### Thirteen Sided Dice

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**Wed Jan 14, 2009 11:03 pm**Recently I rendered the three possible cell-transitive tridecachora using Off files and Stella4D. I started by using the thirteen vertices of their duals, then using a tridecagon to find a set of triangles and complete tetragons (representing tetrahedra) that would form a closed figure - imported this OFF file and take its convex hull - then the dual. There are three possible tridecachora with congruent cells (which make them fair dice) - they are various step tegums.

The first one is the phase 5 tridecachoron (or simply "tridecachoron") - it has double the symmetry of the other two - to get the phase 5 - take a 13 by 13 square - choose the first vertex of the first row, on the next row move to the right 5 units, continue this until the bottom is reached - treat the 13 by 13 grid like an Asteroid game screen, when you move to the right edge, continue on the left edge. This grid is then curved into a duocylinder shape - the thirteen vertices will be the vertices of the dual of the phase 5 tridecachoron. This tridecachoron has 13 cells (8-sided with 4 pentagons and 4 kites), 52 faces (26 isosceles pentagons and 26 kites), 78 edges, and 39 vertices. Attached is a picture of it's net:

The next one is the phase 2 tridecachoron (it can also be treated as a phase 6) - we could call this one the Mobius tridecachoron - for the set of tetragons and triangles seem to form a surface with a Mobius like twist. It's dual's vertices are formed like the above tridecachoron but with a 1-2 L move (knights move) instead of a 1-5 L move. It has 13 cells (12 sided object), 78 faces (13 of each of the following: hendecagon, triangle, 4 types of kites), 130 edges, and 65 vertices. It has an unusual net:

The final one is the phase 3 (also phase 4) tridecachoron. The dual's vertices use a 1-3 L move. It has 13 cells (10-sided), 65 faces (13 of each: octagon, hexagon, kite, 2 types of triangles), 104 edges, and 52 vertices. Here's the net:

With these 3 thirteen sided dice - all we need now is a seriously twisted game - - Hahahahahaha!

I've also got the .stel files if anyone's interested.

The first one is the phase 5 tridecachoron (or simply "tridecachoron") - it has double the symmetry of the other two - to get the phase 5 - take a 13 by 13 square - choose the first vertex of the first row, on the next row move to the right 5 units, continue this until the bottom is reached - treat the 13 by 13 grid like an Asteroid game screen, when you move to the right edge, continue on the left edge. This grid is then curved into a duocylinder shape - the thirteen vertices will be the vertices of the dual of the phase 5 tridecachoron. This tridecachoron has 13 cells (8-sided with 4 pentagons and 4 kites), 52 faces (26 isosceles pentagons and 26 kites), 78 edges, and 39 vertices. Attached is a picture of it's net:

The next one is the phase 2 tridecachoron (it can also be treated as a phase 6) - we could call this one the Mobius tridecachoron - for the set of tetragons and triangles seem to form a surface with a Mobius like twist. It's dual's vertices are formed like the above tridecachoron but with a 1-2 L move (knights move) instead of a 1-5 L move. It has 13 cells (12 sided object), 78 faces (13 of each of the following: hendecagon, triangle, 4 types of kites), 130 edges, and 65 vertices. It has an unusual net:

The final one is the phase 3 (also phase 4) tridecachoron. The dual's vertices use a 1-3 L move. It has 13 cells (10-sided), 65 faces (13 of each: octagon, hexagon, kite, 2 types of triangles), 104 edges, and 52 vertices. Here's the net:

With these 3 thirteen sided dice - all we need now is a seriously twisted game - - Hahahahahaha!

I've also got the .stel files if anyone's interested.