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 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.

CellsFigure Eight
Relaxed Polypents
82
102
1225
14189
161739

8 Cells

10 Cells

12 Cells

Last revised 2023-12-10.

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Col. George Sicherman [ HOME | MAIL ]