Polytope compound

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A polytope compound is an object consisting of an arrangement of several polytopes of the same rank, known as the compound's components.

Definitions[edit | edit source]

Geometric[edit | edit source]

A n -compound is n  polytopes of the same rank arranged in space. In some but not all definitions, it is permissible for elements from different polytopes to occupy the same locations.

Abstract[edit | edit source]

A diagram showing the Hasse diagram of a hexagram with a dotted line showing a partition into two triangles. Note how no solid line crosses the dotted line.

A ranked and bounded poset is a n -compound if there exists a partition of its proper elements into n  non-empty sets such that no two elements from different sets in the partition are comparable, and each set along with the original minimal and maximal elements forms a valid abstract polytope.

An alternative formulation is that a bounded poset is a compound iff every proper section is connected but the poset itself is not connected.

The Hasse diagram of the dyad can be partitioned into two rays, but rays do not satisfy the diamond property so the dyad is not considered a compound.

This partitioning can be thought of as drawing lines on the Hasse diagram connecting the minimal and maximal elements which do not pass through any of the existing connections.

Compounds of rank greater than 1 satisfy the diamond property, but compounds of more than 1 polytope are not connected and thus they are not abstract polytopes themselves.

Compounding operation[edit | edit source]

Given two abstract polytopes A  and B  of the rank n  their compound is the set:

with the operation

If A  and B  are abstract polytopes with rank greater than 1, then the result satisfies the diamond property.

Uniform compounds[edit | edit source]

A polyhedron compound is uniform iff it is finite, isogonal, has a single edge length, and all its faces are regular polygons. Unlike the standard definition of uniform polyhedra, doubled vertices are permitted if they belong to different polyhedra in the compound, but doubled edges and doubled faces are not. Infinitely many uniform compounds can be formed by duplicating uniform prisms and antiprisms about their axes. The remaining uniform polyhedron compounds have been completely enumerated.

If doubled edges and doubled vertices (but not doubled faces) are allowed in the definition of uniform polyhedra, there are five additional uniform compounds which can also be viewed as exotic polyhedroids that violate dyadicity.

Examples[edit | edit source]

See Category:Polytope compounds.

Notes[edit | edit source]