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

Simple pyramids
Schläfli symbol	( ) ∨ {p}
Coxeter diagram
Faces	p triangles, 1 n-gon
Edges	2p
Vertices	p + 1
Symmetry group	[1,p], order 2p
Dual polyhedron	Self-dual
Properties	convex

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.

The coordinates of a regular polygon p pyramid of height h can be given as:

(0,0,h)

(r cos(2*πi/p),r sin(2*πi/p),0), i=1..p

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

Disphenoid

Digonal disphenoid Schläfli symbol { } ∨ { } Coxeter diagram Faces 4 ( )∨{ } Edges 6 Vertices 4 Symmetry group [1,4], order 4 Dual polyhedron Self-dual Properties convex

Tetragonal disphenoid Schläfli symbol 2.{ } = { } ∨ { } Coxeter diagram Faces 4 ( )∨{ } Edges 6 Vertices 4 Symmetry group [[2]] = [2+,4], order 8 Dual polyhedron Self-dual Properties convex

Digonal-polygonal pyramid

Set of digon-polygonal pyramids
Type	Polychoron
Schläfli symbol	{p} ∨ { }
Coxeter diagram
Cells	p+2p+2: p ( )∨{ } 2p { }∨{ } 2 ( )∨{p}
Faces	2+3p: 1 {p} 4p ( )∨{ }
Edges	2p+p+2
Vertices	p+2
Vertex figures	Irr. tetrahedra
Symmetry	[p,2] = [p]×[ ], order 8p
Dual	Self-dual
Properties	convex

Double polygonal pyramids

Set of polygonal pyramids Type Uniform 5-polytope Schläfli symbol {p} v {q} Coxeter diagram 4-face p+q: p { }∨{q} q { }∨{p} Cells p+pq+q: p ( )∨{q} pq { }∨{ } q {p}∨( ) Faces 2+2pq: 1 {p} 1 {q} 2pq ( )∨{ } Edges pq+p+q Vertices p+q Vertex figures Irr. 5-cells Symmetry [p,2,q] = [p]×[p], order 4pq Dual Self-dual Properties convex

Set of polygonal double-pyramids Type Uniform 5-polytope Schläfli symbol 2.{p} = {p} ∨ {p} Coxeter diagram 4-face 2p { }∨{p} Cells 2p+p2: 2p ( )∨{p} p2 { }∨{ } Faces 2+2p2: 2 {p} 2p2 ( )∨{ } Edges p2+2p Vertices 2p Vertex figures Irr. 5-cells Symmetry [[p,2,p]] =[2p,2+,2p], order 8p2 Dual Self-dual Properties convex

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*πi/p),r₁sin(2*πi/p),0,0,-h/2), i=1..p

(0,0,r₂cos(2*πj/q),r₂sin(2*πj/q),h/2), j=1..q

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

Triple pyramids

Set of polygonal triple-pyramids Type 8-polytope Schläfli symbol {p}∨{q}∨{r} Coxeter diagram 7-face p: { }∨{q}∨{r} q: {p}∨{ }∨{r} r: {p}∨{q}∨{ } 6-face p: ( )∨{q}∨{r} q: {p}∨( )∨{r} r: {p}∨{q}∨( ) qr: {p}∨{ }∨{ } pr: { }∨{q}∨{ } pq: { }∨{ }∨{r} 5-faces 1: {p}∨{q} 1: {p}∨{r} 1: {q}∨{r} 2qr: {p}∨{ }∨( ) 2pr: ( )∨{q}∨{ } 2pq: ( )∨{ }∨{r} pqr: { }∨{ }∨{ } 4-faces 2(p+q): { }∨{r} 2(p+r): { }∨{q} 2(q+r): { }∨{p} 3: {p}∨( )∨( ) 3: ( )∨{q}∨( ) 3: ( )∨( )∨{r} 3pqr: ( )∨{ }∨{ } Cells 2(q+r): ( )∨{p} 2(p+r): ( )∨{q} 2(p+q): ( )∨{r} p+q+r: { }∨{ } 3pqr: ( )∨( )∨{ } Faces 1: {p} 1: {q} 1: {r} 6(p+q+r): ( )∨{ } pqr: ( )∨( )∨( ) Edges p+q+r: { } 3(p+q+r): ( )∨( ) Vertices p+q+r: ( ) Vertex figures Irr. 7-simplexes Symmetry [p,2,q,2,r], order 8pqr Dual Self-dual Properties convex

Set of polygonal triple-pyramids Type 8-polytope Schläfli symbol 3.{p} = {p}∨{p}∨{p} Coxeter diagram 7-face 3p: { }∨{p}∨{p} 6-face 3p: ( )∨{p}∨{p} 3p2: { }∨{ }∨{p} 5-faces 3: 2.{p} 6p2: ( )∨{ }∨{p} p3: { }∨{ }∨{ } 4-faces 12p: { }∨{p} 3p: ( )∨( )∨{p} 9p2: ( )∨{ }∨{ } Cells 36p: ( )∨{p} 3p: { }∨{ } 3p3: ( )∨( )∨{ } Faces 3: {p} 18p: ( )∨{ } p3: ( )∨( )∨( ) Edges 3p { } 9p ( )∨( ) Vertices 3p: ( ) Vertex figures Irr. 7-simplexes Symmetry [3[p]3], order 24p3 Dual Self-dual Properties convex

Examples

Vertex figures for truncated 6-simplexes

( )∨{3,3,3}	{ }∨{3,3}	{3}∨{3}
[1,3,3,3]	[2,3,3]	[3,2,3]
truncated 6-simplex	bitruncated 6-simplex	tritruncated 6-simplex

Vertex figures for truncated 6-cubes, truncated 6-orthoplexes

( )∨{3,3,3}	( )∨{3,3,4}	{ }∨{3,3}	{ }∨{3,4}	{3}∨{4}
[1,3,3,3]	[1,3,3,4]	[2,3,3]	[2,3,4]	[3,2,4]
truncated 6-cube	truncated 6-orthoplex	bitruncated 6-cube	bitruncated 6-orthoplex	tritruncated 6-cube

See also

• Pyramid (geometry)

References

• N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms