User:Tomruen/Disphenoid

In geometry, a disphenoid is a tetrahedron whose four faces are congruent acute-angled triangles. 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.

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: The join A∨B will be:
 * 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)
 * 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
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
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.

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
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.

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.

Polytope-simplex di-wedges
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: 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.

Multi-wedge altitudes
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, $$ \binom nk = \frac{n!}{(n-k)!k!}$$. 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.

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.

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

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).

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

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.

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)

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

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

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).

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(&pi;/p))>0.

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.

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).

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(&pi;/p)+1/sin(&pi;/q))>0.

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).

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).

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).

Polyhedron-polygon di-wedge
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).

Equilateral multi-wedges
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 for pyramids.

The altitude of an equilateral join can be computed by h=√(1-r-r). 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 rp=1/[2sin(&pi;/p)]

Polyhedra
For regular and uniform polyhedra:

Polychora
For regular and uniform polychora:

5-polytope
For regular and uniform 5-polytopes: