hyperbolic spirograph

0:00 Tusi couple
The circumference of the small circle is 𝑐 and the circumference of the large circle is 2𝑐. In Euclidean geometry, a point on the small circle moves in a straight line. Can you guess what changes in hyperbolic geometry?
0:12 half the circumference in hyperbolic geometry
In the hyperbolic plane, the circumference of the circle grows exponentially with the radius. Therefore, the small circle is not much smaller than the big circle (the difference between the radii up to ln(2)). The shape will be different, closer to a circle if 𝑐 is large.
1:12 Quarter the circumference in hyperbolic geometry
Here the ratio is 4. In the Euclidean case we get an astroid, but for large 𝑐 we get a shape closer to a triple circle.
2:12 Half the radius in hyperbolic geometry
If the circumferences are not commensurate, we get no period, so only a part is shown -- obtained after 1 full rotation inside the big circle.
3:12 Half the circumference in spherical geometry
Spherical geometry is the opposite of hyperbolic geometry, so the effect is also opposite -- we also get a lens shape in the end, though, just drawn in a different way. The largest red circle is a great circle. (We could go further than that -- the circumference of the would start getting smaller -- but that does not produce any interesting new effects.)
3:32 Circle rolling on a horocycle
3:52 Circle rolling on an equidistant curve
4:12 CIrcle rolling on a line
Source code: github.com/zenorogue/hyperrog...
Twitter thread: / 1358464962088337410

[@Ottmar555]

Love the horocycle one.

[@mymyscellany]

That was beautiful. Thank you.

[@sosasees]

This looks really interesting.
This video was stunning in a literal sense of the word:
My body was in the same position for the video's length of time to see every moment.
I also really like the Color Oscilloscope visual style. It fits really well.

[@katakana1]

0:40 CBS logo

[@katakana1]

He protecc
He attacc
But most importantly
The HyperRogue guy is bacc

[@AttacMage]

I am here just to tell you that you spelled attac wrong >:( ruined my day, unsubbing from you.
I definitely was subscribed beforehand

[@katakana1]

@@AttacMage
attac
c

[@melwugon3687]

You should check out the discord, Zeno is active there

[@Invalid571]

Beautiful animation. 👏 👏 😁

[@WildStar2002]

So beautiful!

[@zacozacoify]

Beautiful

[@FlamingKetchup]

Why in the Half the radius in hyperbolic geometry case the shape is not closed?

[@FlamingKetchup]

I'm guessing it's because the pattern repeats, but misaligned, which if shown would take up too much of the screen?

[@MushookieMan]

@@FlamingKetchup It appears to be changing period in a continuous fashion, sometimes the two circumferences are related by a rational number, and the pattern will repeat. Most of the time, it's probably an irrational relationship.

[@harrympharrison]

There's “no period,” because the ratio is irrational (pi), so yes, it would never ‘join up,’ and so only part of it has been drawn.

[@MushookieMan]

​@@harrympharrison The ratio between the circumferences is not any constant number (including pi). If it was, we would be talking about euclidean space.

[@glitchy9613]

What projection is used for the spherical geometry

[@evilsorosfundedgovernments433]

pretty cool

[@zebrapringlez3141]

Wow

[@rickyardo2944]

Gorgeous! Muchas gracias!

[@Fine_Mouche]

i need half the circumference in hyperbolic geometry to help me draw eyes XD

[@moroni8299]

Interesting

[@-_Nuke_-]

That was a bit hyperbolic of you

[@Austinkeith2010]

Android port when

[@einekartoffel2490]

B

[@mjde9532]

Does anyone else just sees a black screen?