A tetragonal disphenoid is a join of two orthogonal equal segments, { }∨{ }, offset by a third orthogonal direction.
A pyramid is limiting case of a generalized disphenoid or di-wedge, a join of a polytope to a point, like a square pyramid, {4}∨( ), shown here in side a unit cube. It has 1 square base, {4}, and 4 lateral isosceles triangle faces, { }∨( ).

In geometry, a disphenoid (from Greek sphenoeides 'wedgelike') is a tetrahedron whose four faces are congruent acute-angled triangles.[1] It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths.

A general disphenoid or di-wedge can be represented a join A∨B, where A and B are polytopes. rank(A∨B)=rank(A)+rank(B)+1.

A general trisphenoid or tri-wedge can be represented a join A∨B∨C, where A, B, and C are polytopes. rank(A∨B∨C)=rank(A)+rank(B)+rank(C)+2.

A general tetrasphenoid or tetra-wedge joins four polytopes, A∨B∨C∨D. rank(A∨B∨C∨D)=rank(A)+rank(B)+rank(C)+rank(D)+3. Each join operator adds one dimension.

A multi-wedge can be any of them, while a 3D geometric wedge is geometrically topologically different, more representing a quadrilateral and parallel segment offset by an orthogonal dimension.

A limiting case of a disphenoid is a pyramid, joining an n-polytope to a point (a 0-polytope), A∨( ). rank(A∨( ))=rank(A)+1. The join of a sequence of (n+1) joined points, ∨( )∨( )∨...∨( ) makes an n-simplex. For this reason, A join with a point can also be called a pyramid product.[2]

This article mostly offers examples with regular polytopes, while lower symmetry polytopes work identically. It also looks at equilateral multi-wedges which includes some uniform polytopes and johnson solids.

Properties

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The join operator is:

  • Identity element: nullitope: A∨∅ = A
  • Commutative : A∨B = B∨A
  • Associative : (with both join and sums)
    • A∨B∨C = (A∨B)∨C = A∨(B∨C)
    • A∨B+C = (A∨B)+C = A∨(B+C)
  • Supports De Morgan's law with duality: *(A∨B) = (*A)∨(*B)
  • rank(A∨B)=rank(A)+rank(B)+1
  • Vertex figures:
    • verf(A∨A) = verf(A)∨A
    • verf(A∨B) = verf(A)∨B, A∨verf(B)
    • verf(A∨A∨A) = verf(A)∨A∨A
    • verf(A∨B∨C) = verf(A) ∨B∨C, A∨ verf(B) ∨C, A∨B∨ verf(C)

The join A∨B will be:

  • Convex, if A and B are convex.
  • self-dual, if A and B are self-dual, or if A and B are duals.
  • A simplex, if A and B are simplexes.

When looking at vertices and edges alone as a graph, the join A∨B is the union of graphs A and B, and their connecting complete bipartite graph. It has vA+vB vertices, and eA+eA+vA×vB edges.

Multi-wedges have the vertices of all of the element polytopes. Their edges can be seen as the union of the edges of the element polytopes, and all connections of vertices between elements, as defining in a complete multipartite graph. Higher k-faces exist for all element permutations from nullitope to full polytopes joins.

Extended f-vectors

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Pascal's triangle as simplex f-vectors:

( ) (point)
{ } (segment)
{3} (triangle)
{3,3} (tetrahedron)
{3,3,3} (5-cell)
{3,3,3,3} (5-simplex)
{3,3,3,3,3} (6-simplex)
{3,3,3,3,3,3} (7-simplex)

The f-vector counts the number of k-faces in a polytope, 0..n-1. Extended f-vectors can include end elements -1 and n, both 1. f-1=1, a nullitope, and fn=1, the body.

f0 is the number of vertices, f1 the number of edges, etc. Regular polygons, f({p})=(1,p,p,1).

If you join only points, f-vectors sum in simplexes as Pascal's triangle as binomial coefficients. A nullitope has f-vector (1). A point, ( ), has f( )=(1,1). Segment, f({ })=(1,2,1). A triangle has f({3})=(1,3,3,1). A tetrahedron has f({3,3})=(1,4,6,4,1).

A self-dual polytope will have f-vectors are forward-reverse symmetric.

k-faces of A∨B are generated by joins of all i-faces of A, and all (k-i)-faces of B. With i=-1 to k.

  • The number of vertices are the sum of the vertices of each.
  • New edges are edges of A, edges of B, and new edges between vertices of A and vertices of B.
  • New faces are generated by all faces of A, all faces of B, and new faces from edges of A to every vertex of B, and edges of B to each vertex of A
  • Etc

f-vector products

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There are four classes of product operators, working directly on f-vectors. The join include both f-1, and fn. The rhombic sum only includes f-1, and its dual rectangular product only includes fn. The meet includes neither and only applies to flat elements.

For instance a triangle has f-vector (3,3), with 3 vertices (f0) and 3 edges (f1). Extended with the nullitope (f-1) gives f=(1,3,3), extending with the (polygonal interior) body (f2) gives f=(3,3,1), while extending both is f=(1,3,3,1).

The rhombic sum and rectangular product are dual operators, with f-vectors reversed. The join and rhombic sums shares vertex counts, summing vertices in elements. The rectangular and meet products also share vertex counts, being the product of the element vertex counts.

The meet product is the same as Cartesian product if elements are infinite. Meets are not connected unless polytopes are polygons or higher. For finite elements, like {n} with f=(n,n), or toroidal polyhedra {4,4}b,c, {3,6}b,c,{6,3}b,c with f=(n,2n,n), (n,3n,2n), and (2n,3n,n) respectively.

Four product operators on polytopes
Operator
names
Symbols f-vectors Rank Polytope
names
Join[3]
Join product[4]
Pyramid product[5]
A ∨ B
A ⋈ B
A ×1,1 B
(1,fA,1) * (1,fB,1) Rank(A)+Rank(B)+1 A ∨ ( ) = pyramid
A ∨ { } = wedge
A ∨ B = di-wedge
A ∨ B ∨ C = tri-wedge
Sum
"Rhombic sum"[3]
Direct sum[4]
Tegum product[5]
A + B
A ⊕ B
A ×1,0 B
(1,fA) * (1,fB) Rank(A)+Rank(B) A + { } = fusil or bipyramid
A + B = di-fusil or duopyramid or double pyramid
A + B + C = tri-fusil
Product
Rectangular product[3]
Cartesian product[4]
Prism product[5]
A × B
A ×0,1 B
(fA,1) * (fB,1) Rank(A)+Rank(B) A × { } = prism
A × B = duoprism or double prism
A × B × C = tri-prism or triple prism
Meet
Topological product[4]
Honeycomb product[5]
A ∧ B
A □ B
A ×0,0 B
fA * fB Rank(A)+Rank(B)-1 A ∧ { } = meet
A ∧ B = di-meet or double meet
A ∧ B ∧ C = tri-meet or triple meet

A product A*B, with f-vectors fA and fB, fA∨B=fA*fB is computed like a polynomial multiplication polynomial coefficients.

For example for join of a triangle and dion, {3} ∨ { }:

fA(x) = (1,3,3,1) = 1 + 3x + 3x2 + x3 (triangle)
fB(x) = (1,2,1) = 1 + 2x + x2 (dion)
fA∨B(x) = fA(x) * fB(x)
= (1 + 3x + 3x2 + x3) * (1 + 2x + x2)
= 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= (1,5,10,10,5,1) (triangle ∨ dion = 5-cell)

For a join, explicitly:

k-face counts: f(A∨B)k = f(A)-1*f(B)k + f(A)0*f(B)k+ f(A)1*f(B)k-1 + ... + f(A)k*f(B)-1.
k-face sets: (A∨B)k = {(A-1∨Bk), ∀(A0∨Bk), ∀(A1∨Bk-1), ..., ∀(Ak∨B-1)}, where Ai=set of i-faces in A, etc.

Factorization

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We can factorize extended f-vectors or polynomials of any polytope. This factorization can represent a multi-wedge, if the elements are all valid polytopes.

For example, if we factorize fZ=fA*fB*fC, and fA,fB,fC represent valid polytope f-vectors, then Z=A∨B∨C.

A factorized f-vector can fail to represent valid element polytopes. For example a cubic pyramid, f=(1,9,20,18,7,1), can be decomposed into (1,8,12,6,1)*(1,1), as a join of a cube and a point, while a full factorization (1,7,5,1)*(1,1)2 has an invalid polygon element, f=(1,7,5,1). On the other hand, the f-vector is not unique, like an elongated triangular pyramid has f=(1,7,12,7,1)=(1,6,6,1)*(1,1), shared with a hexagonal pyramid, {6}∨( ), so face types also matter.

All convex polyhedra have f-vectors can be factored by (1,1), but don't represent a real pyramids.

Examples
Rank Name f-vector Factorized Joins
-1 Nullitope f=(1) None ∅∨∅ = ∅
0 Point f=(1,1) (1,1) ( )∨∅ = ∅∨( ) = ( )
1 Segment f=(1,2,1) (1,1)2 2⋅( ) = ( )∨( ) = { }
2 Triangle f=(1,3,3,1) (1,1)3 3⋅( ) = ( )∨( )∨( ) = {3}
3 Tetrahedron f=(1,4,6,4,1) (1,1)4 4⋅( ) = ( )∨( )∨( )∨( ) = {3,3}
3 Triangular pyramid (1,3,3,1)*(1,1) {3}∨( ) = {3,3}
3 Digonal disphenoid (1,2,1)2 2⋅{ } = { }∨{ }
4 5-cell f=(1,5,10,10,5,1) (1,1)5 5⋅( ) = ( )∨( )∨( )∨( )∨( ) = {3,3,3}
4 Tetrahedral pyramid f=(1,5,10,10,5,1) (1,4,6,4,1)*(1,1) {3,3}∨( ) = {3,3,3}
5 5-simplex f=(1,6,15,20,15,6,1) (1,1)6 6⋅( ) = ( )∨( )∨( )∨( )∨( )∨( ) = {3,3,3,3}
5 5-cell pyramid f=(1,6,15,20,15,6,1) (1,5,10,10,5,1)*(1,1) {3,3,3}∨( ) = {3,3,3,3}
5 Digonal trisphenoid f=(1,6,15,20,15,6,1) (1,2,1)3 3⋅{ } = { }∨{ }∨{ } = {3,3,3,3}
6 6-simplex f=(1,7,21,35,35,21,7,1) (1,1)7 7⋅( ) = ( )∨( )∨( )∨( )∨( )∨( )∨( ) = {3,3,3,3,3}
7 7-simplex f=(1,8,28,56,70,56,28,8,1) (1,1)8 8⋅( ) = ( )∨( )∨( )∨( )∨( )∨( )∨( )∨( ) = {3,3,3,3,3,3}
7 Digonal tetrasphenoid f=(1,8,28,56,70,56,28,8,1) (1,2,1)4 4⋅{ } = { }∨{ }∨{ }∨{ } = {3,3,3,3,3,3}
4 Cubic pyramid f=(1,9,20,18,7,1) (1,8,12,6,1)*(1,1)
=(1,7,5,1)*(1,1)2
{4,3}∨( )
 
4 Octahedral pyramid f=(1,7,18,20,9,1) (1,6,12,8,1)*(1,1)
=(1,5,7,1)*(1,1)2
{3,4}∨( )
 

11 Johnson solids have f-vectors matching pyramids, while only the first two are real. This demonstrates f-vectors are insufficient from identifying joins. Toroidal polyhedra don't factorized at all.

