From Polytope Wiki
Jump to navigation Jump to search

Blending is an operation where two or more polytopes of the same rank are overlaid, coinciding and congruent facets are removed, and coinciding and noncongruent facets are themselves blended. It is similar to the creation of compounds, but in compounds every facet remains, possibly producing multiple-covers. If no facets are shared between the blended polytopes, a compound is formed.

Some allow blends to affect all elements, not just facets and elements adjacent to them, such as two polyhedra merged at a single vertex.

In general, blends produce polytopoids which may be exotic. For example, the great dirhombicosidodecahedron and the disnub icosahedron (compound of 20 octahedra) blend to form the great disnub dirhombidodecahedron, which has four faces meeting at some edges.

Blending example
Polytopes Resultant blend
3 octagonal prisms oriented differently Small rhombihexahedron
Octagonal prism.png Small rhombihexahedron.png
Six pairs of coincident square faces are removed.

Coincident edges and vertices (belonging to those squares) are merged.

(The octagonal faces intersect one another, but do not coincide.)

In the context of the Johnson solids, a similar operation is referred to as "augmenting." This usually involves a coincident pair of facets being removed, as they end up inside the resultant polytope, and coincident edges and vertices being merged since they lie on the polytope's surface. Diminishing can also be considered a type of blend, in which all but one facet of one polytope is removed.

Blending "augmentation-style"
Polytope 1 Polytope 2 Resultant blend
Dodecahedron Pentagonal pyramid Augmented dodecahedron
Dodecahedron.jpg Pentagonal pyramid.png Augmented dodecahedron.png
The coincident pentagonal faces end up on the inside and are removed.

Ten pairs of coincident edges and vertices end up on the outside and are merged.

An augmentation performed in the opposite direction is sometimes referred to as an "excavation". In layman's terms, the difference is that an augmentation "adds material" while an excavation "subtracts" it. A diminishment that adds material is called a replenishment.