# Honeycomb product

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Honeycomb product | |
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The square tiling is the honeycomb product of two apeirogons (outlined in cyan). | |

Symbol | ^{[1]} |

Rank formula | ^{[note 1]}^{[1]} |

Dimension formula | |

Element formula | ^{[note 2]} |

Dual | Self-dual |

Algebraic properties | |

Algebraic structure | Coummutative semigroup^{[note 3]} |

Associative | Yes |

Commutative | Yes |

Identity | Ray^{[note 4]} |

Annihilator | Point |

Uniquely factorizable | Yes^{[note 5]}^{[1]} |

The **honeycomb product** or **comb product** for short, also known as the **topological product**^{[1]}, is one of four common polytope products along with the prism, tegum and pyramid products. The honeycomb product of two euclidean honeycombs is itself an euclidean honeycomb. The comb product of polytopes other than honeycombs is skew.

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 polytope with itself is regular. For the comb product of two squares is the square duocomb, a regular skew polyhedron.

## Definition[edit | edit source]

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:

## Notes[edit | edit source]

- ↑ For .
- ↑ For .
- ↑ It forms a monoid on partial orders but its identity is not an abstract polytope.
- ↑ Not an abstract polytope.
- ↑ With the exception of the annihilator.

## References[edit | edit source]

- ↑
^{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.

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