Blending (skew polytopes)

Blending
Blending a hexagon (blue) with a dyad (yellow) gives a skew hexagon (green).
Symbol	$\#$
Rank formula	$\max(n,m)$
Dimension formula	$n+m$
Algebraic properties
Algebraic structure	Monoid
Associative	Yes
Commutative	Yes
Identity	Point
Uniquely factorizable	Yes
Idempotent	Yes

Blending is an operation on polytopes. Blending typically produces skew polytopes.

Definition

Geometric definition

For two polytopes of the same rank, 𝓟 and 𝓠, in spaces D  and E  respectively their blend is a polytope in the space $D\times E$ 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 $p_{0}\times q_{0}$ and $p_{1}\times q_{1}$ iff there is an edge between $p_{0}$ and $p_{1}$ in 𝓟 and an edge between $q_{0}$ and $q_{1}$ 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.

• 

  Start with the two base polytopes in orthogonal planes. In this example a 2D hexagon (blue) and a 1D dyad (yellow).

• 

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

• 

  Connect the vertices.

• 

  Select one connected component.

Distinguished generators

For two regular polytopes of the same rank their blend can be defined in terms of their distinguished generators. For a polytopes ${\mathcal {P}}=\langle p_{0},p_{1},\dots ,p_{i}\rangle$ in space ${\mathcal {Q}}=\langle q_{0},q_{1},\dots ,q_{i}\rangle$ in space E , their blend is a polytope in the space $D\times E$ given by the generators $\langle p_{0}q_{0},p_{1}q_{1},\dots ,p_{i}q_{i}\rangle$.

For polytopes of differing ranks the missing generators are simply replaced with the identity.

Properties

• ${\mathcal {P}}\#{\mathcal {Q}}$ is abstractly equal to 𝓟 iff 𝓠 is a quotient of 𝓟.

See also

• Purity
• Parallel product

Bibliography

• McMullen, Peter (2007). "Four-Dimensional Regular Polyhedra" (PDF). Discrete & Computational Geometry. doi:10.1007/s00454-007-1342-7.