Recreational geometry (long)
One of my pastimes is what I call recreational geometry.
Basically, the idea is to take out some paper, a compass, and a straightedge, and do geometric constructions using just those tools. No measurement involved.
For me, that typically means drawing a circle, then dividing it into an arbitrary number of nearly equal parts. That involves drawing lots of arcs with the compass, finding intersections, connecting dots with the straightedge.
I find it very meditative. It's simple and repetitive. And it plays well with some of my disabilities, e.g. essential tremor (which makes drawing freehand a challenge at times).
I've already devised a metric ton of constructions approximating odd-numbered divisions up to like 55. That was fun and took up many hours! I used a program called GeoGebra to do that. It was like a puzzle but without pieces; I used trial and error to come up with the necessary steps to construct each polygon.
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re: Recreational geometry (long)
@Qwyrdo ohhh i love ruler and straightedge constructions. there's something very zen about them, and the resulting illustrations always look really cool, like magical sigils
iirc origami folds can also be used for measurement-free geometric construction? i read a paper about it once
related math
@Qwyrdo @typhlosion this reminds me of a math project I did in undergrad. If you start with a few marked points on a sheet of paper you can't rotate and you can only draw lines of a few fixed angles through marked points, and you are allowed to mark any points that lie at the intersection of two lines, then what does the set of constructible points look like?
If you have three angles, it looks like a lattice. With four or more, the constructible points are dense in the plane and have a rather algebraically nice structure
(The advisor for the project motivated it by relating this question to the way origami folds can be used to find reference points for new folds)
re: Recreational geometry (long)
@typhlosion Yes! Origami can do neat things from what I recall, like trisect an angle (which is impossible to do with compass and straightedge alone).
That said I've never really tried it, myself. I don't really have the right kind of paper to hand, which folds nice and crisply.