Unique Polypents with Full Symmetry and Minimum Perimeter

Introduction

A polypent is made by joining regular pentagons edge to edge. Some polypents have full symmetry. This is the maximum symmetry for polypents, 10-fold rotary plus reflection. The number of cells in such a polypent must be a multiple of 10.

For some numbers of cells, there is a unique fully symmetric polypent with minimum perimeter, including the perimeters of the holes. Here I show such unique polypents for up to 120 cells.

Enumeration

Number of Cells	Minimum Perimeter	Number of Polypents
10	30	1
20	—	0
30	90	2
40	120	2
50	130	1
60	160	1
70	190	2
80	220	9
90	250	22
100	260	5
110	290	14
120	300	1
130	330	9
140	360	18
150	370	3
160	400	6

Polypents

10 Cells, Perimeter 30

50 Cells, Perimeter 130

60 Cells, Perimeter 160

120 Cells, Perimeter 300

Last revised 2023-10-30.

Back to Polyform Catalogues < Polyform Curiosities

Col. George Sicherman [ HOME | MAIL ]

Number of Cells	Minimum Perimeter	Number of Polypents
10	30	1
20	—	0
30	90	2
40	120	2
50	130	1
60	160	1
70	190	2
80	220	9
90	250	22
100	260	5
110	290	14
120	300	1
130	330	9
140	360	18
150	370	3
160	400	6

Number of Cells	Minimum Perimeter	Number of Polypents
10	30	1
20	—	0
30	90	2
40	120	2
50	130	1
60	160	1
70	190	2
80	220	9
90	250	22
100	260	5
110	290	14
120	300	1
130	330	9
140	360	18
150	370	3
160	400	6

Number of Cells	Minimum Perimeter	Number of Polypents
10	30	1
20	—	0
30	90	2
40	120	2
50	130	1
60	160	1
70	190	2
80	220	9
90	250	22
100	260	5
110	290	14
120	300	1
130	330	9
140	360	18
150	370	3
160	400	6