Blending (skew polytopes)
Blending  

Symbol  
Rank formula  
Dimension formula  
Algebraic properties  
Algebraic structure  Monoid 
Associative  Yes 
Commutative  Yes 
Identity  Point 
Uniquely factorizable  Yes^{[note 1]} 
Idempotent  Yes 
Blending is an operation on polytopes. Blending typically produces skew polytopes.
Definition[edit  edit source]
Geometric definition[edit  edit source]
For two polytopes of the same rank, 𝓟 and 𝓠, in spaces D and E respectively their blend is a polytope in the space constructed as follows:
 Start with the Cartesian product of of the vertices of 𝓟 with the vertices of 𝓠.
 Add edges between any two vertices and iff there is an edge between and in 𝓟 and an edge between and in 𝓠.
 Similarly add faces to every set of vertices all incident on the same face in both 𝓟 and 𝓠.
 Repeat as such for all ranks of proper elements.
 From the resulting polytope, select one connected component.
For some pairs of polytope the last step is not guaranteed to give a unique result. However for regular polytopes the result is guaranteed to be unique, thus blending is frequently restricted only to regular polytopes.
For two polytopes of unequal rank, we give the polytope with lesser rank virtual elements of the missing rank incident on all the real elements. For example to blend a dyad with a cube, we give the dyad a virtual face incident on its edge and both its vertices. These virtual elements violate dyadicity, but this presents no problem for the final result.

Add vertices corresponding to the cartesian product of the vertices of the base polytopes.

Connect the vertices.

Select one connected component.
Distinguished generators[edit  edit source]
For two regular polytopes of the same rank their blend can be defined in terms of their distinguished generators. For a polytopes in space D , and in space E , their blend is a polytope in the space given by the generators .
For polytopes of differing ranks the missing generators are simply replaced with the identity.
Properties[edit  edit source]
 is abstractly equal to 𝓟 iff 𝓠 is a quotient of 𝓟.
See also[edit  edit source]
Notes[edit  edit source]
 ↑ See Maschke's theorem
Bibliography[edit  edit source]
 McMullen, Peter (2007). "FourDimensional Regular Polyhedra" (PDF). Discrete & Computational Geometry. doi:10.1007/s0045400713427.