# Comb product

(Redirected from Honeycomb product)

The comb product (also honeycomb product or topological product[1]), is an operation on polytopes, defined on both abstract polytopes and polytopes realized in Euclidean space. Given an input n -polytope P  and m -polytope Q  where both ranks are greater than 1, the comb product produces a new polytope ${\displaystyle P\,\square \,Q}$ of rank n + m - 1. If P is realized in Euclidean d -space and Q  in e -space, the comb product is in Euclidean (d + e)-space. The comb product generally produces skew polytopes even if both operands are planar; however, if both base polytopes are tilings, then the result is also a valid tiling. In particular, the comb product of an n -dimensional hypercubic honeycomb with an m -dimensional hypercubic honeycomb is an (n + m)-dimensional hypercubic honeycomb. This is because an n -dimensional hypercubic honeycomb can be seen as the comb product of n  apeirogons.

Comb product
The square tiling is the honeycomb product of two apeirogons (outlined in cyan).
Symbol${\displaystyle \square }$[1]
Rank formula${\displaystyle n+m-1}$[note 1][1]
Dimension formula${\displaystyle n+m}$
Element formula${\displaystyle (n-2)\times (m-2)+2}$[note 2]
DualSelf-dual
Algebraic properties
Algebraic structureCoummutative semigroup[note 3]
AssociativeYes
CommutativeYes
IdentityRay[note 4]
AnnihilatorPoint
Uniquely factorizableYes[note 5][1]

The comb product is one of four common polytope products along with the prism product, direct sum, and pyramid product. While the other three polytope products are closed over convex polytopes, the comb product's tendency to produce skew polytopes means that it is more or less useless in convex geometry. It is however important in the study of abstract polytopes, tilings, and regular skew polytopes.

The comb product of two polytopes is known as a duocomb, and a multicomb for more than two polytopes. The comb product of a regular polygon with n  copies of itself is regular. For example, the comb product of two squares is the square duocomb, a regular skew polyhedron. These results are all quotients of some hypercubic honeycomb. Other comb products of regular polytopes are not regular.

## Definition

### Abstract polytopes

If ${\displaystyle A}$  is an abstract polytope of rank n  and B  is an abstract polytope of rank m , then the comb product is defined to be:[1]

${\displaystyle A\square B=\left\{(a,b)\mid a\in A,b\in B,{\text{ either }}a{\text{ and }}b{\text{ are proper or the same improper element}}\right\}}$

with the order:

${\displaystyle (a,b)\leq _{A\square B}(a',b')\iff a\leq _{A}a'\land b\leq _{B}b'}$

If ${\displaystyle a\in A}$  and ${\displaystyle b\in B}$  are proper elements, then element ${\displaystyle (a,b)\in A\square B}$  has rank ${\displaystyle {\text{rank}}_{A}(a)+{\text{rank}}_{B}(b)}$ .

### Realizations

If A  and B  are realized in Euclidean spaces, the realization of the corresponding vertices in ${\displaystyle A\square B}$  are formed by concatenating the coordinates of vertices in A with the coordinates of vertices in B.

### Hypertopes

The definition of the comb product is considerably simpler for hypertopes. For two hypertopes A  and B  with types in the natural numbers, then the comb product is:

${\displaystyle (X_{A},*_{A},I_{A},t_{A})\square (X_{B},*_{B},I_{B},t_{B})=\left({\begin{matrix}X_{A}\times X_{B}\\(a,b)*(a',b')\iff a*_{A}a'\land b*_{B}b'\\\{a+b\mid a\in I_{A},b\in I_{B}\}\\(a,b)\mapsto t_{A}(a)+t_{B}(b)\end{matrix}}\right)}$

.

This can be thought of as the Cartesian product with the modification that it merges types whose pairs sum to the same value.

## Properties

The comb product may produce a compound if one of the input polytopes has rank 1 and the other has rank of at least 2. For example, the comb product of a polygon with a dyad is two disconnected copies of the polygon. To avoid this issue, Gleason and Hubard define the comb product only when the input polytopes are of rank 2 or greater.

The comb product is commutative and associative, and admits a unique factorization for every polytope of rank 2 or greater into the comb products of "prime" polytopes. If lower-rank polytopes are allowed, then both the point and the nullitope are annihilators.

If ${\displaystyle ^{*}}$  denotes the dual of an abstract polytope, then ${\displaystyle P\square Q=(P^{*}\square Q^{*})^{*}}$ .

## Notes

1. For ${\displaystyle n,m>1}$ .
2. For ${\displaystyle n,m>1}$ .
3. It forms a monoid on partial orders but its identity is not an abstract polytope.
4. Not an abstract polytope.
5. With the exception of the annihilator.

## References

1. Gleason, Ian; Hubard, Isabel (2018). "Products of abstract polytopes" (PDF). Journal of Combinatorial Theory, Series A. 157: 287–320. doi:10.1016/j.jcta.2018.02.002.