User:Sycamore916/Draft:Regular polyhedron

Convex regular polyhedra
There are five convex regular polyhedra 3D Euclidean space. They are known collectively as the Platonic solids.

The existence and completeness of the Platonic solids has been known since at least the Ancient Greeks. Some sources attribute their discovery to Pythagoreas, however Theaetetus gives the first known description of all five.

Regular star polyhedra
The Kepler-Poinsot polyhedra are a set of 4 non-convex, non-skew, finite polyhedra.

The can all be obtained as stellations of either the dodecahedron or the icosahedron.

Regular tilings
While classical definitions require polytopes to be finite, other definitions may relax this, including regular tilings of the Euclidean plane as regular polyhedra. The regular tilings of the plane can also be considered regular polyhedra; three exist in Euclidean space.

In Euclidean space the internal angles of the faces around a vertex must add up to 360 degrees. This puts a firm restriction on the number of possible non-dense regular tilings.

Hyperbolic tilings
In hyperbolic space the internal angles must exceed 360 degrees. This leads to an infinite number of regular tilings of hyperbolic space. There is a tiling for every pair of convex regular polygons (including the apeirogon), with any pair that does not produce a tiling of spherical or Euclidean tiling producing a hyperbolic tiling.

There are only two infinite sets of star tilings of hyperbolic space: those of the form and their duals, where $p$ is odd (e.g. the stellated heptagonal tiling ).

Petrials
Regular polyhedra can also have skew faces. The Petrie dual or Petrial of a polytope can take any regular polyhedron and transform it into one sharing edges and vertices with the original, but with skew faces. Because of this, there is a Petrie dual to every previous regular polyhedron.

There are multiple extensions to Schläfli symbols which allow the Petrials to be represented. Where the Petrie dual operation can be represented with π, and {p,q}r is defined as a regular map, or equivalently a polyhedron with q p-gons around a vertex, and an r-gonal Petrie polygon. The Petrie dual of a Petrial polyhedron gives the original polytope again.


 * {3,3}π, {4,3}3 - Petrial tetrahedron
 * {4,3}π, {6,3}4 - Petrial cube
 * {3,4}π, {6,4}3 - Petrial octahedron
 * {5,3}π, {10,3}5 - Petrial dodecahedron
 * {3,5}π, {10,5}3 - Petrial icosahedron
 * {5,5/2}π, {6,5/2}5 - Petrial great dodecahedron
 * {5/2,5}π, {6,5}5/2 - Petrial small stellated dodecahedron
 * {3,5/2}π, {10/3,5/2}3 - Petrial great icosahedron
 * {5/2,3}π, {10/3,3}5/2 - Petrial great stellated dodecahedron
 * {4,4}π, {∞,4}4 - Petrial square tiling
 * {3,6}π, {∞,6}3 - Petrial triangular tiling
 * {6,3}π, {∞,3}6 - Petrial hexagonal tiling

Blends
We can also create new regular skew polyhedra by taking the 6 2-dimensional polyhedra (the tilings of the Euclidean plane and their Petrials) and blending them with a 1-dimensional polygon, either a digon or an apeirogon.

Regular skew apeirohedra in 3D hyperbolic space
There are 31 regular skew apeirohedra without self-intersections in 3D hyperbolic space:


 * 14 of these skew polyhedra are compact: {8,10|3}, {10,8|3}, {10,4|3}, {4,10|3}, {6,4|5}, {4,6|5}, {10,6|3}, {6,10|3}, {8,8|3}, {6,6|4}, {10,10|3},{6,6|5}, {8,6|3}, {6,8|3}
 * The other 17 are paracompact: {12,10|3}, {10,12|3}, {12,4|3}, {4,12|3}, {6,4|6}, {4,6|6}, {8,4|4}, {4,8|4}, {12,6|3}, {6,12|3}, {12,12|3}, {6,6|6}, {8,6|4}, {6,8|4}, {12,8|3}, {8,12|3}, and {8,8|4}

There may be many noncompact regular skew apeirohedra, as a list has not yet been enumerated. The total number of regular skew polyhedra in 3D hyperbolic space is currently unknown.

Higher dimensional skews
Just as we can make skew polygons whose vertices lie in 3D space, polyhedra can be made with points lying in 4D space.

Duocombs
Within four dimensions we can take the comb product of any regular polygon with itself to produce a regular 4D skew polyhedron:
 * {4,4|3} - Triangular duocomb
 * {4,4|4} - Square duocomb
 * {4,4|5} - Pentagonal duocomb
 * {4,4|5/2} - Pentagrammic duocomb
 * etc.

Non-prismatic pure polyhedra
Some non-prismatic pure regular skew 4D polyhedra include:
 * {4,6|3}
 * {6,4|3}
 * {4,8|3}
 * {8,4|3}
 * {4,8/3|3}
 * {8/3,4|3}
 * {8,8/3|3}
 * {8/3,8|3}