User:Tomruen/Disphenoid

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

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 v_A+v_B vertices, and e_A+e_A+v_A×v_B 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

{\begin{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\\1\quad 5\quad 10\quad 10\quad 5\quad 1\\1\quad 6\quad 15\quad 20\quad 15\quad 6\quad 1\\1\quad 7\quad 21\quad 35\quad 35\quad 21\quad 7\quad 1\\1\quad 8\quad 28\quad 56\quad 70\quad 56\quad 28\quad 8\quad 1\\\end{array}}

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 f_n=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

There are four classes of product operators, working directly on f-vectors. The join include both f_-1, and f_n. The rhombic sum only includes f_-1, and its dual rectangular product only includes f_n. 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,f_A,1) * (1,f_B,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,f_A) * (1,f_B)	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	(f_A,1) * (f_B,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	f_A * f_B	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 f_A and f_B, f_A∨B=f_A*f_B is computed like a polynomial multiplication polynomial coefficients.

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

f_A(x) = (1,3,3,1) = 1 + 3x + 3x² + x³ (triangle)

f_B(x) = (1,2,1) = 1 + 2x + x² (dion)

f_A∨B(x) = f_A(x) * f_B(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∨B_k), ∀(A₀∨B_k), ∀(A₁∨B_k-1), ..., ∀(A_k∨B_-1)}, where A_i=set of i-faces in A, etc.

Factorization

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 f_Z=f_A*f_B*f_C, and f_A,f_B,f_C 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

{\begin{array}{c}1\quad 4\quad 4\quad 1\\1\quad 5\quad 8\quad 5\quad 1\\1\quad 6\quad 13\quad 13\quad 6\quad 1\\1\quad 7\quad 19\quad 26\quad 19\quad 7\quad 1\\1\quad 8\quad 26\quad 45\quad 45\quad 26\quad 8\quad 1\\1\quad 9\quad 34\quad 71\quad 90\quad 71\quad 34\quad 9\quad 1\\\end{array}}

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)

{\begin{array}{c}1\quad 8\quad 12\quad 6\quad 1\\1\quad 9\quad 20\quad 18\quad 7\quad 1\\1\quad 10\quad 29\quad 38\quad 25\quad 8\quad 1\\1\quad 11\quad 39\quad 67\quad 63\quad 33\quad 9\quad 1\\1\quad 12\quad 50\quad 106\quad 130\quad 96\quad 42\quad 10\quad 1\\1\quad 13\quad 62\quad 156\quad 236\quad 226\quad 138\quad 52\quad 11\quad 1\\\end{array}}

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∨{3^n-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

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 A∨_h(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, ${\binom {n}{k}}={\frac {n!}{(n-k)!k!}}$ . And with multinomial theorem, it is generalized by ${\binom {n}{n-a-b,a,b}}={\frac {n!}{(n-a-b)!a!b!}}$ 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	A∨_hB	n choose 2	${\binom {n}{2}}$	${\binom {n}{n-2,2}}$	${\frac {n!}{(n-2)!2!}}$
tri-wedge+ n≥3	(A∨B)∨_hC	n choose 2+1	${\binom {n}{2}}{\binom {n-2}{1}}$	${\binom {n}{n-3,2,1}}$	${\frac {n!}{(n-3)!2!1!}}$
tetra-wedge+ n≥4	(A∨B)∨_h(C∨D)	n choose 2+2	${\frac {1}{2}}{\binom {n}{2}}{\binom {n-2}{2}}$	${\frac {1}{2}}{\binom {n}{n-4,2,2}}$	${\frac {1}{2}}{\frac {n!}{(n-4)!2!2!}}$
tetra-wedge+ n≥4	(A∨B∨C)∨_hD	n choose 3+1	${\binom {n}{3}}{\binom {n-3}{1}}$	${\binom {n}{n-4,3,1}}$	${\frac {n!}{(n-4)!3!1!}}$
penta-wedge+ n≥5	(A∨B∨C)∨_h(D∨E)	n choose 3+2	${\binom {n}{3}}{\binom {n-3}{2}}$	${\binom {n}{n-5,3,2}}$	${\frac {n!}{(n-5)!3!2!}}$
penta-wedge+ n≥5	(A∨B∨C∨D)∨_hE	n choose 4+1	${\binom {n}{4}}{\binom {n-4}{1}}$	${\binom {n}{n-5,4,1}}$	${\frac {n!}{(n-5)!4!1!}}$
hexa-wedge+ n≥6	(A∨B∨C)∨_h(D∨E∨F)	n choose 3+3	${\frac {1}{2}}{\binom {n}{3}}{\binom {n-3}{3}}$	${\frac {1}{2}}{\binom {n}{n-6,3,3}}$	${\frac {1}{2}}{\frac {n!}{(n-6)!3!3!}}$
	(A∨B∨C∨D)∨_h(E∨F)	n choose 4+2	${\binom {n}{4}}{\binom {n-4}{2}}$	${\binom {n}{n-6,4,2}}$	${\frac {n!}{(n-6)!4!2!}}$
	(A∨B∨C∨D∨E)∨_hF	n choose 5+1	${\binom {n}{5}}{\binom {n-5}{1}}$	${\binom {n}{n-6,5,1}}$	${\frac {n!}{(n-6)!5!1!}}$

