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Fair Dice and Dartboards

So in those two pages of big dice collections, I'e found these ones of the four infinite axial families:
  • Bipyramids: 3-8, 14, 16, 18
  • Trapezohedra: 3-7, 9, 11, 13, 17, 25
  • Prisms: 3-20
  • Antiprisms: 2-6, 8, 10
The small index values are well-represented, while the larger ones are scattered.

Prismatic and antiprismatic dice are sometimes called rolling-pin and rolling-log dice. Many of them have rounded or pyramidal endcaps, to keep the dice from landing on them. The regular icosahedron is a 5-antiprism with pyramidal endcaps.

I spent some time searching for "rpg dice" and "d&d dice" and "crystal dice" in shapefile archives for 3D printers. I found some Braille dice and lots of dice boxes, including some icosahedral ones. DOTS RPG Project has Braille-dotted versions of a 4-prism, a cube, an octahedron, a 5-trapezohedron, a dodecahedron, and an icosahedron: d4, d6, d8, d10, d12, d20, and Screw Top D20 Dice Box by StormCrow13 - Thingiverse

A common variation of 5-trapezohedron dice is to have two of them for do, one as the ones digit: 0, 1, 2, ..., 9, and one as the tens digit: 0, 10, 20, ..., 90.

With them added in, I have
  • Bipyramids: 3-8, 10, 14, 16, 18
  • Trapezohedra: 3-7, 9, 11, 13, 17, 25
  • Prisms: 3-20
  • Antiprisms: 2-10
It seems to me that one may be able to write a program that generates shapefiles for 3D-printed dice.
 
3D printing is not like to Create a fair object. The center of gravity will not be precisely placed at the geometric center. The material is typically plastic.

On a past thread I looked at commercial gambling dice. The depth of the dimples indicating numbers on a side side is adjusted so equal mass is removed from each side.
 
Standard cubical dice have some interesting arrangements of dots on their faces. Here is a complete list of dot arrangements with additional highly-symmetric ones:
Code: 
. . .   . . .   . . O   . . O   O . O   O . O   O O O   O O O   O O O   O O O
. . .   . O .   . . .   . O .   . . .   . O .   . . .   . O .   O . O   O O O
. . .   . . .   O . .   O . .   O . O   O . O   O O O   O O O   O O O   O O O

  0       1       2       3       4       5       6       7       8       9
  D4      D4      D2d     D2d     D4      D4      D2x     D2x     D2      D2
Their symmetries are the symmetry group of the square, D4, and two subgroups, which I call D2x and D2d.
  • D4: rotations: 0d, 90d, 180d, 270d, reflections: both axes, both diagonals
  • D4x: rotations: 0d, 180d, reflections: both axes
  • D4d: rotations: 0d, 180d, reflections: both diagonals
Cubical dice usually have numbers on opposite sides adding up to 7, but there are nevertheless 24 = 16 possible arrangements of their faces:
  • 2: mirror image of the die
  • 2: rotation of (2) by 90d or axial reflection
  • 2: rotation of (3) by 90d or axial reflection
  • 2: rotation of (6) by 90d or diagonal reflection
 
On the subject of symmetries, standard playing cards have these symmetries:
Num
D
SHC
A
D2
D1v
2
D2
D2
3
D2
D1v
4
D2
D2
5
D2
D1v
6
D2
D1v
7
D1v
D1v
8
D2
D1v
9
D2
D1v
10
D2
D2
D = diamonds, SHC = spades, hearts, clubs

All face cards have symmetry C2
  • C2: Rotations: 0d, 180d
  • D1v: Rotation: 0d, Reflection: around vertical line, in horizontal direction
  • D2: Rotations: 0d, 180d, Reflections: both axes of line
 
I will now describe the symmetry groups that I'd introduced. They are two-dimensional rotation-reflection or point groups, and there are two infinite families of them. What I call them:
  • C(n), n-fold rotation
  • D(n), C(n) with reflections
C(n) is the maximal pure-rotation subgroup of D(n).

