Blending



Blending is a polytope operation in which two or more polytopes of the same rank are overlapped onto one another in such a way that some of their facets coincide. In general, this means that some of the elements of a polytope's facet (most importantly the vertices) can all be mapped to the corresponding elements of the other polytope's facet. The polytopes undergoing the blend are positioned in such a way that each pair of facets that are mapped together, and each pair of corresponding elements within them, are in the same location. These facets are said to coincide. Coinciding facets that are congruent are removed, and coinciding and noncongruent facets are themselves blended. Elements of these removed facets that are in the same place are merged together. In most cases, the diamond property is maintained, because the neighbors of one polytope's removed facet become joined at a ridge with those of the other polytope.

Blending 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.

The term "internal blend" or "inner blend" may be used when the centers of the blended-together polytopes are on the same side of the hyperplane that contained the removed facet. When the polytopes' centers are on opposite sides, it may be called an "external blend" or "outer blend."

Augmentation
In the context of the Johnson solids, several specific varieties of blending are given names. Key among them is augmentation. For two polytopes, this involves a coincident pair of congruent facets being removed, as they end up inside the resultant polytope. Coincident edges, vertices, and other subdimensional elements are 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. Consider the diminished icosahedron as a blend of an icosahedron and pentagonal pyramid. All of the pyramid's triangular faces become coincident with the icosahedron's faces and are thus removed (note that some sub-dimensional elements must be removed in addition to the coincident facets, unlike in augmentation), leaving only one of the pyramid's facets behind: its pentagon base. Diminishing can be thought of as the "subtraction" to augmenting's "addition".

An augmentation performed in the opposite direction is sometimes referred to as an "excavation", as seen in the excavated dodecahedron. 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, as seen in the trireplenished great icosahedron. Excavations and diminishings would technically be classified as inner blends, because both of the used polytopes' centers lie on the same side of the coincident facet's hyperplane.