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

1-dimensions

Point di-wedge

( )∨( ) 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

Point tri-wedge

( )∨( )∨( ) 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

{ }∨( ) 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

Point tetra-wedge

( )∨( )∨( )∨( ) 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

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	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

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

Polyhedral pyramid

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	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	pen	(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)	[5,3,1] =	240	{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	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

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	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

Segment tri-wedge

{ }∨{ }∨{ } 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

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	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)			[3,3,3,1] =	120		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}∨( )	[4,3,3,1] =	384	{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}∨( )	[5,3,3,1] =	14400	{3,3,5}∨( )	Not equilateral

Polyhedral prism pyramids
Construction	Name	BSA	f-vector	Verf	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}×{ }	[5,3,2,1] =	480	({5,3}+{ })∨( )	Not equilateral

duoprism pyramids
Construction		Name	BSA	f-vector	Verf	Symmetry	Order	Dual
{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,2,q,1] =	4pq	p-q duoprismatic pyramid

Polygon di-wedge

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	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]	2×36	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]	2×64	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]	2×100	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]	2×144	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]	2×4p²	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

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	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	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)	[3,2,2,1] =	24	{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)	[4,2,2,1] =	32	{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)	[5,2,2,1] =	40	{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)	[6,2,2,1] =	48	{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,2,2,1] =	8p	{p}×{ }∨{ }

6-dimension

Segment-segment-segment-point tetra-wedge

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

Polygon-segment-segment tri-wedge
Construction	name	BSA	f-vector	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

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	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)	[4,3,3,3,1] =	3840	{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)	[3,3,3^1,1,1] =	3840		Equilateral

Polychoric prism pyramid
Construction	name	BSA	f-vector	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)	[3,3,3,2,1] =	240		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)	[4,3,3,2,1] =	768	({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)	[5,3,3,2,1] =	28800	({3,3,5}+{ })∨( )	Not equilateral

Polyhedral-polygon duoprism pyramid
Construction	name	BSA	f-vector	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	-		[5,3,2,3,1] =	360	({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	-		[5,3,2,4,1] =	480	({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	-		[5,3,2,p,1] =	120p	({3,5}+{p})∨( )	Not equilateral

Duoprism-prism duoprism pyramid
Construction	name	BSA	f-vector	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	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

Polychoron-segment di-wedge
Construction	name	BSA	f-vector	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)	[3,3,3,2,1] =	240		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)	[4,3,3,2,1] =	768	{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)	[5,3,3,2,1] =	28800	{3,3,5}∨{ }	Not equilateral

Polyhedral-prism-segment di-wedge
Construction	name	BSA	f-vector	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)	[4,3,2,2,1] =	192	({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)	[5,3,2,2,1] =	480	({3,5}+{ })∨{ }	Not equilateral

Duoprism-segment di-wedge
Construction	name	BSA	f-vector	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

Polyhedron-polygon wedge
Construction	name	BSA	f-vector	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)	[4,3,2,3,1] =	288	{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)	[4,3,2,4,1] =	384	{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)	[4,3,2,p,1] =	96p	{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)	[5,3,2,p,1] =	120p	{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	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

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 A∨_hB.

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

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

For regular p-gon has r_p=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

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

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

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

1 dimension

1 dimensions
Class	Pyramid
Form	( )∨( )
Image
r_1,2	r₁=0 r₂=0
h	1

2 dimensions

2 dimensions
Class	Pyramid
Form	{ }∨( ) ={3}
Image
r_1,2	r₁=1/2 r₂=0
h	√(3/4)

3 dimensions

3 dimensions
Class	Pyramids			Scalene
Form	{3}∨( ) tet ={3,3}	{4}∨( ) squippy	{5}∨( ) peppy	{ }∨{ } tet ={3,3}
Image
r_1,2	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

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
r_1,2	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
r_1,2	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
r_1,2	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

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
r_1,2	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

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
r_1,2	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
r_1,2	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

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

Different Products, occuring with Polytopes, pyramid product
Polytope Names and Constructions Wendy Krieger