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User:Tomruen/Generalized pyramid

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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

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Simple pyramids
Square Pyramid
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.

Coordinates

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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

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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

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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

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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

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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

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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

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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

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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

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References

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