# Johnson solid V E F Matched pyramid f-vectors
J1 Square pyramid 5 8 5 Square pyramid, {4}∨( )
J2 Pentagonal pyramid 6 10 6 Pentagonal pyramid, {5}∨( )
J7 Elongated triangular pyramid 7 12 7 hexagonal pyramid, {6}∨( )
J26 Gyrobifastigium 8 14 8 Heptagonal pyramid, {7}∨( )
J8 Elongated square pyramid 9 16 9 Octagonal pyramid, {8}∨( )
J64 Augmented tridiminished icosahedron 10 18 10 Enneagonal pyramid, {9}∨( )
J9 Elongated pentagonal pyramid 11 20 11 Decagonal pyramid, {10}∨( )
J55 Parabiaugmented hexagonal prism 14 26 14 13-gonal pyramid, {13}∨( )
J56 Metabiaugmented hexagonal prism 14 26 14 13-gonal pyramid, {13}∨( )
J91 Bilunabirotunda 14 26 14 13-gonal pyramid, {13}∨( )

Polytope-simplex di-wedges

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f-vector series for joins with simplices:
2D: {4}∨∅ (square)
3D: {4}∨( ) (square pyramid)
4D: {4}∨{ } (square-segment_di-wedge)
5D: {4}∨{3} (square-triangle_di-wedge)
6D: {4}∨{3,3} (square-tetrahedron di-wedge)
7D: {4}∨{3,3,3} (square-5-cell_di-wedge)
f-vector series for joins with simplices:
3D: {4,3}∨∅ (cube)
4D: {4,3}∨( ) (cubic pyramid)
5D: {4,3}∨{ } (cube-segment di-wedge)
6D: {4,3}∨{3} (cube-triangle di-wedge)
7D: {4,3}∨{3,3} (cube-tetrahedron di-wedge)
8D: {4,3}∨{3,3,3} (cube-5-cell di-wedge)

Wedges of the form A∨( )∨( )∨...∨( ) = A∨n+1⋅( ) = A∨{3n-1}, as a join by a n-simplex.

We can represent as f-vectors as f(A∨n+1⋅( ))=f(A)*(1,1)n+1 .

This family of wedges has a special property like Pascal's triangle, where each new row has f-vector as neighboring sums of previous row f-vector, starting with A. A∨{ } will have f-vectors of sums, but 2 levels down, and A∨{3} is expressed as sums 3 levels down, A∨{3,3} sums 4 levels down, etc.

These polytopes are self-dual if A is self-dual, i.e. f-vectors are forward-reverse symmetric.

Multi-wedges with points have special names by Jonathan Bowers:[6] The names come from BSA names of simplices: 2D (scal), 3D:tet, 4D:pen, 5D:hix, 6D:hop, 7D:oca, 8D:ene, 9D: day, 10D: ux, with suffix -ene.[7]

Join Name Dim Examples
A∨( ) A-ic pyramid 3D {4}∨( ) is a square pyramid {3}∨( ) is a triangular pyramid, same as tetrahedron.
A∨( )∨( ) = A∨{ } A-ic scalene 4D {4}∨{ } is a square scalene {3}∨{ } is a triangular scalene, same as 5-cell.
A∨( )∨( )∨( ) = A∨{3} A-ic tettene 5D {4}∨{3} is a square tettene {3}∨{3} is a triangular tettene same as 5-simplex.
A∨( )∨( )∨( )∨( ) = A∨{3,3} A-ic pennene 6D {4}∨{3,3} is a square pennene {3}∨{3,3} is a triangular pennene (or tetrahedral tettene), a 6-simplex.
A∨( )∨( )∨( )∨( )∨( ) = A∨{3,3,3} A-ic hixene 7D {4}∨{3,3,3} is a square hixene {3}∨{3,3,3} is a triangular hixene (or 5-cell tettene), a 7-simplex.
A∨{3,3,3,3} A-ic hoppene 8D {4}∨{3,3,3,3} is a square hoppene {3}∨{3,3,3,3} is a triangular hoppene (or 5-simplex tettene), a 8-simplex.
A∨{3,3,3,3,3} A-ic ocaene 9D {4}∨{3,3,3,3,3} is a square ocaene {3}∨{3,3,3,3,3} is a triangular ocaene (or 6-simplex tettene), a 9-simplex.
A∨{3,3,3,3,3,3} A-ic eneene 10D {4}∨{3,3,3,3,3,3} is a square eneene {3}∨{3,3,3,3,3,3} is a triangular eneene (or 7-simplex tettene), a 10-simplex.
A∨{3,3,3,3,3,3,3} A-ic dayene 11D
A∨{3,3,3,3,3,3,3,3} A-ic uxene 12D

Multi-wedge altitudes

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A tri-wedge A∨B∨C has 6 altitudes: A∨B, A∨C, B∨C, (A∨B)∨C, (A∨C)∨B, (B∨C)∨A.

Joining three or more polytopes allows multiple orthogonal altitudes. Explicit parentheses are needed to differentiate (A∨B)hC from Ah(B∨C), with highest level join altitude being expressed, h, with altitude h.

Multi-wedges can be evaluated in any order of evaluation, as long as the sum of the square of the circum-radius of the polytope elements are less than 1.

We can determine the counts by combinations, . And with multinomial theorem, it is generalized by for 3 partitions where n>a+b.

Altitude, h, case count for n-wedge by pairwise partitioning. If the partition sizes are equal, like 2+2 or 3+3, the combinations are cut in half.

n-wedge Form Combinations Counts
di-wedge+
n≥2
AhB n choose 2
tri-wedge+
n≥3
(A∨B)hC n choose 2+1
tetra-wedge+
n≥4
(A∨B)h(C∨D) n choose 2+2
(A∨B∨C)hD n choose 3+1
penta-wedge+
n≥5
(A∨B∨C)h(D∨E) n choose 3+2
(A∨B∨C∨D)hE n choose 4+1
hexa-wedge+
n≥6
(A∨B∨C)h(D∨E∨F) n choose 3+3
(A∨B∨C∨D)h(E∨F) n choose 4+2
(A∨B∨C∨D∨E)hF n choose 5+1

A tri-wedge A∨B∨C has 6 altitudes: A∨B, A∨C, B∨C, (A∨B)∨C, (A∨C)∨B, and (B∨C)∨A.

For example, if A, B, and C are points, it makes a triangle. The first three altitudes correspond to the edge lengths of the triangle, and the next 3 correspond to the 3 altitudes of the triangle.

A tetra-wedge has 6 altitudes A∨B, 12 altitude of form (A∨B)∨C, 3 altitude of form (A∨B)∨(C∨D), and 4 altitudes of form (A∨B∨C)∨D.

For example, if all 4 polytopes are points, this corresponds to a tetrahedron, having with 6 edge lengths, 12 altitudes on the 4 triangular faces, 3 digonal disphenoid altitude of opposite edges, and 4 triangular pyramid altitudes.

Lists by dimension

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1-dimensions

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Point di-wedge

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( )∨( ) is segment, { }, full symmetry [ ], order 2. f=(1,1)2=(1,2,1)

Segment
Construction Name BSA f-vector Verfs Coordinates Image Symmetry Order Dual Notes
( )∨( )= 2⋅( )
= { }
Point di-wedge
Segment
- (1,1)2
=(1,2,1)
2: ( ) ([1,0])
(±1)
[1]+ = 1 Self-dual Equilateral { }

2-dimensions

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Point tri-wedge

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( )∨( )∨( ) is a general triangle, no symmetry. f-1...2=(1,3,3,1)=(1,1)3.

If the 3 points can be commuted the symmetry increases to an equilateral triangle. It can be seen with coordinates in 3D ([1,0,0]), coordinate permutations (1,0,0), (0,1,0), and (0,0,1).

Point tri-wedge
Construction Name BSA f-vector Verf Coordinates Image Symmetry Order Dual Notes
( )∨( )∨( ) = 3⋅( ) Point tri-wedge
Triangle
Equilateral triangle
triang (1,1)3
=(1,3,3,1)
3: ( )∨( ) ([1,0,0]) [1,1]+ =
[3,1] =
1
6
Self-dual Equilateral {3}

Segment pyramid

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{ }∨( ) can express an isosceles triangle, symmetry [ ], order 2. f=(1,3,3,1)=(1,1)3.

Segment pyramid
Construction Name BSA f-vector Verfs Coordinates Image Symmetry Order Dual Notes
{ }∨( ) Segment-point di-wedge
isosceles triangle
triang (1,2,1)*(1,1)
=(1,3,3,1)
3: ( )∨( ) ([1,0]), (0,0) [1,1] = 2 Self-dual Equilateral {3}
h=√(3/4)=0.866

3-dimensions

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Point tetra-wedge

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( )∨( )∨( )∨( ) is a general tetrahedron, no symmetry implied. f-1...3=(1,4,6,4,1)=(1,1)4. If all four points can be permuted.

Interchanging the vertices with all permutations increases symmetry to the regular tetrahedron, {3,3}, order 4! = 24.

Point tetra-wedge
Construction Name BSA f-vector Verfs Coordinates Image Symmetry Order Dual Notes
( )∨( )∨( )∨( )= 4⋅( ) Point tetra-wedge
tetrahedron
Regular tetrahedron
tet (1,1)4
=(1,4,6,4,1)
4: ( )∨( )∨( ) ([1,0,0,0]) [1,1,1]+ = 1 Self-dual Regular {3,3}

Polygonal pyramid

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A polygonal-point di-wedge or p-gonal pyramid, {p}∨( ), symmetry [p,1], order 2p. f=(1,p,p,1)*(1,1)=(1,1+p,2p,1+p,1)

Polygonal pyramid
Construction Name BSA f-vector Verfs Coordinates Image Symmetry Order Dual Notes
{3}∨( ) Triangular pyramid
= tetrahedron
tet (1,3,3,1)*(1,1)
=(1,4,6,4,1)
3: { }∨( )
1: {3}
([1,0,0]), (0,0,0) [3,1] = 6 Self-dual Equilateral {3,3}
h=√(2/3)=0.8165
{4}∨( ) Square pyramid squippy
J1
(1,4,4,1)*(1,1)
=(1,5,8,5,1)
4: { }∨( )
1: {4}
(±1,±1,1), (0,0,0) [4,1] = 8 Self-dual Equilateral
h=√(1/2) = 0.7071
{5}∨( ) Pentagonal pyramid peppy
J2
(1,5,5,1)*(1,1)
=(1,6,10,6,1)
5: { }∨( )
1: {5}
(x,y,1), (0,0,0) [5,1] = 10 Self-dual Equilateral
h=√((3-√5)/8) = 0.3090
{6}∨( ) Hexagonal pyramid Flat
-
(1,6,6,1)*(1,1)
=(1,7,12,7,1)
6: { }∨( )
1: {6}
([0,1,2]), (0,0,0) [6,1] = 12 Self-dual Equilateral only if degenerate
h=0
{p}∨( ) p-gonal pyramid Flat
-
(1,p,p,1)*(1,1)
=(1,1+p,2p,1+p,1)
p: { }∨( )
1: {p}
[p,1] = 2p Self-dual

Segment di-wedge

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A digonal disphenoid or segment-segment di-wedge. f=(1,4,6,4,1)=(1,1)4.

