November 23, 2006
How to Make a Pentagon from a Circle
I think I mentioned at some point that it was easy to make an elegant pentagon from a circle of paper. It is, but still, it’s not intuitive. Making polygons from circles has a lot in common with compass-and-straightedge work. Except paper is easier to work with and the straightedge can be marked up like crazy.
Pentagon from a Circle Sequenced Crease Pattern
Based on a method described by H.W. Richards in 1893 – not that I’ve seen this. I read about it at MathWorld.
Wow, that seems like a really complex way to build a pentagon. But indeed, it is elegant. I love geometry done with compass and straightedge, it feels “right”… much less insulting to the paper muse than plotting angles in Illustrator.
My opinion, anyhow!
November 24th, 2006 at 9:12 amComplex, sirrah! I think your Swiss Army knife is perhaps missing the Ockham’s Razor blade.
Find me a method with fewer than eight steps to elegance. I will wait.
*cough*
November 24th, 2006 at 10:38 amHey there! Nice rose, and yes, pentagons are fun. But if I had to teach such a pentagon to a group of people (say, for your rose), I’d opt to use the Fujimoto angle approximation method to divide 360 degrees into 5 equal (or close-enough-for-all-practical-purposes) parts. Do you know this? It works exactly like his method for approximating 1/5ths from a strip of paper, but you do it for angles. So you start with one crease from the center to the side (a radius of the circle) and call this the 0 degree crease. Then guess where the 72 degree crease would be, perhaps by making a pinch on the circle’s border. Then you have two “angles” in the circle — one approx 72 deg and one approx 360-72 = 288 deg. Divide the 288 deg angle in half with another pinch on the border of the circle. This makes a pinch at roughly the 72 + 288/2 = 216 deg mark, except now the error you had has been divided in half. Then bring the 216 deg pinch and the 0 deg mark together to get a pinch for approx the 288 deg mark. Then work your way around the other way of the circle — in the end you’ll have a much better approximation of the 72 deg pinch (your initial error will have been divided into 16ths — barely visible!).
Maybe it’s me, but I find that a lot easier and kinda fun.
November 25th, 2006 at 9:02 pmThus, I am answered. (It never hurts to have a mathematician dropping by.) That does sound easier and I certainly will give it a try.
I was trying to imagine teaching that rose today – there are a lot of stumbling blocks to get over, though I know people would enjoy collapsing the spiral. Perhaps at the next Convention.
November 25th, 2006 at 10:37 pmThere is a sequence for folding a pentagon from a square, which may be handy for some. If folded perfectly, you get a mathematically accurate pentagon at the end. Irregularities of folding something real, by a human folder, will give you results more or less accurate.
The diagram can be found here.
January 7th, 2007 at 12:14 am