Moss's Egg

[sdi]

The highlighted curve is called Moss's Egg, though I'm not certain who Moss is. (The link cites Dixon's Mathographics—I ordered a copy, perhaps it'll tell me.) I was surprised to find a way to construct it using only 8 circles! This is because circles naturally want to generate √3's, so usually it'd take extra effort to generate the √2 proportions the egg demands, but it turned out to only require going a single circle out of my way.

Comments

[violetcabra]

If I may, I applaud these proofs you are doing. They are well beyond my capacities to follow along, and I am deeply impressed! Also, I hope you are feeling better!

[sdi]

Thank you! I'm having a lot of fun with these—it's really difficult to think in circles rather than lines, since you sorta have to come at every problem sideways. I'm slowly gaining some intuition for the problem.

I think the biggest trick to following along on paper is to work out, arithmetically, the radius of each circle drawn. Shapes always have a characteristic number associated with them, and once you've managed to work out the numbers involved, it gets a lot easier to follow: √3's make hexagons, √2's make squares, ϕ's or 1/ϕ's make pentagons...

And thank you also for the well wishes! I'm still sick, but I've at least been learning a lot...

[boccaderlupo]

Agreed with violetcabra. These posts are beyond my grasp, but they are captivating images and the process itself seems interesting.

Maybe you've addressed already, but wondering if you are "tracing" this back to the "shape" of the soul, according to the ancients, as a kind of ritual meditative technique...?

Axé

[sdi]

If you're asking "do you have a purpose in all these?" the answer is "no, I'm just playing." :) Life has been harsh and having the capacity for play is rare and precious: you know my opinions on She-Who-Delights-In-Play, so I take it as a sacred act and try to enjoy myself! The way with these things is that playing tends to lead one to deeper insight, so maybe I'll have something deeper to say about it sometime. (Or maybe not! The play is enough.)

If you're asking "why circles?" the answer is that I was really surprised to discover the Mohr-Mascheroni Theorem—anything you can do with compasses and straightedge, you can do with compasses alone—and wanted to develop an intuition for why. But in playing around with it, I've found that compass-only constructions have a very different feel to them; normal construction feel like you're using a set of tools in known ways; but compass-only constructions are much more mysterious, more like a puzzle. I suppose the mystery of it fascinates me.

Also, I like how the space is less explored. People far smarter than I have been doing "normal" geometry for millennia: it's very well understood, and there's 40 constructions for anything. If you search around for compass-only constructions, though, it's hard to find much of anything. So if you want to know how to draw something, the only way is to sort it out yourself!

also like any straight male I like round things more than straight things

[sdi]

It is worth noting that this construction of Moss's Egg implies another nice, simple construction of a square (e.g. Napoleon's problem):