Segment di-wedge
Construction name BSA f-vector Verfs Coordinates Image Symmetry Order Dual Notes
{ }∨{ } = 2⋅{ } Segment di-wedge
Digonal disphenoid
tet (1,2,1)2
=(1,4,6,4,1)
4: { }∨( ) (±1,0,-1), (0,±1,+1) [2,1] =
[[2],1]=[4,2+]
4
8
Self-dual Equilateral {3,3}
h=1/√2

The symmetry can double to [4,2+], order 8, by mapping edges to each other by a rotoreflection.

4-dimensions

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Polyhedral pyramid

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In 4-dimensions, a polyhedron-point di-wedge or a polyhedral pyramid is a 4-polytope with a polyhedron base and a point apex, written as a join, with a regular polyhedron, {p,q}∨( ), with symmetry [p,q,1]. It is self-dual.

If the polyhedron, {p,q}, has (v,e,f) vertices, edges, and faces, {p,q}∨( ) will have v+1 vertices, v+e edges, e+f faces, and f+1 cells. f=(1,v,e,f,1)*(1,1)=(1,v+1,v+e,e+f,f+1,1).

Polyhedral pyramid
Construction Name BSA f-vector Verfs Coordinates Image Symmetry Order Dual Notes
{ }∨{ }∨( ) = 5⋅( ) Segment-segment-point tri-wedge
Digonal disphenoid pyramid
= 5-cell
pen (1,2,1)2*(1,1) =(1,1)5
=(1,5,10,10,5,1)
{ }∨{ }
{ }∨( )∨( )
([1,0],0,0,-1),(0,0,[1,0],1),
(0,0,0,0,0)
[1,1,1]+ = 4 Self-dual Equilateral {3,3,3}
{3,3}∨( ) Tetrahedron-point di-wedge
Tetrahedral pyramid
= 5-cell
(1,4,6,4,1)*(1,1)
=(1,5,10,10,5,1)
{3}∨( )
{3,3}
([1,0,0,0],1), (0,0,0,0,0) [3,3,1] = 24 Self-dual Equilateral {3,3,3}
{4,3}∨( ) Cubic pyramid
cubic pyramid
cubpy
K-4.26
(1,8,12,6,1)*(1,1)
=(1,9,20,18,5,1)
{3}∨( )
{4,3}
(±1,±1,±1,1), (0,0,0,0) [4,3,1] = 96 {3,4}∨( ) Equilateral
{3,4}∨( ) Octahedral pyramid
Octahedral pyramid
octpy
K-4.3
(1,6,12,8,1)*(1,1)
=(1,7,18,20,9,1)
{4}∨( )
{3,4}
([±1,0,0], 1), (0,0,0,0) {4,3}∨( ) Equilateral
r{3,4}∨( ) Cuboctahedral pyramid (1,12,24,14,1)*(1,1)
=(1,13,36,38,15,1)
([±1,±1,0],1), (0,0,0,0) r{3,4}∨( ) Equilateral if flat
h=0
t{3,4}∨( ) Truncated octahedral pyramid - (1,24,36,14,1)*(1,1)
=(1,25,60,50,15,1)
{ }∨( )∨( )
t{3,4}
([0,1,2,3]), (0,0,0,0) dtr{3,4}∨( ) Not equilateral
{5,3}∨( ) Dodecahedral pyramid - (1,20,30,12,1)*(1,1)
=(1,21,50,42,13,1)
{5}∨( )
{5,3}
(x,y,z,1), (0,0,0,0) [5,3,1] = 240 {3,5}∨( ) Not equilateral
{3,5}∨( ) Icosahedral pyramid
Icosahedral_pyramid
ikepy
K-4.84
(1,12,30,20,1)*(1,1)
=(1,13,42,50,21,1)
{5}∨( )
{3,5}
(x,y,z,1), (0,0,0,0) {5,3}∨( ) Equilateral
s{2,8}∨( ) Square antiprism pyramid
Square antiprismatic pyramid
squappy
K-4.17.1
(1,8,16,10,1)*(1,1)
=(1,9,24,26,11,1)
Equilateral
s{2,10}∨( ) pentagonal antiprism pyramid
Pentagonal antiprismatic pyramid
pappy
K-4.80.1
(1,10,20,12,1)*(1,1)
=(1,11,30,32,13,1)
Equilateral
J11∨( ) Gyroelongated pentagonal pyramid pyramid gyepippy
K-4.85
(1,11,25,16,1)*(1,1)
=(1,12,36,41,17,1)
Equilateral
J62∨( ) Metabidiminished icosahedron pyramid mibdipy
K-4.87
(1,10,20,12,1)*(1,1)
=(1,11,30,32,13,1)
Equilateral
J63∨( ) Tridiminished icosahedron pyramid teddipy
K-4.88
(1,9,15,8,1)*(1,1)
=(1,10,24,23,9,1)
Equilateral
Prism pyramids
Construction Name BSA f-vector Verf Image Symmetry Order Dual Notes
{3}×{ }∨( ) Triangular prismatic pyramid
Triangular_prismatic_pyramid
trippy
K-4.7
(1,6,9,5,1)*(1,1)
=(1,7,15,14,6,1)
{ }×{ }∨( ) [3,2,1] = 12 ({3}+{ })∨( ) Equilateral
{4}×{ }∨( )
= {4,3}∨( )
square prismatic pyramid
= Cubic pyramid
cubpy
K-4.26
(1,8,12,6,1)*(1,1)
=(1,9,20,18,7,1)
{ }×{ }∨( )
{4}×{ }
[4,2,1] = 16 ({4}+{ })∨( ) Equilateral
{5}×{ }∨( ) Pentagonal prismatic pyramid
Pentagonal_prismatic_pyramid
pippy
K-4.141
(1,10,15,7,1)*(1,1)
=(1,11,25,22,8,1)
{ }×{ }∨( )
{5}×{ }
[5,2,1] = 20 ({5}+{ })∨( ) Equilateral
{6}×{ }∨( ) Hexagonal prismatic pyramid - (1,12,18,8,1)*(1,1)
=(1,13,30,26,9,1)
{ }×{ }∨( )
{6}×{ }
[6,2,1] = 20 ({6}+{ })∨( ) Not equilateral
{p}×{ }∨( ) p-gonal prismatic pyramid - (1,2p,3p,2+p,1)*(1,1)
=(1,2p+1,5p,2+4p,3+p,1)
{ }×{ }∨( )
{p}×{ }
[p,2,1] = 4p ({p}+{ })∨( )

Polygon-segment di-wedge

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In 4-dimensions, a polygon-segment di-wedge or polygonal pyramid pyramid is a 4-polytope with p-gonal base and a segment apex, written as a join, with a regular polygon, {p}∨{ }, with symmetry [p,2,1]. It is self-dual.

They can be drawn in perspective projection into the envelope of a p-gonal bipyramid, with an added edge down the bipyramid axis. {p}∨{ } has p+2 vertices, 1+3p edges, 1 p-gonal faces and 3p triangles, and 2 p-gonal pyramidal cells, and p tetrahedral cells. f=(1,p,p,1)*(1,1)2=(1,2+p,1+3p,1+3p,2+p,1)

The join can be equilateral for real altitude h=√(0.5-0.25/sin(π/p))>0.

Polygon-segment di-wedge
Construction Name BSA f-vector Verfs Facets Coordinates Image Symmetry Order Dual Notes
{3}∨{ }
= {3}∨( )∨( )
Triangle-segment di-wedge
Triangular pyramid pyramid
Triangular scalene
pen
K-4.1.1
(1,3,3,1)*(1,1)2
=(1,5,10,10,5,1)
3: { }∨{ }
2: {3}∨( )
3: { }∨{ }
2: {3}∨( )
([1,0,0],0,0), (0,0,0,[1,0]) [3,1,1] =
[3,2,1] =
6
12
Self-dual Equilateral {3,3,3}
h=√(5/12)
{4}∨{ }
= {4}∨( )∨( )
Square-segment di-wedge
Square pyramid pyramid
Square scalene
squasc
K-4.4
(1,4,4,1)*(1,1)2
=(1,6,13,13,6,1)
4: { }∨{ }
2: {4}∨( )
4: { }∨{ }
2: {4}∨( )
(±1,±1,0,0), (0,0,[1,0]) [4,1,1] =
[4,2,1] =
16 Self-dual Equilateral
h=1/2
{5}∨{ }
= {5}∨( )∨( )
Pentagon-segment di-wedge
Pentagonal pyramid pyramid
Pentagonal scalene
pesc
K-4.86
(1,5,5,1)*(1,1)2
=(1,7,17,17,7,1)
5: { }∨{ }
2: {5}∨( )
5: { }∨{ }
2: {5}∨( )
(x,y,0,0), (0,0,[1,0]) [5,1,1] =
[5,2,1] =
20 Self-dual Equilateral
h=0.026393202
{6}∨{ }
= {6}∨( )∨( )
Hexagon-segment di-wedge
Hexagonal pyramid pyramid
Hexagonal scalene
- (1,6,6,1)*(1,1)2
=(1,8,20,20,8,1)
6: { }∨{ }
2: {6}∨( )
6: { }∨{ }
2: {6}∨( )
([0,1,2],0,0), (0,0,0,[1,0]) [6,1,1] =
[6,2,1] =
24 Self-dual Not equilateral
{p}∨{ }
= {p}∨( )∨( )
p-gon-segment di-wedge
p-gonal pyramid pyramid
p-gonal scalene
- (1,p,p,1)*(1,1)2
=(1,2+p,1+3p,1+3p,2+p,1)
p: { }∨{ }
2: {p}∨( )
p: { }∨{ }
2: {p}∨( )
[p,1,1] =
[p,2,1] =
4p Self-dual

5-dimension

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Segment tri-wedge

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{ }∨{ }∨{ } is a tri-wedge in 5-dimensions, a lower dimensional form of a 5-simplex. It is self-dual. f=(1,2,1)3=(1,1)6=(1,6,15,20,15,6,1)

It has symmetry [2,2,1,1], order 8. The symmetry order can increase by a factor of 6 by interchanging segments, [3[2,2],1,1] or [4,3,1,1], order 48.

