Honeycomb product

The honeycomb product or comb product for short, also known as the topological product[1], is one of four polytope products along with the prism, tegum and pyramid products. The honeycomb product of two euclidean honeycombs is itself an euclidean honeycomb.

Honeycomb product
Square tiling with outlined edges.svg
The square tiling is the honeycomb product of two apeirogons (outlined in cyan).
Rank formula[note 1][1]
Dimension formula
Element formula[note 2]
Algebraic properties
Algebraic structureCoummutative semigroup[note 3]
IdentityRay[note 4]
Uniquely factorizableYes[note 5][1]

The comb product of two polytopes is known as a duocomb, and a multicomb for more than two polytopes. Polygonal multicombs are regular polytopes, for example the square duocomb.


If   is an abstract polytope of rank   and   is an abstract polytope of rank  , then the honeycomb product is defined to be:[1]


with the order:



  1. For  .
  2. For  .
  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.


  1. 1.0 1.1 1.2 1.3 1.4 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.