Turning to three-dimensional ones, there are seven infinite families of them, and seven additional ones. The seven infinite families are wraparound versions of the seven frieze groups, and I call them axial groups. The seven additional ones I call quasi-spherical groups. Using Schoenflies notation, the axial ones are, with axis vertical:
  • C(n): n-fold axial rotation
  • C(n,h): C(n) with horizontal-plane reflections
  • S(2n): C(n) with alternating horizontal-plane reflections -- alternating between horizontally reflected and not
  • C(n,v): C(n) with vertical-plane reflections
  • D(n): C(n) with 180-degree rotations with axes in the horizontal plane
  • D(n,h): D(n) with horizontal-plane reflections
  • D(n,d): D(2n) with alternating horizontal-plane reflections
S(2n) might more also be called C(n,s) or C(n,d).

The quasi-spherical ones are:
  • T: tetrahedron rotations
  • Th: T with inversions
  • Td: T with tetrahedron reflections
  • O: octahedron rotations
  • Oh: O with inversions / reflections
  • I: icosahedron rotations
  • Ih: I with inversions / reflections
The rotation-reflection groups with their maximal rotation subgroups:
  • C(n,h), S(2n), C(n,v): C(n)
  • D(n,h), D(n,d): D(n)
  • Th, Td: T
  • Oh: O
  • Ih: I
 
I thought of listing which of these polyhedra have which symmetry group, but I decided on something that may be more interesting: the operations in the symmetry groups.


I'll start with the tetrahedral group, T. A regular tetrahedron is symmetric over these rotations:
  • No rotation (identity)
  • Axis between opposite-side edges, angle 180d -- 3 of them
  • Axis between opposite-side vertex and face, angle 120d -- 8 of them
The second one is divided between clockwise at vertex and counterclockwise at vertex, splitting those 8 rotations into 2 sets of 4 rotations. Adding them up gives 1 + 3 + 4 + 4 = 12.

Adding reflections gives the group Td. The reflections may be combined with rotations. Here they are:
  • Along the line between two neighboring vertices -- 6 of them
  • Along the line between opposite-side edges, followed by a 90d rotation around that line -- 6 of them
This gives 6 + 6 = 12 reflections, equal to the number of rotations, for a total of 24.

The rotations and reflections make all possible permutations of a tetrahedron's vertices. The rotations make even ones and reflections make odd ones. Even and odd refers to the number of interchanges that the permutations contain.


There is another extension of T, Th, with inversions: move every point to the opposite side of the center. It is not a symmetry of the tetrahedron, though it is of the volleyball pattern.


The next one up is octahedral symmetry. Its rotation symmetry is O, and it contains these rotations of an octahedron. For a cube, interchange vertices and faces.
  • No rotation (identity)
  • Axis between opposite-side edges, angle 180d -- 6 of them
  • Axis between opposite-side faces, angle 120d -- 8 of them
  • Axis between opposite-side vertices, angles 90d and 180d -- 6 and 3 of each
Adding them up gives 1 + 6 + 8 + 6 + 3 = 24 of them

Adding inversions gives reflections, optionally with rotations, giving group Oh.
  • Pure inversion
  • Reflection along line between opposite-side edges -- 6 of them
  • Axis between opposite-side faces, angle 60d -- 8 of them
  • Reflection along line between opposite-side vertices, optionally with 90d rotation -- 3 and 6 of each
Adding them up gives 24 of them, for a grand total of 48.


