82°Posted: June 1, 2018
This is a 100% recycled yak fur story.
Molly asked me a thoughtful question while I was in the middle of something else and I said “Just a second, I need to rotate my brain.” A moment later, I said OK and she asked me if my brain was all rotated now. I replied, “Well, 82° out of 90°, close enough.” Molly quipped “And since when did you ever do things square?”
But then I nerdsniped myself. I started to wonder about 82° angles. If a polygon with 90° angles is a square, what sort of polygon do you get if you turn 82° at each corner? It’s not one of your basic shapes, because:
- 120° angles make a triangle (3 sides, since 360° ÷ 120° = exactly 3)
- 90º angles make a square (4 sides; 360° ÷ 90° = exactly 4)
- 82° angles make a ??? (?? sides; 360° ÷ 82° = 4.3902439024???)
- 72° angles make a pentagon (5 sides; 360° ÷ 72° = exactly 5)
It must be some kind of … star shape, 82 doesn’t go evenly into 360, which means that you’d have to spirograph-around the circle more than once to come back to where you started.
But what kind of star? How many points would there be on a star with 82° angles at each point? I decided that I had to know, and that I wanted to see what the star looked like, not just find out the numeric answer. The numeric answer is the GCD of 82 and 360, and I could figure that out, but then where’s my picture of the star? I decided to write a quick program to draw stars with angles of ‘N’ degrees at each corner, and to print out how many points there were on the star.
So ten minutes later, I had refreshed my Logo skills. Luckily, Logo is secretly sort of a dialect of LISP, and I’m very comfortable with LISP-like languages. Here’s the code I finally came up with:
to anglestar :angle clearscreen penup forward 150 pendown make "count 1 setheading :angle while (heading > 0) [ forward 200 right :angle setheading round modulo heading 360 make "count sum :count 1 ] forward 200 show :count end anglestar 82
And it printed “180”, meaning that this is a 180-point star. Or, as Molly pointed out, more of a bike wheel than a star really. An imaginary bike wheel.
I decided to try other angles since I now had this great program. With N=80°, you get a nine-pointed star, because 80° x 9 = 720°, which is twice around the 360° circle:
And with N=81° you get a 40-pointed star:
Although as Molly pointed out, this one isn’t really a ‘star’, either. This one is more of an imaginary bicycle gear, a lot like this real 40-toothed bicycle gear:
Oh, and after all this, Molly very patiently re-asked me her original question again, and I answered her without yak-further delay.