Polytopes: Part II

This is the second part of a four part post. For the first part, see here.

The space in which all possible planes reside is the third dimension. In three dimensional space, one can define six directions: up, down, left, right, forward and back. The structures in this space are known as polyhedra (singular: polyhedron). Polyhedra are constructed with polygons, with a certain number of polygons at each vertex. Regular polyhedra have a certain number of identical regular polygons at each vertex. By extending the system from previously, these are denoted {p,q}.

One can construct the polygon {p,q} by having q regular polygons {p} at each vertex. As with polygons, situations with p≤2 are trivial, and q≤2 also has this property. This is due to the fact that any polyhedron {p,1} equals {p}, and any {p,2} is two overlapping polygons {p}, again trivially equaling {p}.

For any {p,q}, the total angular measure around each vertex is q internal angles of the regular polygon {p}. Since the internal angle of a regular polygon with p≥3 is (1-2/p)180º, the total angular measure around each vertex is (1-2/p)q180º. If this value is less than 360º, than the polyhedron curves inward, and eventually intersects itself, forming a finite polyhedron. Therefore, for a finite polyhedron,

(1-2/p)q180<360
(1-2/p)q<2
1-2/p<2/q
1<2/p+2/q
1/p+1/q>1/2

The final formula sets a limit on the number of possible finite regular polyhedra (assuming p,q>2), and only five pairs of {p,q}, where p and q are whole numbers, meet these requirements. These are called the Platonic Solids:

{3,3}
1/3+1/3=2/3>1/2
The above pair {3,3} defines a polyhedron with three triangles at each vertex. The resulting figure has four triangular faces, six lines, four vertices, and is known as the tetrahedron. It is pictured below.


{3,4}
1/3+1/4=7/12>1/2
The pair {3,4} defines another polyhedron with four triangles at each vertex. The resulting figure has eight triangular faces twelve lines, six vertices, and is known as the octahedron. It is pictured below.


{4,3}
1/4+1/3=7/12>1/2
The pair {4,3} defines another polyhedron with three squares at each vertex. The resulting figure has six square faces, twelve lines, eight vertices, and is known as the cube, or hexahedron. It is pictured below.


{3,5}
1/3+1/5=8/15>1/2
The pair {3,5} defines yet another polyhedron with five triangles at each vertex. The resulting figure has twenty triangular faces, thirty lines, twelve vertices, and is known as the icosahedron. It is pictured below.


{5,3}
1/5+1/3=8/15>1/2
The pair {5,3} defines a final polyhedron with three pentagons at each vertex. The resulting figure has twelve pentagonal faces, thirty lines, twenty vertices, and is known as the dodecahedron. It is pictured below.


The above polyhedrons are the only convex regular polyhedrons that are finite. They are finite due to the fact that they are tilings of a sphere, and therefore exist on the elliptic plane in elliptic geometry. Elliptic space is simply space with positive curvature. For more information on the three geometries, see here. In addition, convex regular polyhedra in elliptic 2-space or Euclidean 3-space (depending on point of view because elliptic and hyperbolic planes in n dimensions can only be accurately represented in n+1 dimensions) also have the property that the number of vertices plus the number of faces minus the number of sides always equals two. Also, for the above polyhedra, if the number of vertices and the number of faces are switched, the result is another polyhedron, known as the original polyhedron's dual. The dual of the tetrahedron is the tetrahedron, since both numbers are the same, and the other four come in dual pairs, with the cube's dual being the octahedron and the icosahedron's dual being the dodecahedron. This is equivalent to switching the numbers in {p,q} to {q,p}. As a result, if a polyhedron is finite, its dual will be also.

Another case to consider is when 1/p+1/q=1/2. In this case, the curvature is exactly zero, and the polyhedron tiles the Euclidean plane, and the polyhedron is known as a tiling. Since the flat Euclidean Universe is infinite, so are polyhedra that tile it. There are only three regular tilings of the Euclidean plane:

{3,6}