Finally, icosahedral symmetry. Its rotation symmetry is I, and it contains these rotations of an icosahedron. For a dodecahedron, interchange vertices and faces.
  • No rotation (identity)
  • Axes between opposite-side edges, angle 180d -- 15 of them
  • Axes between opposite-side faces, angle 120d -- 20 of them
  • Axes between opposite-side vertices, angles 72d and 144d -- 12 each of them, making 24
Adding them up gives 1 + 15 + 20 + 24 = 60 of them

Adding inversions gives reflections, optionally with rotations, giving group Ih.
  • Pure inversion
  • Reflection along line between opposite-side edges -- 15 of them
  • Axis between opposite-side faces, angle 60d -- 20 of them
  • Reflection along line between opposite-side vertices, with rotation angles 36d and 108d -- 12 each of them, making 24
Adding them up gives 60 of them, for a grand total of 120.

That's impressive symmetry.


Turning to the polyhedra that I've described, most of them have the maximum possible symmetries for their configurations.
  • n-bipyramids, n-prisms: D(n,h)
  • n-trapezohedra, n-antiprisms: D(n,d)
  • Tetrahedron and related: Td
  • Octahedron, cube, and related: Oh
  • Icosahedron, dodecahedron, and related: Ih
These symmetries are also shared with the relatives of the Platonic solids, relatives with pyramid faces, rhombus faces, kite faces, and various truncations. The exceptions among the relatives are the pentagon-face ones and the snub ones. These are not reflection symmetric, and thus have only pure rotation symmetry: O and I.

The tetrahedral counterparts of those last ones are deformed regular dodecahedra and icosahedra, and there are various other deformed semiregular shapes with partial symmetry.
 
I must add that the flat-plane tilings also have symmetries, and theirs are among the 17 wallpaper groups.

The triangular and hexagonal regular tilings and most of their derived ones have symmetry p6m, while their pentagon-face and snub ones have symmetry p6 (no reflections).

The square regular tiling and most of its derived ones have symmetry p4m, while its pentagon-face and snub ones have symmetry p4 (no reflections).

The picket-fence tiling and its dual have symmetry p2mg.

If one wants a quick takeaway, it is that regular and semiregular polyhedra and tilings are highly symmetrical shapes.


I have been unable to find any simple way of generating the elements of the symmetry groups of hyperbolic tilings. Nothing like what I can find for the polyhedral and flat-plane cases.
 
The remaining semiregularity possibility is isotoxal or edge-transitive.

This feature constrains the vertices and faces to at most two types each, because there is one vertex on each end of an edge, and one face on each side.

Calculating the Euler characteristic X for each edge gives Xe = X/E. It has the equality and inequalities that one would expect for each type of surface curvature: >0 for polyhedron (spherical tiling), =0 for flat-plane tiling, and <0 for hyperbolic-plane tiling. It is:

Xe = 1/r1 + 1/r2 + 1/n1 + 1/n2 - 1

where each of an edge's two vertices has ranks r1 and r2, and each face has number of sides n1 and n2. If there is only one type of vertex, then r1 = r2 = r, its rank. If there is only one type of face, then n1 = n2 = n, its number of sides.

We first consider only one type each. This gives us the regular polyhedra and tilings:

Xe = 2*(1/r + 1/n - 1/2)

Then two types of vertices and one type of face (isohedral or face-transitive). This means that n must be even, since the vertex types alternate around a face. This gives us

Xe = 1/r1 + 1/r2 + 2/n - 1

The lowest possible value of n is 4, and it gives

Xe = 1/r1 + 1/r2 - 1/2

This is the regular-polyhedron expression again, and it gives rhombus-face shapes derived from regular-polygon ones. Each rhombus's vertices are

(vertex) - (face center) - (vertex) - (face center)

So each rhombus-face shape is derived from both a regular-polygon one and its dual.

Its dual (isogonal or vertex-transitive) is a cross between a regular-polygon shape and its dual, like the cuboctahedron, a cross between a cube and an octahedron.

The next possible value of n is 6, and it gives

Xe = 1/r1 + 1/r2 - 2/3

The only non-hyperbolic shape has r1 = r2 = 3, the hexagonal tiling. Its dual is the triangular tiling.

All higher values of n give hyperbolic tilings.

