User:Tomruen/Generalized pyramid

A simple polygonal pyramid is a 3-polytope constructed the joining of a point with a polygon, ∨ {p}.

A digonal-digonal pyramid or disphenoid is a 3-polytope constructed the joining of a two orthogonal digons, { } ∨ { }.

A digonal-polygonal pyramid is a 4-polytope constructed the joining of a digon with an orthogonal polygon, { } ∨ {p}.

A double-polygonal pyramid is a 5-polytope the joining of 2 orthogonal polygons, {p} ∨ {q}. The regular 5-simplex can be constructed as 3 generalized pyramids: ∨{3,3,3}, {}∨{3,3}, and {3}∨{3}.

A triple-polygonal pyramid is a 7-polytope the joining of 3 orthogonal polygons, {p} ∨ {q} ∨ {r}. The regular 7-simplex can be constructed as 7 generalized triple-pyramid forms: ∨∨{3,3,3,3,3}, ∨{ }∨{3,3,3,3}, ∨{3,3}∨{3,3}, ∨{3}∨{3,3,3}, { }∨{ }∨{3,3,3}, { }∨{3}∨{3,3}, and {3}∨{3}∨{3}.

Simple pyramid
A simple pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with polygonal base. A pyramid with an n-sided base has n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.

Coordinates
The coordinates of a regular polygon p pyramid of height h can be given as:
 * (0,0,h)
 * (r cos(2*&pi;i/p),r sin(2*&pi;i/p),0), i=1..p

Edges are define between pairs of vertices from the first set are connected to the second set.

Geometry
A p-q pyramid can be seen as two regular planar polygons of p and q sides with the same center and orthogonal orientations in 4 dimensions, and offset by a 5th dimension. Along with the p and q edges of the two polygons, all permutations of vertices in one polygon to vertices in the other form edges. All connecting faces are triangles, connecting cells are tetrahedra, and connecting 4-faces are 5-cells.

It has two vertex figures, both 5-cell, with 1 of 10 edges generated from one of the polygons.

Coordinates
The coordinates of a regular polygon p-q pyramid of height h can be given as:
 * (r1cos(2*&pi;i/p),r1sin(2*&pi;i/p),0,0,-h/2), i=1..p
 * (0,0,r2cos(2*&pi;j/q),r2sin(2*&pi;j/q),h/2), j=1..q

Edges are define between pairs of vertices from the first set are connected to the second set.