The third image in the last post is a faceting of the icosidodecahedron. In that faceting, the faces used are equilateral triangles, star pentagons, and golden rectangles. To make these two new images, starting with that particular faceting of the icosidodecahedron, I rendered its triangles and star pentagons invisible, leaving only the thirty golden rectangles. It’s shown twice below, simply because I wanted to show it using two different coloring-schemes.
I would not be able to create images like this without the use of my favorite computer program, Stella 4d, written by a friend of mine who lives in Australia. You can try this program yourself, as a free trial download, at http://www.software3d.com/Stella.php.
It would really be a pain to count the faces of this polyhedron, in order to verify that there are 270 of them. Fortunately, it isn’t necessary to do so. The polyhedron above is made of rhombus-shaped panels which correspond to the thirty faces of the rhombic triacontahedron. Each of these panels contains nine faces: one square, surrounded by eight triangles. Since (9)(30) = 270, it is therefore possible to see that this polyehdron has 270 faces, without actually going to the trouble to count them, one at a time.
The software I used to make this polyhedron may be found at http://www.software3d.com/Stella.php, and is called Stella 4d. With Stella 4d, a single mouse-click will let you see the dual of a polyhedron. Here’s the dual of the one above.
This polyhedron is unusual, in that it has faces with nine sides (enneagons, or nonagons), as well as fifteen sides (pentadecagons). However, these enneagons and pentadecagons aren’t regular — yet — but they will be in the next post.
For a d2, number a cube’s faces with three ones and three twos.
For a d3, number a cube’s faces 1,1,2,2,3,3.
The standard d4 is a Platonic solid, the tetrahedron. Double-numbered (two ones, two twos, etc.) octahedra are sometimes used as d4s, also.
For a d5, an icosahedron can be renumbered with four each of the numbers one through 5. Double-numbering a pentagonal dipyramid or pentagonal trapezohedron also works.
The d6 is the familiar cube.
For a d7, one option is to roll an octahedron, but re-roll 8 each time it comes up.
For a d9, one option is to roll a d10, but reroll 10s.
For d10s, pentagonal dipyramids and pentagonal trapezohedra both work. There is also the option of double-numbering an icosahedron.
For a d11, one option is to roll a d12, but reroll 12s.
For a d12, the Platonic and rhombic dodecahedra both work.
For a d13, roll a d14, but reroll 14s.
For a d14, one option is to roll a d7 and a d2, then add 7 to the d7 result iff the d2 shows 2. Another is to roll a d15, but reroll 15s.
For a d15, simply double-number the thirty faces of a rhombic triacontahedron.
For a d16, roll a d2 and a d8 together, using the d8 result, but adding 8 to the d8 result iff the d2 result is 2.
For d17, roll a d18, but reroll 18s.
For d18, roll a d2 and a d9 together, using the d9 result, but adding 9 to the d9 result iff the d2 result is 2.
For d19, roll a d20, but reroll 19s.
For d20, icosahedra are used.
For d21, one option is to roll a d24, rerolling any result from 22-24. Another is roll a d7 and a d3 together, using the d7 result, but adding 7 to this d7 result iff the d3 result is 2, but adding 14 to the d7 result iff the d3 result is 3.
For d22, one option is to roll a d24, rerolling any 22s and 24s. Another is to roll a d2 and a d11 together, using the d11 result, but adding 11 to this d11 result iff the d2 result is 2.
For d23, use a d24, and re-roll 24s.
Options for the d24 include the triakis octahedron, the tetrakis cube, the deltoidal icositetrahedron, and the pentagonal icositetrahedron, all of which are Catalan solids (duals of the Archimedeans). Another d24 can be made by rolling a d2 and a d12 together, and using the d12 result, but adding 12 to this d12 result iff the d2 result is 2.
For a d25, roll two distinguishabale d5s, called d5a and d5b. The 1-25 random number is (d5a)+ (5)(d5b-1).
For a d26, roll a d13 and a d2, then add 13 to the d13 result if the d2 shows 2. Another is to roll a d15, but reroll 15s. The first option may require two different d2s, so they will have to be distinguishable, in that case, or one d2 must be rolled twice, each for a different purpose.
For a d27, roll a d9 and d3 together. The result is (d9) + (9)(d3-1).
For a d28, roll a d14 and d2 together, using the d14 result, but adding 14 to it iff the d2 result is 2. Another option: roll a d30, but reroll results of 28 or 29. The first option may require two different d2s, so they will have to be distinguishable, in that case, or one d2 must be rolled twice, each for a different purpose.
For a d29, roll a d30, and reroll 30s.
The most common d30 is a rhombic triacontahedron. After the Platonic solids and the d10, these d30s are the most commonly available example of polyhedral dice.