ZenoRogue | hyperbolic spirograph @ZenoRogue | Uploaded February 2021 | Updated October 2024, 13 minutes ago.
0:00 Tusi couple
The circumference of the small circle is π and the circumference of the large circle is 2π. 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 π 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 π 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/hyperrogue/blob/master/rogueviz/hypocycloid.cpp
Twitter thread: twitter.com/ZenoRogue/status/1358464962088337410
0:00 Tusi couple
The circumference of the small circle is π and the circumference of the large circle is 2π. 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 π 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 π 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/hyperrogue/blob/master/rogueviz/hypocycloid.cpp
Twitter thread: twitter.com/ZenoRogue/status/1358464962088337410