Segment tri-wedge
Construction name BSA f-vector Verfs Image Symmetry Order Dual Notes
{ }∨{ }∨{ } = 3⋅{ } = 6⋅( ) Segment tri-wedge
= 5-simplex
hix (1,2,1)3
=(1,6,15,20,15,6,1)
{ }∨{ }∨( ) [2,2,1,1] =
[3[2,2],1,1] = [4,3,1,1] =
8
24
Self-dual Equilateral {3,3,3,3}

Polychoral pyramid

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In 5-dimensions, a polychoron-point di-wedge or polychoral pyramid is a 5-polytope pyramid, with a polychoron base and a point apex, written as a join, with a regular polyhedron, {p,q,r}∨( ), with symmetry [p,q,r,1].

A polychoral pyramid with base f-vector=(v,e,f,c) will have new f-vector=(1,v,e,f,c,1)*(1,1)=(1+v,v+e,e+f,f+c,1+c).

Polychoral pyramid
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3,3,3}∨( ) 5-cell pyramid hix (1,5,10,10,5,1)*(1,1)
=(1,6,15,20,15,6,1)
{3,3}∨( )
{3,3,3}
1: {3,3,3}
5: {3,3}∨( )
[3,3,3,1] = 120 Self-dual Equilateral {3,3,3,3}
r{3,3,3}∨( ) Rectified 5-cell pyramid rappy (1,15,60,80,45,12,1)*(1,1)
=(1,16,75,140,125,57,13,1)
Equilateral
{3,3,4}∨( ) 16-cell pyramid hexpy (1,8,24,32,16,1)*(1,1)
=(1,9,32,56,48,17,1)
{3,4}∨( )
{3,3,4}
1: {3,3,4}
16: {3,3}∨( )
[4,3,3,1] = 384 {4,3,3}∨( ) Equilateral
{4,3,3}∨( ) Tesseractic pyramid - (1,16,32,24,8,1)*(1,1)
=(1,17,48,56,32,9,1)
{4,3}∨( )
{4,3,3}
1: {4,3,3}
16: {3,3}∨( )
{3,3,4}∨( ) Not equilateral
{3,4,3}∨( ) 24-cell pyramid - (1,24,96,96,24,1)*(1,1)
=(1,25,120,192,120,25,1)
{4,3}∨( )
{3,4,3}
1: {3,4,3}
24: {3,4}∨( )
[3,4,3,1] = 1152 Self-dual Not equilateral
{3,3,5}∨( ) 600-cell pyramid - (1,120,720,1200,600,1)*(1,1)
=(1,121,840,1920,1800,601,1)
{3,5}∨( )
{3,3,5}
1: {3,3,5}
120: {3,3}∨( )
[5,3,3,1] = 14400 {5,3,3}∨( ) Not equilateral
{5,3,3}∨( ) 120-cell pyramid - (1,600,1200,720,120,1)*(1,1)
=(1,601,1800,1920,840,121,1)
{3,3}∨( )
{5,3,3}
1: {5,3,3}
600: {5,3}∨( )
{3,3,5}∨( ) Not equilateral
Polyhedral prism pyramids
Construction Name BSA f-vector Verf Image Symmetry Order Dual Notes
{3,3}×{ }∨( ) Tetrahedral prismatic pyramid tepepy (1,8,16,14,6,1)*(1,1)
=(1,9,24,30,20,7,1)
{3}×{ }∨( )
{3,3}∨( )
{3,3}×{ }
[3,3,2,1] = 48 Tetrahedral bipyramid pyramid Equilateral
{4,3}×{ }∨( )
= {4,3,3}∨( )
Cubic prismatic pyramid
= Tesseract pyramid
- (1,16,32,24,8,1)*(1,1)
=(1,17,48,56,32,9,1)
{3}×{ }
{4,3}∨( )
{4,3}×{ }
[4,3,2,1] = 192 ({3,4}+{ })∨( )
= 16-cell pyramid
Not equilateral
{3,4}×{ }∨( )
r{3,3}×{ }∨( )
Octahedral prismatic pyramid opepy (1,12,30,16,10,1)*(1,1)
=(1,13,42,46,26,11,1)
{4}×{ }∨( )
{3,4}∨( )
{3,4}×{ }
({4,3}+{ })∨( ) Equilateral
r{3,4}×{ }∨( ) Cuboctahedral prismatic pyramid - (1,24,60,52,16,1)*(1,1)
=(1,25,84,112,68,17,1)
Not equilateral
{5,3}×{ }∨( ) Dodecahedral prismatic pyramid - (1,40,80,54,14,1)*(1,1)
=(1,41,120,134,68,15,1)
{3}×{ }∨( )
{5,3}∨( )
{5,3}×{ }
[5,3,2,1] = 480 ({3,5}+{ })∨( ) Not equilateral
{3,5}×{ }∨( ) Icosahedral prismatic pyramid - (1,24,72,70,22,1)*(1,1)
=(1,25,96,142,92,23,1)
{3,5}∨( )
{3,5}×{ }
({5,3}+{ })∨( ) Not equilateral
duoprism pyramids
Construction Name BSA f-vector Verf Symmetry Order Dual Notes
{3}×{3}∨( ) {3}×{3} 3-3 duoprismatic pyramid - (1,9,18,15,6,1)*(1,1)
=(1,7,27,24,21,7,1)
{3}×{ }∨( )
{ }×{3}∨( )
{3}×{3}
[3,2,3,1] = 36 ({3}+{3})∨( )
{3}×{4}∨( ) {3}×{4} 3-4 duoprismatic pyramid - (1,12,24,19,7,1)*(1,1)
=(1,8,36,31,26,8,1)
{3}×{ }∨( )
{ }×{4}∨( )
{3}×{4}
[3,2,4,1] = 48 ({3}+{4})∨( )
{4}×{4}∨( ) {4}×{4} tesseractic pyramid - (1,16,32,24,8,1)*(1,1)
=(1,9,48,40,32,9,1)
{4}×{ }∨( )
{ }×{4}∨( )
{4}×{4}
[4,2,4,1] = 64 ({4}+{4})∨( )
{p}×{q}∨( ) {p}×{q} p-q duoprismatic pyramid - (1,pq,2pq,pq+p+q,p+q,1)*(1,1)
=(1,1+p+q,3pq,p+q+2pq,2p+2q+pq,1+p+q,1)
{p}×{ }∨( )
{ }×{q}∨( )
{p}×{q}
[p,2,q,1] = 4pq p-q duopyramid pyramid
({p}+{q})∨( ) {p}+{q} p-q duopyramidal pyramid - (1,p+q,pq+p+q,2pq,pq,1)*(1,1)
=(1,1+p+q,2p+2q+pq,p+q+2pq,3pq,1+p+q,1)
{p}+{q}∨( )
{ }+{q}∨( )
{p}+{ }
p-q duoprismatic pyramid

Polygon di-wedge

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In 5-dimensions, a polygon di-wedge is a 5-polytope with a p-gonal base and a q-gonal base, written as a join, {p}∨{q}. It is self-dual. It has symmetry [p,2,q,1], order 4pq, double if p=q

{p}∨{q} has p+q vertices, p+q+pq edges, 2+2pq faces, and p+q+pq cells, and p+q hypercells. f-1...5=(1,p,p,1)*(1,q,q,1)=(1,p+q,p+q+pq,2+2pq,p+q+pq,p+q,1).

The join can be equilateral for real altitude h=√(1-0.25(1/sin(π/p)+1/sin(π/q))>0.

Polygon di-wedge
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3}∨{3} = 2⋅{3}= 6⋅( ) Triangle di-wedge
= 5-simplex
hix (1,3,3,1)2 =(1,1)6
=(1,6,15,20,15,6,1)
6: {3}∨{ } 6: {3}∨{ } [[3,2,3],1] 36 Self-dual Equilateral {3,3,3,3}
h=1/√3
{3}∨{4}
= {4}∨( )∨( )∨( )
Triangle-square di-wedge
Square pyramid pyramid pyramid
Square tettenes
squete (1,3,3,1)*(1,4,4,1)
=(1,7,19,26,19,7,1)
4: {3}∨{ }
3: { }∨{4}
4: {3}∨{ }
3: { }∨{4}
[3,2,4,1] 48 Self-dual Equilateral
h=1/√6
{3}∨{5}
= {5}∨( )∨( )∨( )
Triangle-pentagon di-wedge
Pentagonal pyramid pyramid pyramid
(1,3,3,1)*(1,5,5,1)
=(1,8,23,32,23,8,1)
5: {3}∨{ }
3: { }∨{5}
5: {3}∨{ }
3: { }∨{5}
[3,2,5,1] 60 Self-dual Not equilateral
{4}∨{4} = 2⋅{4} Square di-wedge Flat
4g=perp4g
(1,4,4,1)2
=(1,8,24,34,24,8,1)
8: {4}∨{ } 8: {4}∨{ } [[4,2,4],1] 64 Self-dual Equilateral only if degenerate
h=0
{4}∨{5} Square-pentagon di-wedge - (1,4,4,1)*(1,5,5,1)
=(1,9,29,42,29,9,1)
5: {4}∨{ }
4: { }∨{5}
5: {4}∨{ }
4: { }∨{5}
[4,2,5,1] 80 Self-dual Not equilateral
{5}∨{5} = 2⋅{5} Pentagon di-wedge - (1,5,5,1)2
=(1,10,35,52,35,10,1)
10: {5}∨{ } 10: {5}∨{ } [[5,2,5],1] 100 Self-dual Not equilateral
{3}∨{6}
= {6}∨( )∨( )∨( )
Triangle-hexagon di-wedge
Hexagonal pyramid pyramid pyramid
Hexagonal tettenes
- (1,3,3,1)*(1,6,6,1)
=(1,9,27,38,27,9,1)
6: {3}∨{ }
3:{ }∨{6}
6: {3}∨{ }
3: { }∨{6}
[3,2,6,1] 72 Self-dual Not equilateral
{4}∨{6} Square-hexagon di-wedge - (1,4,4,1)*(1,6,6,1)
=(1,10,34,50,34,10,1)
6: {4}∨{ }
4: { }∨{6}
6: {4}∨{ }
4: { }∨{6}
[4,2,6,1] 96 Self-dual Not equilateral
{5}∨{6} Pentagon-hexagon di-wedge - (1,5,5,1)*(1,6,6,1)
=(1,11,41,62,41,11,1)
6: {5}∨{ }
5: { }∨{6}
6: {5}∨{ }
5: { }∨{6}
[5,2,6,1] 120 Self-dual Not equilateral
{6}∨{6} = 2⋅{6} Hexagon di-wedge - (1,6,6,1)2
=(1,12,48,74,48,12,1)
12: {6}∨{ } 12: {6}∨{ } [[6,2,6],1] 144 Self-dual Not equilateral
{p}∨{q} p-q-gon di-wedge - (1,p,p,1)*(1,q,q,1)
=(1,p+q,p+q+pq,2+2pq,p+q+pq,p+q,1)
q: {p}∨{ }
p: { }∨{q}
q: {p}∨{ }
p: { }∨{q}
[p,2,q,1] 4pq Self-dual
{p}∨{p} = 2⋅{p} p-gon di-wedge - (1,p,p,1)2
=(1,2p,(2+p)p,2+2p2,(2+p)p,2p,1)
2p: {p}∨{ } 2p: {p}∨{ } [[p,2,p],1] 4p2 Self-dual

A vertex-edge graph for the pyramid can be drawn with a p+q vertex polygon, partitioning them into a p-gon, a q-gon, with one each between each vertex of the p-gon to a vertex of the q-gon.

