Polytope compound

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

= Definition =

Abstract polytopes


An bounded poset is a $n$-compound if there exists a partition of it's non-minimal and maximal 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.



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 are always valid abstract polytopes.

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

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

with the operation

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

If $A$ and $B$ are abstract polytopes with rank > 1 then the result is also an abstract polytope.

= Examples =

See Category:Polytope compounds.


 * The hexagram is a compound of two triangles.
 * The great cube is a compound of three square dihedra.
 * The dyad is a compound of two rays, but the ray is not a polytope so the dyad is not considered a compound.