# Figure Eight Polypent Loops

## Introduction

A *polypent* is a plane figure made by joining regular pentagons
edge to edge.
A polypent is *cyclic*
if its cells are joined in a closed loop.
The cells of a polypent may not overlap one another,
so the interior of a cyclic polypent must be homeomorphic to an annulus.
For examples see Catalogue of Cyclic Polypents.

We may relax the definition of a polypent to let its cells overlap.
A relaxed polypent may be cyclic without being homeomorphic to an annulus.
In particular it may be homeomorphic to a *figure eight,*
crossing itself so that the winding number of the loop is zero.

Here I show all Figure Eight Cyclic Relaxed Polypents
with up to 12 cells.
As with cyclic strict polypents,
the number of cells must be an even number.
This is because successive cells point alternately up and down.

## Enumeration

This table counts only cyclical relaxed polypents
with winding number zero.
For example, a cycle that makes a double clockwise full turn
and a single counterclockwise full turn is not counted
as a figure eight polypent.

Cells | Figure Eight Relaxed Polypents |

8 | 2 |

10 | 2 |

12 | 25 |

14 | 189 |

16 | 1739 |

## 8 Cells

## 10 Cells

## 12 Cells

Last revised 2023-12-10.

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Col. George Sicherman
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