Polyhedron-segment di-wedge

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A polyhedron-segment di-wedge, if regular as {p,q}∨{ } or {p,q}∨( )∨( ), is a join of a polyhedron and a segment, or a polyhedral pyramid pyramid in 5 dimensions. It has symmetry [p,q,2,1]. Its dual, if regular, is {q,p}∨{ }.

A {3,3}∨{ } is a lower symmetry 5-cell, symmetry [3,3,2,1], order 48.

If the polyhedron, {p,q}, has f=(v,e,f), then f({p,q}∨{ })=(v,e,f)*(1,1)2=(1,v+2,1+2v+e,v+2e+f,1+e+2f,2+f).

Polyhedron-segment di-wedge
Construction Name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3,3}∨{ }
= {3,3}∨( )∨( )
Tetrahedron-segment di-wedge
Tetrahedral pyramid pyramid
Tetrahedral scalene
hix (1,4,6,4,1)*(1,1)2
=(1,6,15,20,15,6,1)
4: {3}∨{ }
2: {3,3}∨( )
4: {3}∨{ }
2: {3,3}∨( )
[3,3,2,1] 48 Self-dual Equilateral {3,3,3,3}
t{3,3}∨{ }
= t{3,3}∨( )∨( )
Truncated tetrahedron-segment di-wedge
Truncated tetrahedral pyramid pyramid
Truncated tetrahedral scalene
- (1,8,18,12,1)*(1,1)2
=(1,10,35,56,43,14,1)
[3,3,2,1] 48 Not equilateral
{3,4}∨{ }
= {3,4}∨( )∨( )
Octahedron-segment di-wedge
Octahedral pyramid pyramid
Octahedral scalene
octasc (1,6,12,8,1)*(1,1)2
=(1,8,25,38,29,10,1)
6: {4}∨{ }
2: {3,4}∨( )
8: {3}∨{ }
2: {3,4}∨( )
[4,3,2,1] 96 {4,3}∨{ } Equilateral
{4,3}∨{ }
= {4,3}∨( )∨( )
Cube-segment di-wedge
Cubic pyramid pyramid
Cubic scalene
Flat
cubasc
(1,8,12,6,1)*(1,1)2
=(1,10,29,38,25,8,1)
8: {3}∨{ }
2: {4,3}∨( )
6: {4}∨{ }
2: {4,3}∨( )
[4,3,2,1] 96 {3,4}∨{ } Equilateral only if degenerate
t{4,3}∨{ }
= t{4,3}∨( )∨( )
Truncated cube-segment di-wedge
Truncated cubic pyramid pyramid
Truncated cubic scalene
- (1,24,36,14,1)*(1,1)2
=(1,26,61,110,65,16,1)
[4,3,2,1] 96 Not equilateral
t{3,4}∨{ }
= t{3,4}∨( )∨( )
Truncated octahedron-segment di-wedge
Truncated octahedral pyramid pyramid
Truncated octahedral scalene
- (1,24,36,14,1)*(1,1)2
=(1,26,85,110,65,16,1)
[4,3,2,1] 96 Not equilateral
r{3,4}∨{ }
= r{3,4}∨( )∨( )
Cuboctahedron-segment di-wedge
Cuboctahedral pyramid pyramid
Cuboctahedral scalene
- (1,12,24,14,1)*(1,1)2
=(1,14,49,74,53,16,1)
[4,3,2,1] 96 {4,3}∨{ } Not equilateral
rr{3,4}∨{ }
= rr{3,4}∨( )∨( )
Rhombicuboctahedron-segment di-wedge
Rhombicuboctahedral pyramid pyramid
Rhombicuboctahedral scalene
- (1,26,48,24,1)*(1,1)2
=(1,28,101,146,97,26,1)
[4,3,2,1] 96 Not equilateral
sr{3,4}∨{ }
= sr{3,4}∨( )∨( )
Snub cube-segment di-wedge
Rhombicuboctahedral pyramid pyramid
Snub cube scalene
- (1,24,60,38,1)*(1,1)2
=(1,26,109,182,137,40,1)
[(4,3)+,2,1] 48 Not equilateral
{3,5}∨{ }
= {3,5}∨( )∨( )
Icosahedron-segment di-wedge
Icosahedral pyramid pyramid
Icosahedral scalene
- (1,12,30,20,1)*(1,1)2
=(1,14,55,92,71,22,1)
12: {5}∨{ }
2: {3,5}∨( )
20: {3}∨{ }
2: {3,5}∨( )
[5,3,2,1] 240 {5,3}∨{ } Not equilateral
{5,3}∨{ }
= {5,3}∨( )∨( )
Dodecahedron-segment di-wedge
Dodecahedral pyramid pyramid
Dodecahedral scalene
- (1,20,30,12,1)*(1,1)2
=(1,22,71,92,55,14,1)
20: {3}∨{ }
2: {5,3}∨( )
12: {5}∨{ }
2: {5,3}∨( )
[5,3,2,1] 240 {3,5}∨{ } Not equilateral
Prism-segment di-wedges
Construction Name BSA f-vector Verf Image Symmetry Order Dual Notes
{3}×{ }∨{ }
{3}×{ }∨( )∨( )
Triangular prism-segment di-wedge
Triangular prism scalene
trippasc (1,6,9,5,1)*(1,1)2
=(1,8,22,29,20,7,1)
[3,2,2,1] = 24 ({3}+{ })∨{ } Equilateral
({3}+{ })∨{ }
({3}+{ })∨( )∨( )
Triangular bipyramid-segment di-wedge
Triangular bipyramidal scalene
- (1,5,9,6,1)*(1,1)2
=(1,7,20,29,16,8,1)
{3}×{ }∨{ } Not equilateral
{4}×{ }∨{ }
= {4,3}∨{ }
Square prism-segment di-wedge
= Cube-segment di-wedge
square prism scalene
Flat
cubasc
(1,6,12,8,1)*(1,1)2
=(1,10,29,38,25,8,1)
[4,2,2,1] = 32 ({4}+{ })∨{ } Equilateral only if degenerate
({4}+{ })∨{ }
= {3,4}∨{ }
square bipyramid-segment di-wedge
= Octahedron-segment di-wedge
square bipyramid scalene
octasc (1,8,12,6,1)*(1,1)2
=(1,8,25,38,21,10,2)
{4}×{ }∨{ } Equilateral
{5}×{ }∨{ }
{5}×{ }∨( )∨( )
Pentagonal prism-segment di-wedge
Pentagonal prism scalene
- (1,10,15,7,1)*(1,1)2
=(1,12,26,47,30,9,1)
[5,2,2,1] = 40 ({5}+{ })∨{ } Not equilateral
({5}+{ })∨{ } Pentagonal bipyramid-segment di-wedge
Pentagonal bipyramidal scalene
- (1,7,15,10,1)*(1,1)2
=(1,9,30,47,26,12,1)
{5}×{ }∨{ } Not equilateral
{6}×{ }∨{ }
{6}×{ }∨( )∨( )
Hexagonal prism-segment di-wedge
Hexagonal prism scalene
- (1,12,18,8,1)*(1,1)2
=(1,14,31,56,35,10,1)
[6,2,2,1] = 48 ({6}+{ })∨{ } Not equilateral
({6}+{ })∨{ } Hexagonal bipyramid-segment di-wedge
Hexagonal bipyramidal scalene
- (1,8,18,12,1)*(1,1)2
=(1,10,35,56,31,14,1)
{6}×{ }∨{ } Not equilateral
{p}×{ }∨{ }
{p}×{ }∨( )∨( )
p-gonal prism-segment di-wedge
p-gonal prismatic scalene
- (1,2p,3p,2+p,1)*(1,1)2
=(1,2+2p,1+5p,2+9p,5+5p,4+p,1)
[p,2,2,1] = 8p ({p}+{ })∨{ }
({p}+{ })∨{ } p-gonal bipyramid-segment di-wedge
p-gonal bipyramidal scalene
- (1,2+p,3p,2p,1)*(1,1)2
=(1,4+p,5+5p,2+9p,1+5p,2+2p,1)
{p}×{ }∨{ }

6-dimension

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Segment-segment-segment-point tetra-wedge

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Segment-segment-segment-point tetra-wedge
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{ }∨{ }∨{ }∨( ) = 7⋅( ) Segment-segment-segment-point tetra-wedge hop (1,2,1)3*(1,1) =(1,1)7
=(1,7,21,35,35,21,7,1)
[2,2,2,2,1] = 8 Self-dual Equilateral 6-simplex

Polygon-segment-segment tri-wedge

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Polygon-segment-segment tri-wedge
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3}∨{3,3}
{3}∨{ }∨{ }
{3}∨( )∨( )∨( )∨( )
triangle-tetrahedron di-wedge
triangle-segment-segment tri-wedge
Triangle pennene
hop (1,3,3,1)*(1,1)4
=(1,3,3,1)*(1,2,1)2
=(1,7,21,35,35,21,7,1)
[3,2,2,2,1] = 48 Self-dual Equilateral {3,3,3,3,3}
{4}∨{3,3}
{4}∨{ }∨{ }
{4}∨( )∨( )∨( )∨( )
square-tetrahedron di-wedge
square-segment-segment tri-wedge
Square pennene
squepe (1,4,4,1)*(1,1)4
=(1,4,4,1)*(1,2,1)2
=(1,8,26,45,45,26,8,1)
[4,2,2,2,1] = 64 Self-dual Equilateral
{5}∨{3,3}
{5}∨{ }∨{ }
{5}∨( )∨( )∨( )∨( )
Pentagon-tetrahedron di-wedge
Pentagon-segment-segment tri-wedge
Pentagon pennene
- (1,5,5,1)*(1,1)4
=(1,5,5,1)*(1,2,1)2
=(1,9,31,55,55,31,9,1)
[5,2,2,2,1] = 80 Self-dual Not equilateral
{6}∨{3,3}
{6}∨{ }∨{ }
{6}∨( )∨( )∨( )∨( )
Hexagon-tetrahedron di-wedge
Hexagon-segment-segment tri-wedge
Hexagon pennene
- (1,6,6,1)*(1,1)4
=(1,6,6,1)*(1,2,1)2
=(1,10,36,65,65,36,10,1)
[6,2,2,2,1] = 96 Self-dual Not equilateral
{p}∨{3,3}
{p}∨{ }∨{ }
{p}∨( )∨( )∨( )∨( )
p-gon-tetrahedron di-wedge
p-gon-segment-segment tri-wedge
p-gon pennene
- (1,p,p,1)*(1,1)4
=(1,p,p,1)*(1,2,1)2
=(1,p,p,1)*(1,2,1)2
[p,2,2,2,1] = 16p Self-dual

Polyteron pyramid

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In 6-dimensions, a polyteron-point di-wedge or polyteric pyramid is a 6-polytope pyramid, with a polyteron base and a point apex, written as a join, with a regular polyteron, {p,q,r,s}∨( ), with symmetry [p,q,r,s,1].

