# Polytope compound

A polytope compound is an object consisting of an arrangement of several polytopes of the same rank.

# Definition

## Geometric

A n-compound is n polytopes of the same rank arranged in space such that they share a common center. Compounds of interest are usually arranged as to maximize symmetry.

## Abstract

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

Given two abstract polytopes A and B of the rank n their compound ${\displaystyle A + B}$ is the set:

${\displaystyle \{a\mid a\in A, 0\leq\text{rank}(a)\} \cup \{b\mid b\in B, \text{rank}(b)< n\} }$

with the operation

${\displaystyle x \leq_{A+B} y \text{ iff } \text{rank}(x)=0 \text{ or } \text{rank}(y)=n \text{ or } (x,y\in A\text{ and } x\leq_A y) \text{ or } (x,y\in B\text{ and } x\leq_B y) }$

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

# Regular compounds

Regular compounds are compounds that are transitive on every rank of element. This is the definition of weakly regular, and it is a weaker notion than regular which requires full flag transitivity.

There are 5 regular polyhedron compounds.

Regular polyhedra compounds
Name Picture Compound of
Stella octangula[note 1] 2 tetrahedra
Chiricosahedron[note 2] 5 tetrahedra
Icosicosahedron 10 tetrahedra
Rhombihedron 5 cubes
Icosicosahedron 5 octahedra

# Notes

1. Stella octangula is flag transitive.
2. Chiricosahedron is chiral and thus its pair is also a regular compound.