Wednesday, November 8, 2017

Polyhedra

We make use of in-game puzzles. To prosper in the Athenopolis campaign, familiarity with a fair amount of polyhedra lore is necessary.

Platonic Solids

Polyhedral dice come in the shapes of the five Platonic solids:
• tetrahedron or triangular pyramid
• hexahedron or cube
• octahedron
• dodecahedron
• icosahedron
In the original edition of the game the colors of the dice were yellow, orange, green, blue, and white respectively.

How many vertices does an icosahedron have? You can pull out a die and try to count them, but it is better to observe that there 20 faces, and each has 3 vertices, for a total of 60 (vertex, face) pairs. However, each vertex is shared by 5 faces, so the number of vertices is 60 / 5 = 12.

A similar line of reasoning can be used to show the number of edges is 30.

In the Timaeus (c. 360 BCE), Plato associates the tetrahedron, octahedron, icosahedron, and cube with classical elements fire, air, water, and earth respectively. The dodecahedron is said to be the shape of the universe. Aristotle later associates the dodecahedron with the element aether.

The Platonic solids are the only five convex regular polyhedra. Euclid seems to argue something to this effect in Book XIII, Proposition 18 of The Elements (c. 300 BCE).  One translation of his claim is:
No other figures, besides the five said figures, can be constructed which is contained by equilateral and equiangular figures equal to one another.

Face, Edge, and Vertex Transitive

Imagine we had a case for the model of a polyhedron that fit it snugly. The polyhedron is face-transitive if we can place any face of the polyhedron to any face of the case and it still fits. A consequence of being face transitive is that every face is the same shape and size.

The polyhedron is edge-transitive if we can place any edge of the polyhedron to any inside angle of the case and it still fits. A consequence of being edge transitive is that every edge is the same length.

The polyhedron is vertex-transitive if we can place any vertex of the polyhedron to any corner of the case and it still fits. A consequence of being vertex transitive is that every vertex touches the same number of faces and edges.

Alternative names for face-transitive, edge-transitive, and vertex-transitive are isohedral, isotoxsal, and isogonal, respectively.

A polyhedron which is face, edge, and vertex-transitive is called a regular polyhedron.

Alternating Groups and Symmetry Groups

Let's reserve the letters V, E, and F  for the number of vertices, edges, and faces of a polyhedron, respectively. The vector (V, E, F) is called the face counting vector or f-vector of a polyhedron. For example the f-vector of the cube is (8, 12, 6).

Returning to the model of a polyhedron and its case, each way we can put the polyhedron in its case represents an element of the alternating group of the polyhedron.

Since Platonic solids are vertex transitive there is at least one group element for each of the V vertices. Each vertex is connected to 2 × E / V edges, and if we assign one of the model edges to a case edge the positions of the rest of the vertices and edges is determined. Thus the size of the alternating group is V × (2 × E / V) = E × 2.

The symmetric group allows for reflections as well as rotations. If the mirror image of a polyhedron also fits in its case, its symmetric group is twice the size of its alternating group.

VEF|A||S|
tetrahedron4641224
cube81262448
octohedron61282448
dodecahedron20301260120
icosahedron12302060120

The symmetry groups of the cube and octohedron are the same size. They are in fact isomorphic. The same is true of the dodecahedron and the icosahedron.

Other Polyhedra

Let's show that the tetrahedron with an f-vector of (4, 6, 4) has the smallest possible number of vertices, edges, and faces for a polyhedron.

A polyhedron must have at least 4 vertices since any 3 points are contained in a plane.

We can rule out a polyhedron with 3 faces by noting that an edge must have exactly two faces. For the polyhedron to have a vertex on the edge, the third face must intersect the vertex but not the rest of the edge. It follows  the polyhedron doesn't have any additional vertices, but this contradicts the fact that a polyhedron must have at least 4 vertices.

Each edge belongs to exactly two faces, but each face has three or more edges. The constraint 2E ≥ 3F  follows, and because a polyhedron must have at least 4 faces it must also have at least 6 edges.

For every value of F ≥ 4 we can construct a polyhedron called a pyramid with F faces by starting with a (F – 1)-polygon and connecting each vertex with a point not in the plane of the polygon. The f-vector of the polyhedron is (F, 2F – 2, F). A tetrahedron is triangular pyramid and the Great Pyramid of Giza is a square pyramid. The formula for the f-vector shows that we have also constructed a polyhedron for every V ≥ 4.

A pyramid is right if the line connecting the center of the polygon to the off vertex is perpendicular to the plane of the polygon. Otherwise the pyramid is oblique.

Similarly a prism is a way of constructing a polyhedron for each F ≥ 5 with an f-vector of (2F – 4, 3F – 6, F). The cube is a square prism. Note that a decagon prism has an f-vector of (20, 30, 12) which is the same as the f-vector of a dodecahedron. The example shows that the f-vector of a polyhedron doesn't necessarily tell us the number of edges each face has.