A polyteral pyramid with base f-vector=(v,e,f,c,h) will have new f-vector=(1,v,e,f,c,h,1)*(1,1)=(1+v,v+e,e+f,f+c,c+h,1+h).

Polyteric pyramid
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3,3,3,3}∨( ) 5-simplex pyramid hop (1,6,15,20,15,6,1)*(1,1)
=(1,7,21,35,35,21,7,1)
[3,3,3,3,1] = 120 Self-dual Equilateral {3,3,3,3,3}
r{3,3,3,3}∨( ) rectified 5-simplex pyramid rixpy (1,10,30,30,10,1)*(1,1)
=(1,11,40,60,40,11,1)
Equilateral
2r{3,3,3,3}∨( ) birectified 5-simplex pyramid dotpy (1,20,90,120,60,12,1)*(1,1)
=(1,21,110,210,180,72,13,1)
Equilateral
{3,3,3,4}∨( ) 5-orthoplex pyramid tacpy (1,10,40,80,80,32,1)*(1,1)
=(1,11,50,120,160,112,33,1)
[4,3,3,3,1] = 3840 {4,3,3,3}∨( ) Equilateral
{4,3,3,3}∨( ) Penteractic pyramid - (1,32,80,80,40,10,1)*(1,1)
=(1,33,112,160,120,50,11,1)
{3,3,3,4}∨( ) Not equilateral
h{4,3,3,3}∨( ) Demipenteractic pyramid hinpy (1,16,80,160,120,26,1)*(1,1)
=(1,17,96,240,280,146,27,1)
[3,3,31,1,1] = 3840 Equilateral
Polychoric prism pyramid
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3,3,3}×{ }∨( ) 5-cell prism pyramid penppy (1,10,25,30,20,7,1)*(1,1)
=(1,11,35,55,50,27,8,1)
[3,3,3,2,1] = 240 ({3,3,3}+{ })∨( ) Equilateral
r{3,3,3}×{ }∨( ) Rectified 5-cell prism pyramid rappip∨( )
rappippy
(1,10,25,30,20,7,1)*(1,1)
=(1,11,35,55,50,27,8,1)
Equilateral
{3,3,4}×{ }∨( ) 16-cell prism pyramid hexippy (1,16,56,88,64,18,1)*(1,1)
=(1,17,72,144,152,82,19,1)
[4,3,3,2,1] = 768 ({4,3,3}+{ })∨( ) Equilateral
{4,3,3}×{ }∨( ) 5-cube pyramid - (1,10,40,80,80,32,1)*(1,1)
=(1,11,50,120,160,112,33,1)
({3,3,4}+{ })∨( ) Not equilateral
{3,4,3}×{ }∨( ) 24-cell prism pyramid - (1,26,144,288,216,48,1)*(1,1)
=(1,27,170,432,504,264,49,1)
[3,4,3,2,1] = 2304 ({3,4,3}+{ })∨( ) Not equilateral
{3,3,5}×{ }∨( ) 600-cell prism pyramid - (1,602,2400,3120,1560,240,1)*(1,1)
=(1,603,3002,5520,4680,1800,241,1)
[5,3,3,2,1] = 28800 ({5,3,3}+{ })∨( ) Not equilateral
{5,3,3}×{ }∨( ) 120-cell prism pyramid - (1,122,960,2640,3000,1200,1)*(1,1)
=(1,123,1082,3600,5640,4200,1201,1)
({3,3,5}+{ })∨( ) Not equilateral
Polyhedral-polygon duoprism pyramid
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3,3}×{3}∨( ) tetrahedron-triangle duoprism pyramid tratetpy (1,12,30,34,21,7,1)*(1,1)
=(1,13,42,64,55,28,8,1)
[3,3,2,3,1] = 144 (({3,3}+{3})∨{ }) Equilateral
{3,3}×{4}∨( ) tetrahedron-square duoprism pyramid squatet∨( )
squatetpy
(1,16,40,44,26,8,1)*(1,1)
=(1,17,56,84,70,34,9,1)
[3,3,2,4,1] = 192 (({3,3}+{4})∨{ }) Equilateral
{3,3}×{p}∨( ) tetrahedron-p-gon duoprism pyramid - [3,3,2,p,1] = 48p (({3,3}+{p})∨{ })
{3,4}×{3}∨( ) Octahedron-triangle duoprism pyramid troctpy (1,18,54,66,39,11,1)*(1,1)
=(1,19,72,120,105,50,12,1)
[4,3,2,3,1] = 288 ({4,3}+{3})∨{ } Equilateral
{3,4}×{4}∨( ) octahedron-square duoprism pyramid Flat
squoct∨( )
squoctpy
(1,16,40,44,26,8,1)*(1,1)
=(1,17,56,84,34,9,1)
[4,3,2,4,1] = 384 ({4,3}+{4})∨{ }
{3,4}×{p}∨( ) octahedron-p-gon duoprism pyramid - [4,3,2,p,1] = 96p ({4,3}+{p})∨{ } Equilateral if flat
h==0
{4,3}×{3}∨( ) Cube-triangle duoprism pyramid - [4,3,2,3,1] = 96*3 ({3,4}+{3})∨( ) Not equilateral
{4,3}×{4}∨( ) Cube-square duoprism pyramid - [4,3,2,4,1] = 96*4 ({3,4}+{4})∨( ) Not equilateral
{4,3}×{p}∨( ) Cube-p-gon duoprism pyramid - [4,3,2,p,1] = 96p ({3,4}+{p})∨( ) Not equilateral
{3,5}×{3}∨( ) icosahedron-triangle duoprism pyramid - [5,3,2,3,1] = 360 ({5,3}+{3})∨( ) Not equilateral
{5,3}×{3}∨( ) dodecahedron-triangle duoprism pyramid - ({3,5}+{3})∨( ) Not equilateral
{3,5}×{4}∨( ) icosahedron-square duoprism pyramid - [5,3,2,4,1] = 480 ({5,3}+{4})∨( ) Not equilateral
{5,3}×{4}∨( ) dodecahedron-square duoprism pyramid - ({3,5}+{4})∨( ) Not equilateral
{3,5}×{p}∨( ) icosahedron-p-gon duoprism pyramid - [5,3,2,p,1] = 120p ({5,3}+{p})∨( ) Not equilateral
{5,3}×{p}∨( ) dodecahedron-p-gon duoprism pyramid - ({3,5}+{p})∨( ) Not equilateral
Duoprism-prism duoprism pyramid
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3}×{3}×{ }∨( ) 3-3 duoprism prism pyramid tratrip∨( )
tratrippy
(1,18,45,48,27,8,1)*(1,1)
=(1,19,63,93,75,35,9,1)
[3,2,3,2,1] = 72 ({3}+{3}+{ })∨( ) Equilateral!
{3}×{4}×{ }∨( ) 3-4 duoprism prism pyramid tracube∨( )
tracubepy
(1,24,60,62,33,9,1)*(1,1)
=(1,25,84,122,95,42,10,1)
[3,2,4,2,1] = 96 ({3}+{4}+{ })∨( ) Not equilateral
{4}×{4}×{ }∨( ) 5-cube pyramid - (1,32,80,80,40,10,1)*(1,1)
=(1,33,112,160,120,50,11,1)
[4,2,4,2,1] = 128 ({4}+{4}+{ })∨( ) Not equilateral
{p}×{q}×{ }∨( ) p-q duoprism prism pyramid - [p,2,q,2,1] = 8pq ({p}+{q}+{ })∨( )

A polygon-polygon di-wedge pyramid, {p}∨{q}∨( ), has f-vector (1,p,p,1)*(1,q,q,1)*(1,1)=(1,1+p+q,2p+2q+pq+2+p+q+3pq,2+p+q+3pq+2p+2q+pq,1+p+q,1).

Polygon-polygon di-wedge pyramid
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3}∨{3}∨( ) triangle di-wedge pyramid hop (1,3,3,1)2*(1,1)
=(1,7,21,35,35,21,7,1)
[[3,2,3],1,1] = 72 Self-dual Equilateral {3,3,3,3,3}
{3}∨{4}∨( ) triangle-square di-wedge pyramid squete∨( )
squetepy
(1,3,3,1)*(1,4,4,1)*(1,1)
=(1,8,26,45,26,8,1)
[3,2,4,1,1] = 48 Self-dual Equilateral
{4}∨{4}∨( ) square di-wedge pyramid Flat
4g=perp4g∨( )
(1,4,4,1)2*(1,1)
=(1,9,32,58,32,9,1)
[[4,2,4],1,1] = 128 Self-dual Equilateral only if degenerate
{p}∨{p}∨( ) p-gon di-wedge pyramid - (1,p,p,1)2*(1,1) [[p,2,p],1,1] = 8p2 Self-dual
{p}∨{q}∨( ) Polygon-polygon di-wedge pyramid - (1,p,p,1)*(1,q,q,1)*(1,1) [p,2,q,1,1] = 4pq Self-dual

