same as the second from right one of the paper models at the top. The whole thing was painted to look like an eyeball. Funny thing is, I saw that prop on somebody's cubicle at work about a year later. I guess somebody involved in the show decided it was worth keeping.
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