Tetrahex-Pentahex Pair Hex-Convex Shapes

Introduction

A polyhex is said to be hex-convex if every line joining the centers of two of its cells lies in its interior. Here are the smallest known hex-convex polyhexes that can be formed with copies of a tetrahex and a pentahex, using at least one of each.

Nomenclature

Tetrahexes

Pentahexes

Table of Results

Each figure shows the number of cells in the corresponding tiling.

	5A	5C	5D	5E	5F	5H	5I	5J	5K	5L	5N	5P	5Q	5R	5S	5T	5U	5V	5W	5X	5Y	5Z
4I	34	60	9	19	78	75	9	42	26	9	19	14	42	18	52	110	22	23	66	56	14	40
4J	17	26	9	21	21	21	13	13	17	13	14	9	18	21	13	30	13	17	18	18	9	17
4O	9	14	9	13	78	213	14	13	18	13	9	14	18	13	13	108	18	9	—	13	9	141
4Q	9	17	9	13	13	21	17	9	21	13	9	9	17	18	21	29	13	18	18	17	13	18
4S	18	18	39	24	58	—	54	22	—	18	18	14	—	70	—	—	18	30	—	—	24	—
4U	17	18	13	9	18	26	31	18	18	19	13	14	18	26	69	60	13	27	114	13	14	28
4Y	182	—	54	—	—	—	—	18	—	44	19	19	—	—	—	—	19	—	—	—	28	—

Navigation

[9 Cells] [13 Cells] [14 Cells] [17 Cells] [18 Cells] [19 Cells] [21 Cells] [22 Cells] [23 Cells] [24 Cells]
[26 Cells] [27 Cells] [28 Cells] [29 Cells] [30 Cells] [31 Cells] [34 Cells] [39 Cells] [40 Cells] [42 Cells]
[44 Cells] [52 Cells] [54 Cells] [56 Cells] [58 Cells] [60 Cells] [66 Cells] [69 Cells] [70 Cells]
[75 Cells] [78 Cells] [108 Cells] [110 Cells] [114 Cells] [141 Cells] [182 Cells] [213 Cells]

Solutions

These minimal known solutions are not necessarily unique.

9 Cells

13 Cells

14 Cells

17 Cells

18 Cells

19 Cells

21 Cells

22 Cells

23 Cells

24 Cells

26 Cells

27 Cells

28 Cells

29 Cells

30 Cells

31 Cells

34 Cells

39 Cells

40 Cells

42 Cells

44 Cells

52 Cells

54 Cells

56 Cells

58 Cells

60 Cells

66 Cells

69 Cells

70 Cells

75 Cells

78 Cells

108 Cells

110 Cells

114 Cells

141 Cells

182 Cells

213 Cells

Last revised 2025-08-30.

Back to Polyhex Tiling < Polyform Tiling < Polyform Curiosities

Col. George Sicherman [ HOME | MAIL ]

	5A	5C	5D	5E	5F	5H	5I	5J	5K	5L	5N	5P	5Q	5R	5S	5T	5U	5V	5W	5X	5Y	5Z
4I	34	60	9	19	78	75	9	42	26	9	19	14	42	18	52	110	22	23	66	56	14	40
4J	17	26	9	21	21	21	13	13	17	13	14	9	18	21	13	30	13	17	18	18	9	17
4O	9	14	9	13	78	213	14	13	18	13	9	14	18	13	13	108	18	9	—	13	9	141
4Q	9	17	9	13	13	21	17	9	21	13	9	9	17	18	21	29	13	18	18	17	13	18
4S	18	18	39	24	58	—	54	22	—	18	18	14	—	70	—	—	18	30	—	—	24	—
4U	17	18	13	9	18	26	31	18	18	19	13	14	18	26	69	60	13	27	114	13	14	28
4Y	182	—	54	—	—	—	—	18	—	44	19	19	—	—	—	—	19	—	—	—	28	—

	5A	5C	5D	5E	5F	5H	5I	5J	5K	5L	5N	5P	5Q	5R	5S	5T	5U	5V	5W	5X	5Y	5Z
4I	34	60	9	19	78	75	9	42	26	9	19	14	42	18	52	110	22	23	66	56	14	40
4J	17	26	9	21	21	21	13	13	17	13	14	9	18	21	13	30	13	17	18	18	9	17
4O	9	14	9	13	78	213	14	13	18	13	9	14	18	13	13	108	18	9	—	13	9	141
4Q	9	17	9	13	13	21	17	9	21	13	9	9	17	18	21	29	13	18	18	17	13	18
4S	18	18	39	24	58	—	54	22	—	18	18	14	—	70	—	—	18	30	—	—	24	—
4U	17	18	13	9	18	26	31	18	18	19	13	14	18	26	69	60	13	27	114	13	14	28
4Y	182	—	54	—	—	—	—	18	—	44	19	19	—	—	—	—	19	—	—	—	28	—

	5A	5C	5D	5E	5F	5H	5I	5J	5K	5L	5N	5P	5Q	5R	5S	5T	5U	5V	5W	5X	5Y	5Z
4I	34	60	9	19	78	75	9	42	26	9	19	14	42	18	52	110	22	23	66	56	14	40
4J	17	26	9	21	21	21	13	13	17	13	14	9	18	21	13	30	13	17	18	18	9	17
4O	9	14	9	13	78	213	14	13	18	13	9	14	18	13	13	108	18	9	—	13	9	141
4Q	9	17	9	13	13	21	17	9	21	13	9	9	17	18	21	29	13	18	18	17	13	18
4S	18	18	39	24	58	—	54	22	—	18	18	14	—	70	—	—	18	30	—	—	24	—
4U	17	18	13	9	18	26	31	18	18	19	13	14	18	26	69	60	13	27	114	13	14	28
4Y	182	—	54	—	—	—	—	18	—	44	19	19	—	—	—	—	19	—	—	—	28	—