User:Tomruen/Disphenoid
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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.