Finally, two types of both vertices and faces. This makes r1, r2, n1, and n2 all even, and the only non-hyperbolic one has r1 = r2 = n1 = n2 = 4, the square tiling.
 
There is a curious consequence of my investigation of regular and semiregular polyhedra and tilings. It is that the hyperbolic plane has a rich structure of semiregular tilings that is only hinted at in the polyhedral and flat-plane cases. All the regular and semiregular tilings of those cases can easily be extended from those cases into the hyperbolic case, and there seems to be an infinite number of families of additional semiregular tilings.

Mitchell Porter mentions some additional hyperbolic tilings in Semi-Regular Tilings of the Plane Part 1: Introduction and Historical Background and some additional pages.

I will specialize to face-transitive (isohedral) semiregular polyhedra here; vertex-transitive (isogonal) ones are their duals.

In his page on General Theorems, he mentions his Parity Lemma. That has two parts:

1. Consider vertices with types a, b, c. If (a,b) only occurs in (a,b,c) and (b,c) likewise, and if b has odd rank, then types a and c must be the same.

2. If a vertex type b only occurs in (a,b,b,c) for types a and c, and if b's rank is odd, then either types a = b or c = b or both.

So a regular shape with vertex ranks (y of x) has these related ones:
(x of y) -- its dual regular one
(x, 2y, 2y) and (2x, 2x, y) -- pyramids
(4, 2x, 2y) -- split pyramids
(x, y, x, y) -- rhombi
(4, x, 4, y) -- kites
(3, 3, x, 3, q) -- pentagons
(3, x, 3, x, 3, y/2) for y even and >= 6 -- hexagons

MP also mentions these ones in his page on Hyperbolic Results, which I rewrite in my notation:

Semiregular tiling with vertex ranks (r1, r2, ..., r(n)): Xf = 1/r1 + 1/r2 + ... + 1/r(n) - n/2 + 1

An "alternating tiling" with ranks (r1, r2, r1, r2, .... r1, r2) -- the only non-hyperbolic ones are the cube (rhombi on tetrahedron), the rhombic dodecahedron (rhombi on octahedron), the square tiling, and the hexagonal tiling.

For x even, m >= 2, and y, there is one with (m of x, y) -- the only non-hyperbolic ones are the bipyramids, kites on octahedron, and the square tiling.

For x even, m, y, and z, there is one with (m of x, y, m of x, z) -- the only non-hyperbolic ones are kites on tetrahedron and octahedron and icosahedron and triangular tiling, and the square tiling. Kites on tetrahedron = rhombic dodecahedron = rhombi on oxtahedron.

MP has a lot of nice references, but some of them may be hard to find outside some well-stocked research-university library.
 
Returning to dice, I've thought of a hack that someone else might already have thought of. To use a die as a virtual die with a smaller number of faces. Doing that can get an odd number of virtual faces, something otherwise only possible for prismatic dice or else by rerolling for certain numbers. A d6 can act like a d5 by rerolling whenever one gets a 6.

To use a d(n) die as a d(m) die, one divides each roll's value by m and takes the remainder, thus doing modulo-m arithmetic. This will produce evenly distributed values only if m evenly divides n. Thus, a d6 die can act like a d2 by using evenness vs. oddness of the number, or like a d3 by dividing by 3.

Turning to axial dice, n-prisms are d(n), n-antiprisms, n-bipyramids, and n-trapezohedra are d(2n), and the latter three can be used as virtual d(n). Looking at the dice that have been made, one can do odd numbers 3 - 19 with prisms, and 3 - 13, 17, 25 with the other three kinds, giving 3 - 19, 25.


The Platonic solids have numbers of faces 4, 6, 8, 12, 20, with divisors 1, 2, 3, 4, 5, 6, 8, 10, 12, 20.

The Catalan solids have numbers of faces 12, 24, 30, 48, 60, 120, with divisors 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 120 -- including all the Platonic-solid divisors.

The Platonic solids can do odd numbers 3, 5, and the Catalan ones 3, 5, 15.
 
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