Polychoron-segment di-wedge

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Polychoron-segment di-wedge
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3,3,3}∨{ } 5-cell-segment di-wedge
5-cell scalene
hop (1,5,10,10,5,1)*(1,1)2 =(1,1)5
=(1,7,21,35,35,21,7,1)
[3,3,3,2,1] = 240 Self-dual Equilateral {3,3,3,3,3}
r{3,3,3}∨{ } Rectified 5-cell-segment di-wedge
Rectified 5-cell scalene
rapesc (1,10,30,30,10,1)*(1,1)2
=(1,12,51,100,100, 51,12,1)
Equilateral
{3,3,4}∨{ } 16-cell-segment di-wedge
16-cell scalene
hexasc (1,8,24,32,16,1)*(1,1)2
=(1,10,41,88,104,65,18,1)
[4,3,3,2,1] = 768 {4,3,3}∨{ } Equilateral
{4,3,3}∨{ } Tesseract-segment di-wedge
Tesseract scalene
- (1,16,32,24,8,1)*(1,1)2
=(1,18,65,104,88,91,10,1)
{3,4,3}∨{ } Not equilateral
{3,4,3}∨{ } 24-cell-segment di-wedge
24-cell scalene
- (1,24,96,96,24,1)*(1,1)2
=(1,26,145,312,312,150,26,1)
[3,4,3,2,1] = 2304 Self-dual Not equilateral
{3,3,5}∨{ } 600-cell-segment di-wedge
600-cell scalene
- (1,120,720,1200,600,1)*(1,1)2
=(1,122,961,2760,3720,2401,602,1)
[5,3,3,2,1] = 28800 {5,3,3}∨{ } Not equilateral
{5,3,3}∨{ } 120-cell-segment di-wedge
120-cell scalene
- (1,600,1200,720,120,1)*(1,1)2
=(1,602,2401,3720,2760,961,122,1)
{3,3,5}∨{ } Not equilateral
Polyhedral-prism-segment di-wedge
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3,3}×{ }∨{ } Tetrahedral prism-segment di-wedge
Tetrahedral-prismatic scalene
tepasc (1,8,16,14,6,1)*(1,1)2
=(1,10,33,54,50,27,8,1)
[3,3,2,2,1] = 96 (({3,3}+{ })∨{ }) Equilateral
{3,4}×{ }∨{ } Octahedral prism-segment di-wedge
Octahedral-prismatic scalene
opepy∨( )
opesc
(1,12,30,28,10,1)*(1,1)2
=(1,14,55,100,96,49,12,1)
[4,3,2,2,1] = 192 ({4,3}+{ })∨{ } Equilateral if degenerate
h=0
{4,3}×{ }∨{ }
={4,3,3}∨{ }
Tesseract-segment di-wedge
Cubic-prismatic scalene
- (1,16,32,24,8,1)*(1,1)2
=(1,18,65,104,88,41,10,1)
({3,4}+{ })∨{ } Not equilateral
{3,5}×{ }∨{ } Icosahedral prism-segment di-wedge
Icosahedral-prismatic scalene
- (1,24,72,70,22,1)*(1,1)2
=(1,26,121,238,234,115,24,1)
[5,3,2,2,1] = 480 ({5,3}+{ })∨{ } Not equilateral
{5,3}×{ }∨{ } Dodecahedral prism-segment di-wedge
Dodecahedral-prismatic scalene
- (1,22,70,72,24,1)*(1,1)2
=(1,24,115,234,238,121,26,1)
({3,5}+{ })∨{ } Not equilateral
Duoprism-segment di-wedge
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3}×{3}∨{ } 3-3 duoprism-segment di-wedge
3-3 duoprism scalene
triddipasc (1,9,18,15,6,1)*(1,1)2
=(1,11,37,60,54,28,8,1)
[3,2,3,2,1] = 72 ({3}+{3})∨{ } Equilateral
{3}×{4}∨{ } 3-4 duoprism-segment di-wedge
3-4 duoprism scalene
Flat
tisdippy∨( )
tisdipasc
(1,12,24,19,7,1)*(1,1)2
=(1,14,49,79,69,34,9,1)
[3,2,4,2,1] = 96 ({3}+{4})∨{ } Equilateral if degenerate
{4}×{4}∨{ } Tesseract-segment di-wedge
Tesseract duoprism scalene
- (1,16,32,24,8,1)*(1,1)2
=(1,18,65,104,88,41,10,1)
[4,2,4,2,1] = 128 ({4}+{4})∨{ } Not equilateral
{p}×{q}∨{ } p-q duoprism-segment di-wedge
p-q duoprism scalene
- (1,pq,2pq,p+q+pq,p+q,1)*(1,1)2 [p,2,q,2,1] = 8pq ({p}+{q})∨{ }

Polyhedron-polygon di-wedge

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Polyhedron-polygon wedge
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3,3}∨{3}
{3,3}∨( )∨( )∨( )
tetrahedron di-wedge
tetrahedral tettenes
triangle pennene
hop (1,4,6,4,1)*(1,3,3,1)
=(1,7,21,35,35,21,7,1)
[3,3,2,3,1] = 144 Self-dual Equilateral
{3,3}∨{4} tetrahedron-square di-wedge
square pennene
squepe (1,4,6,4,1)*(1,4,4,1)
=(1,8,26,45,45,26,8,1)
[3,3,2,4,1] = 192 Self-dual Equilateral
{3,3}∨{5} tetrahedron-pentagon di-wedge
pentagon pennene
(1,4,6,4,1)*(1,5,5,1)
=(1,9,31,55,55,31,9,1)
[3,3,2,5,1] = 240 Self-dual Not equilateral
{3,3}∨{6} tetrahedron-hexagon di-wedge
hexagon pennene
(1,4,6,4,1)*(1,6,6,1)
=(1,10,36,65,65,36,10,1)
[3,3,2,6,1] = 288 Self-dual Not equilateral
{3,3}∨{p} tetrahedron-p-gon di-wedge
p-gon pennene
trip∨{p} (1,4,6,4,1)*(1,p,p,1) [3,3,2,p,1] = 48p Self-dual
{3,4}∨{3}
{3,4}∨( )∨( )∨( )
octahedron-triangle di-wedge
octahedral tettenes
octepe (1,6,12,8,1)*(1,3,3,1)
=(1,9,33,63,67,39,11,1)
[4,3,2,3,1] = 288 {4,3}∨{3} Equilateral
{4,3}∨{3}
{4,3}∨( )∨( )∨( )
Cube-triangle di-wedge
cubic tettenes
(1,8,12,6,1)*(1,3,3,1)
=(1,11,39,67,63,33,9,1)
{3,4}∨{3} Not equilateral
{3,4}∨{4} octahedron-square di-wedge oct∨{4} (1,6,12,8,1)*(1,4,4,1)
=(1,10,40,81,87,48,12,1)
[4,3,2,4,1] = 384 {4,3}∨{4} Not equilateral
{4,3}∨{4} Cube-square di-wedge (1,8,12,6,1)*(1,4,4,1)
=(1,12,48,87,81,40,10,1)
{3,4}∨{4} Not equilateral
{3,4}∨{p} octahedron-p-gon di-wedge oct∨{p} (1,6,12,8,1)*(1,p,p,1) [4,3,2,p,1] = 96p {4,3}∨{p}
{4,3}∨{p} Cube-p-gon di-wedge oct∨{p} (1,8,12,6,1)*(1,p,p,1) {3,4}∨{p} Not equilateral
{3,5}∨{p} icosahedron-p-gon di-wedge ike∨{p} (1,12,30,20,1)*(1,p,p,1) [5,3,2,p,1] = 120p {5,3}∨{p} Not equilateral
{5,3}∨{p} dodecahedron-p-gon di-wedge doe∨{p} (1,20,30,12,1)*(1,p,p,1) {3,5}∨{p} Not equilateral

A polygonal-prism-polygon di-wedge, {p}×{ }∨{q},has f-vector as (1,2p,3p,2+p,1)*(1,q,q,1)=(1,2p+q,3p+q+2pq,3+p+5pq,1+2p+2q+4pq,1+5p+q,3+p,1).

Polygonal-prism-polygon di-wedge
Construction name BSA f-vector Verfs Facets Image Symmetry Order Dual Notes
{3}×{ }∨{3} triangular prism-triangle di-wedge
triangular prism tettenes
trippete (1,6,9,5,1)*(1,3,3,1)
=(1,9,30,51,49,27,8,1)
[3,2,2,3,1] = 72 ({3}+{ })∨{3} Equilateral
{3}×{ }∨{4} triangular prism-square di-wedge trip∨{4} (1,6,9,5,1)*(1,4,4,1)
=(1,10,37,66,63,33,9,1)
[3,2,2,4,1] = 96 ({3}+{ })∨{4} Not equilateral
{3}×{ }∨{5} triangular prism-pentagon di-wedge trip∨{5} (1,6,9,5,1)*(1,5,5,1)
=(1,11,44,81,77,39,10,1)
[3,2,2,5,1] = 120 ({3}+{ })∨{5} Not equilateral
{3}×{ }∨{6} triangular prism-hexagon di-wedge trip∨{6} (1,6,9,5,1)*(1,6,6,1)
=(1,12,51,96,91,45,11,1)
[3,2,2,6,1] = 144 ({3}+{ })∨{6} Not equilateral
{4}×{ }∨{3}
={4,3}∨{3}
cube-triangle di-wedge
cubic tettenes
cubasc∨( ) (1,8,12,6,1)*(1,3,3,1)
=(1,11,39,67,63,24,7,1)
[4,2,2,3,1] = 96 ({4}+{ })∨{3} Not equilateral
{4}×{ }∨{4}
={4,3}∨{4}
cube-square di-wedge cube∨{4} (1,8,12,6,1)*(1,4,4,1)
=(1,12,48,67,81,25,1)
[4,2,2,4,1] = 128 ({4}+{ })∨{4} Not equilateral
{4}×{ }∨{5}
={4,3}∨{5}
cube-pentagon di-wedge cube∨{5} (1,8,12,6,1)*(1,5,5,1)
=(1,13,57,107,99,47,11,1)
[4,2,2,5,1] = 160 ({4}+{ })∨{5} Not equilateral
{4}×{ }∨{6}
={4,3}∨{6}
cube-hexagon di-wedge cube∨{6} (1,8,12,6,1)*(1,6,6,1)
=(1,14,66,127,117,54,12,1)
[4,2,2,6,1] = 192 ({4}+{ })∨{6} Not equilateral
{5}×{ }∨{3} pentagonal prism-triangle di-wedge
pentagonal prismatic tettenes
- (1,10,15,7,1)*(1,3,3,1)
=(1,13,48,83,77,39,10,1)
[5,2,2,3,1] = 120 ({5}+{ })∨{3} Not equilateral
{5}×{ }∨{4} pentagonal prism-square di-wedge - (1,10,15,7,1)*(1,4,4,1)
=(1,14,59,108,99,47,11,1)
[5,2,2,4,1] = 160 ({5}+{ })∨{4} Not equilateral
{5}×{ }∨{5} pentagonal prism-pentagon di-wedge - (1,10,15,7,1)*(1,5,5,1)
=(1,15,70,133,121,55,12,1)
[5,2,2,5,1] = 200 ({5}+{ })∨{5} Not equilateral
{5}×{ }∨{6} pentagonal prism-hexagon di-wedge - (1,10,15,7,1)*(1,6,6,1)
=(1,16,81,158,143,63,13,1)
[5,2,2,6,1] = 240 ({5}+{ })∨{6} Not equilateral
{6}×{ }∨{3} hexagonal prism-triangle di-wedge
hexagonal prismatic tettenes
- (1,12,18,8,1)*(1,3,3,1)
=(1,15,57,99,91,45,11,1)
[6,2,2,3,1] = 144 ({6}+{ })∨{3} Not equilateral
{6}×{ }∨{4} hexagonal prism-square di-wedge - (1,12,18,8,1)*(1,4,4,1)
=(1,16,70,129,117,54,12,1)
[6,2,2,4,1] = 192 ({6}+{ })∨{4} Not equilateral
{6}×{ }∨{5} hexagonal prism-pentagon di-wedge - (1,12,18,8,1)*(1,5,5,1)
=(1,17,83,159,143,63,13,1)
[6,2,2,5,1] = 240 ({6}+{ })∨{5} Not equilateral
{6}×{ }∨{6} hexagonal prism-hexagon di-wedge - (1,12,18,8,1)*(1,6,6,1)
=(1,18,96,189,169,72,14,1)
[6,2,2,6,1] = 288 ({6}+{ })∨{6} Not equilateral
{p}×{ }∨{q} p-gonal prism-q-gon di-wedge - (1,2p,3p,2+p,1)*(1,q,q,1) [p,2,2,q,1] = 8pq ({p}+{ })∨{q}

Equilateral multi-wedges

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A vertex-edge join of a blue square and red triangle, {3}∨{4}, with green 12 green edges between all pairs of vertices across polygons. The green edges form a complete bipartite graph. It has 3+4 vertices, 3+4+12 edges, 1 triangle {3}, 1 square {4}, 12+12 triangles { }∨( ), 12 disphenoids { }∨{ }, 4 triangular pyramids, and 3 square pyramids {3}∨( ), 4 {3}∨{ }, and 3 {4}∨{ }. It can be made equilateral with altitude h=1/√6.[8]

A join, A∨B, is equilateral if:

  • A and B are both uniform, and if circumradii, r, of A and B are both less edge length by adjusting the join altitude and relative sizes of A and B.
  • May also be a CRF polytope, a convex regular-faced polytope, and Convex segmentotopes[9] for pyramids.

