Jabe
Joined: 12 Jan 2008 Posts: 46 Location: Somewhere between Texas and the Fourth Dimension

Posted: Wed Jan 14, 2009 11:03 pm Post subject: Thirteen Sided Dice 


Recently I rendered the three possible celltransitive 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 (8sided 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 12 L move (knights move) instead of a 15 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 13 L move. It has 13 cells (10sided), 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. _________________ May the Fourth (dimension) be with you. 
