higher-order sphericons

After seeing this hexa-sphericon rolling yiff I started playing around with similar derivative shapes.

The sphericon can be understood as a square that is revolved around an axis that passes through two opposite corners. That forms two cones, which are then cut in half, with one half rotated ninety degrees, and glued back together. So “higher order” sphericons can be produced the same way, only instead of a square to begin with we use a higher numbered n-gon. This page illustrates the cross sections & results for various polygons.

Instead of revolving the square around an axis passing through the corners (which produces two cones), we can instead revolve it around an axis passing through the middle of two opposite edges (which produces a cylinder). However this only works for polygons with an even number of sides, for polygons with an odd number of sides (triangle, pentagon, heptagon, ...) the only axis of symmetry will pass through both a point and an edge.

After revolving, when we cut the shape in half and one half, we're given additional new options with higher numbered polygons. Triangles & squares only give one option each (120 degrees & 90 degrees respectively), but for a pentagon we can rotate one half either 36 or 72 degrees. (There are two more options that are mirror images of these as well.)

a sphericon made with a 12-sided polygon

This version has exactly one flat “strip” that forms a complete loop (in green), and two sharp edges dividing it (red and blue), though I don't know how to figure out what path it would trace out if you were to roll it (like the hexasphericon linked above).

a sphericon made with a 12-sided polygon

We can rotate the 12-sided polygon 5 different ways in 30 degree increments (the sixth rotation would be 180 degrees and be back to where it started). Rotating it 90 degrees results in three separate flat loops (orange, green, and blue).