Uploaded July 2021 | Updated September 2026, 1 week ago
We can subdivide every face of a dodecahedron into 10 triangles, making a 120-sided fair die (The Dice Lab sells dice like this).
Here, every 3-dimensional face of the 120-cell is subdivided into 120 tetrahedra, as explained above. Thus, we obtain a four-dimensional 14400-sided die.
The boundary (3-dimensional surface) is three-dimensional, so we can have the "inner view" of the die. Here the inner view is approximated using spherical geometry. The opposite faces sum to 14401; because of the weirdness of spherical geometry, they are seen in the same spot. (1/2)
Here is a similar scene without the approximation: twitter.com/ZenoRogue/status/1415331248785006593 (people think there are compression artifacts -- not sure whether they see actual compression artifacts or think that the weird effects caused by the Euclidean approximation of spherical geometry are compression artifacts!) or in HyperRogue 11.2d with commandline parameters: "-noplayer -geo 120c -coxeter 7 -canvas-random 5 -ray-do -ray-cells 99999 ray_fixed_map=1 -gfake-euc"
We can subdivide every face of a dodecahedron into 10 triangles, making a 120-sided fair die (The Dice Lab sells dice like this).
Here, every 3-dimensional face of the 120-cell is subdivided into 120 tetrahedra, as explained above. Thus, we obtain a four-dimensional 14400-sided die.
The boundary (3-dimensional surface) is three-dimensional, so we can have the "inner view" of the die. Here the inner view is approximated using spherical geometry. The opposite faces sum to 14401; because of the weirdness of spherical geometry, they are seen in the same spot. (1/2)
Here is a similar scene without the approximation: twitter.com/ZenoRogue/status/1415331248785006593 (people think there are compression artifacts -- not sure whether they see actual compression artifacts or think that the weird effects caused by the Euclidean approximation of spherical geometry are compression artifacts!) or in HyperRogue 11.2d with commandline parameters: "-noplayer -geo 120c -coxeter 7 -canvas-random 5 -ray-do -ray-cells 99999 ray_fixed_map=1 -gfake-euc"