A prism is right if the off-polygon faces are perpendicular to the polygon faces. Otherwise the prism is oblique.

Polyhedral Dice

If we want the die to be fair, then the die should be face transitive. The Platonic solids are face transitive, but pyramids and prisms in general are not. Recall that a triangular pyramid is a tetrahedron and a square prism can be a cube.

There is a type of polyhedron called a bipyramid which yields an infinite family of face transitive polyhedra. The vertices of a regular n-polygon is connected to two vertices off the plane of the polygon on opposite sides. For the polyhedron to be face transitive, the vertices must be equidistant from and perpendicular to the center of the n-polygon. The figure is (n + 2, 3n, 2n) for n ≥ 3. The octohedron is a bipyramid.

The pentagonal bipyramid gives us a way to construct a 10-sided die, but most 10-sided dice are a different polyhedron called the pentagonal trapezohedron. The faces are quadrilaterals and in particular kites. The f-vector is (2n + 2, 4n, 2n) for n ≥ 3. The triangular trapezohedron has a f-vector of (8, 12, 6) which is the same as the cube.

The trapezohedron shows that face transitive does not imply edge transitive or vertex transitive. Bipyramids other than the square bipyramid or octahedron also show this.

The tetrahedron notwithstanding, for a polyhedron to be useful as a die we want each face to be parallel to a face on the opposite side so that there is always a clear "up".  Insisting that a bipyramid or trapezohedron have parallel faces places a constraint on the distance of the off-polygon vertices from the polygon. Update: or so I thought, but in fact the trapezohedron or bipyramid can have parallel faces regardless of the ratio of off-polygon vertex distance to polygon side length. And it should be emphasized that for a bipyramid to have parallel faces the polygon must have an even number of sides. The trapezohedron does not have this limitation.

The Catalan solids are another set of 13 face transitive polyhedra. One of these, the rhombic triacontahedron is used for 30-sided dice.

Euler's Characteristic

For a convex polyhedron, the following formula for the number of vertices, edges, and faces holds:
V – E + F = 2

Polytope

A regular polygon is a two dimensional analog of a Platonic solid.

Polytopes generalize polygons (in two dimensions) and polyhedra (in three dimensions) to n-dimensions.

A 4-polytope has 0-dimensional vertices, 1-dimensional edges, 2-dimensional faces, and 3-dimensional cells. The number of each can be represented by a f-vector with four elements: (V, E, F, C).

An n-polytope has n – 1 dimensional facets, n – 2 dimensional ridges, n – 3 dimensional peaks.

Sub-polytopes of any dimension can be called faces. If necessary to indicate the dimension they are called j-faces.

The extended f-vector of a polytope includes extra values on either end. One value, for dimension –1, represents the empty set. The other value, for dimension n, represents the polytope itself. These values are always 1. The cube has extended f-vector (1, 8, 12, 6, 1).

The formula for Euler's characteristic extends to convex polytopes and motivates our use of extended f-vectors. If fi is the number of faces of dimension t, then

$$\sum_{i=–1}^n (–1)^i f_i = 0$$

Lattices and Flags

When studying a polyhedron such as the cube above we can create a graph called the face lattice of the polyhedron which shows which edges belong to which faces and which vertices belong to which edges.

At the top of the graph we put the entire polyhedron and at the bottom we put the empty set. This makes the graph a mathematical lattice. Every two elements in the lattice have a unique least element which contains both of them and a unique greatest element contained by both of them. The extended f-vector is the number of elements in each row of the lattice from the bottom up.

Two polyhedra are combinatorially equivalent if their face lattices are isomorphic.  The cube and the triangular trapezohedron are examples of combinatorially equivalent polyhedra. The dodecagon prism and the dodecahedron are not combinatorially equivalent even though their f-vectors are the same.

A flag is a sequence of "faces" of an n-polytope of increasing dimension: F- 1, F0, F1, ..., Fn, where each face is contained in the face that follows it in the sequence. The F-1 face is the "empty face" and the Fn face is the entire n-polytype.

A regular polytope is one where the symmetry group of the polytope acts transitively on the flags. This implies that the polytope is face, edge, and vertex transitive.

Ways to Number a Platonic Solid

The number of ways we can put n different numbers on the n faces of a Platonic solid is n! divided by the size of the alternating group of the solid.

 tetrahedron 2 cube 30 octohedron 1680 dodecahedron 7,983,360 icosahedron 40,548,366,802,944,000

Six-sided dice are often numbered with the constraint that opposite sides must add to 7. With this constraint there are only two ways to number a six-sided die. If you hold a six-sided die so that the sides for 1, 2, and 3 are showing, if the numbers increase in a counter-clockwise direction the die is said to be right-handed, which is the way Western dice are numbered.

Net

A polyhedron net is a set of edge-connected polygons, which can be folded into a polyhedron. Polyhedra can be constructed by printing a net on paper, cutting it out, and then folding the edges to create the shape.