The altitude of an equilateral join can be computed by h=√(1-r2
A
-r2
B
). The specific altitude can be given with the join symbol as AhB.

An altitude h=0 becomes geometric degenerate, but topologically fine. For instance an equilateral hexagonal pyramid, {6}∨( ), can be seen as a polyhedron in 2D with a regular hexagon connected to a central point. The 6 equilateral lateral triangle faces coincide with the hexagonal base.

Circumradii

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Regular, and single ringed uniform polyhedra have all vertices on a single n-sphere. This radius is called the circumradii, given for a polytope with unit edge length.

Polygon

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For regular p-gon has rp=1/[2sin(π/p)]

{ } {3} {4} {5} {6}
r 1/2
=0.5000
√(1/3)
=0.5773
√(1/2)
=0.7071
√((5+√5)/10)
=0.8506
1

Polyhedra

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For regular and uniform polyhedra:

{3,3}
tet
{3,4}
oct
{4,3}
cube
{3,5}
ike
s{2,8}
squap
s{2,10}
pap
{3}×{ }
trip
{5}×{ }
ipe
r{3,4}
co
t{3,3}
tut
{5,3}
doe
Image
r √(3/8)
=0.6124
√(1/2)
=0.7071
√(3/4)
=0.8660
√((5+√5)/8)
=0.9511
√((4+√2)/8)
=0.8227
√((5+√5)/8)
=0.9511
√(7/12)
=0.7638
√((15+2√5)/20)
=0.9867
1 √(11/8)
=1.1726
√((9+3√5)/8)
=1.4013

Polychora

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For regular and uniform polychora:

{3,3,3}
pen
r{3,3,3}
rap
{3,3,4}
hex
{3,3}×{ }
tepe
{3,4}×{ }
ope
{3}×{3}
triddip
{3}×{4}
tisdip
{4,3,3}
tes
{4,3}×{ }
{4}×{4}
{3,4,3}
ico
r{3,3,4}
{3,3,5}
ex
{5,3,3}
hi
Image
r √(2/5)
=0.6325
√(3/5)
=0.7746
√(1/2)
=0.7071
√(5/8)
=0.7906
√(3/4)
=0.8660
√(2/3)
=0.8165
√(5/6)
=0.9129
1 1 (1+√5)/2
=1.6180
√(7+3√5)
=3.7025

5-polytope

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For regular and uniform 5-polytopes:

{3,3,3,3}
hix
r{3,3,3,3}
rix
2r{3,3,3,3}
dot
{3,3,3,4}
tac
h{4,3,3,3}
hin
{3,3,3}×{ }
penp
{3,3,4}×{ }
hexip
{3}×{3}×{ }
tratrip
{3,3}×{3}
tratet
{3,3}×{4}
squatet
{3,4}×{3}
troct
r{3,3,3}×{ }
rappip
{3,4}×{4}
squoct
{4,3}×{3}
tracube
{4}×{3}×{ }
{4,3,3,3}
pent
{4,3,3}×{ }
{4,3}×{4}
Image
r √(5/12)
=0.6455
√(2/3)
=0.8165
√(3/4)
=0.8660
√(1/2)
=0.7071
√(5/8)
=0.7906
√(13/20)
=0.8062
√(3/4)
=0.8660
√(11/12)
=0.9574
√(17/24)
=0.8416
√(7/8)
=0.9354
√(5/6)
=0.9129
√(17/20)
=0.9220
1 √(13/12)
=1.0408
√(5/4)
=1.1180

Equilateral solutions by dimension

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1 dimension

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1 dimensions
Class Pyramid
Form ( )∨( )
Image
r1,2 r1=0
r2=0
h 1

2 dimensions

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2 dimensions
Class Pyramid
Form { }∨( ) ={3}
Image
r1,2 r1=1/2
r2=0
h √(3/4)

3 dimensions

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3 dimensions
Class Pyramids Scalene
Form {3}∨( )
tet
={3,3}
{4}∨( )
squippy
{5}∨( )
peppy
{ }∨{ }
tet
={3,3}
Image
r1,2 r1=√(1/3)
r2=0
r1=√(1/2)
r2=0
r1=√((5+√5)/10)
r2=0
r1=1/2
r2=1/2
h √(2/3) √(1/2) √((5-√5)/10) √(1/2)

4 dimensions

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4 dimension pyramids
Form {3,3}∨( )
pen
={3,3,3}
{4,3}∨( )
cubpy
{3,4}∨( )
octpy
s{2,8}∨( )
squappy
{3}×{ }∨( )
trippy
{4}×{ }∨( )
cubpy
{5}×{ }∨( )
pippy
Images
r1,2 r1=√(3/8)
r2=0
r1=√(3/4)
r2=0
r1=√(1/2)
r2=0
r1=√((4+√2)/8)
r2=0
r1=√(7/12)
r2=0
r1=√(3/4)
r2=0
r1=√((7+√5)/8)
r2=0
h √(5/8) √(1/4) √(1/2) √((4-√2)/8) √(5/12) √(1/4) √((1-√5)/8)


4 dimension diminished icosahedral pyramids
Form {3,5}∨( )
ikepy
s{2,10}∨( )
pappy
J11∨( )
gyepip∨( )
gyepippy
J62∨( )
mibdipy
J63∨( )
teddipy
Images
r1,2 r1=√((5+√5)/8)
r2=0
r1=√((5+√5)/8)
r2=0
r1=√((5+√5)/8)
r2=0
r1=√((5+√5)/8)
r2=0
r1=√((5+√5)/8)
r2=0
h √((3-√5)/8) √((3-√5)/8) √((3-√5)/8) √((3-√5)/8) √((3-√5)/8)
4 dimension scalenes
Form {3}∨{ }
pen
={3,3,3}
{4}∨{ }
squippypy
{5}∨{ }
peppypy
Images
r1,2 r1=√(1/3)
r2=1/2
r1=√(1/2)
r2=1/2
r1=√((5+√5)/10)
r2=1/2
h √(1/12) 1/2 √((5-2√5)/20)

5 dimensions

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5 dimensions
Class Pyramids Scalenes Tettenes
Form {3,3,3}∨( )
hix
= {3,3,3,3}
r{3,3,3}∨( )
rappy
{3,3,4}∨( )
hexpy
{3,3}×{ }∨( )
tepepy
{3,4}×{ }∨( )
opepy
{3,3}∨{ }
hix
= {3,3,3,3}
{3}×{ }∨{ }
trippasc
{3}∨{3}
hix
= {3,3,3,3}
{4}∨{3}
squete
Images
r1,2 r1=√(2/5)
r2=0
r1=√(3/5)
r2=0
r1=√(1/2)
r2=0
r1=√(5/8)
r2=0
r1=√(3/4)
r2=0
r1=√(3/8)
r2=1/2
r1=√(7/12)
r2=1/2
r1=√(1/3)
r2=√(1/3)
r1=√(1/2)
r2=√(1/3)
h √(3/5) √(2/5) √(1/2) √(3/8) √(1/4) √(3/8)

6 dimensions

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6 dimension pyramids
Form {3,3,3,3}∨( )
hop
{3,3,3,3,3}
r{3,3,3,3}∨( )
rixpy
2r{3,3,3,3}∨( )
dotpy
{3,3,3,4}∨( )
tacpy
{3,3,3}×{ }∨( )
penppy
r{3,3,3}×{ }∨( )
rappip
{3,3,4}×{ }∨( )
hexippy
{3,3}×{3}∨( )
tratetpy
[{3,3}×{4}]∨( )
squatet
{3,4}×{3}∨( )
troctpy
[{3}×{3}×{ }]∨( )
tratrip
({3}∨{3})∨( )
hop
{3,3,3,3,3}
({3}∨{4})∨( )
squete
Images
r1,2 r1=
r2=0
r1=
r2=0
r1=
r2=0
r1=
r2=0
r1=
r2=0
r1=
r2=0
r1=
r2=
r1=
r2=
r1=
r2=
r1=
r2=
r1=
r2=
r1=
r2=
r1=
r2=
h
More 6 dimensions
Class Scalenes Tettenes Pennenes
Form {3,3,3}∨{ }
hop
{3,3,3,3,3}
r{3,3,3}∨{ }
rapesc
{3,3,4}∨{ }
hexasc
{3,3}×{ }∨{ }
tepasc
{3,4}×{ }∨{ }
opepy
{3}×{3}∨{ }
triddipasc
{3,3}∨{3}
hop
{3,3,3,3,3}
{3,4}∨{3}
octepe
{3}×{ }∨{3}
trippete
{3,3}∨{4}
squepe
Images
r1,2 r1=
r2=1/2
r1=
r2=1/2
r1=
r2=1/2
r1=
r2=1/2
r1=
r2=1/2
r1=
r2=1/2
r1=
r2=
r1=
r2=
r1=
r2=
r1=
r2=
h

References

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  1. ^ Coxeter, H. S. M. (1973), Regular Polytopes (3rd ed.), Dover Publications, p. 15, ISBN 0-486-61480-8
  2. ^ Pyramid_product
  3. ^ a b c Geometries and TransformationNorman Johnson, 2018, 11.3 Pyramids, Prisms, and Antiprisms, p.163
  4. ^ a b c d Products of abstract polytopes Ian Gleason and Isabel Hubard, 2016
  5. ^ a b c d https://bendwavy.org/klitzing/explain/product.htm
  6. ^ https://bendwavy.org/klitzing/explain/axials.htm#pyramid
  7. ^ https://bendwavy.org/klitzing/explain/product.htm#simplex
  8. ^ https://polytope.miraheze.org/wiki/Square_tettene
  9. ^ https://bendwavy.org/klitzing/explain/axials.htm

See also

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