# Demiregular tiling

In geometry, the demiregular tilings are a set of Euclidean tessellations made from 2 or more regular polygon faces. Different authors have listed different sets of tilings. A more systematic approach looking at symmetry orbits are the 2-uniform tilings of which there are 20. Some of the demiregular ones are actually 3-uniform tilings.

## 20 2-uniform tilings

Grünbaum and Shephard enumerated the full list of 20 2-uniform tilings in Tilings and patterns, 1987:
2-uniform tilings :
| cmm, 2*22 (4$^4$; 3$^3$.4$^2$)$_1$ | cmm, 2*22 (4$^4$; 3$^3$.4$^2$)$_2$ | pmm, *2222 (3$^6$; 3$^3$.4$^2$)$_1$ | cmm, 2*22 (3$^6$; 3$^3$.4$^2$)$_2$ | cmm, 2*22 (3.4$^2$.6; (3.6)$^2$)$_2$ | pmm, *2222 (3.4$^2$.6; (3.6)$^2$)$_1$ | pmm, *2222 ((3.6)$^2$; 3$^2$.6$^2$) |
| p4m, *442 (3.12.12; 3.4.3.12) | p4g, 4*2 (3$^3$.4$^2$; 3$^2$.4.3.4)$_1$ | pgg, 2× (3$^3$.4$^2$; 3$^2$.4.3.4)$_2$ | p6m, *632 (3$^6$; 3$^2$.6$^2$) | p6m, *632 (3$^6$; 3$^4$.6)$_1$ | p6, 632 (3$^6$; 3$^4$.6)$_2$ | cmm, 2*22 (3$^2$.6$^2$; 3$^4$.6) |
| p6m, *632 (3$^6$; 3$^2$.4.3.4) | p6m, *632 (3.4.6.4; 3$^2$.4.3.4) | p6m, *632 (3.4.6.4; 3$^3$.4$^2$) | p6m, *632 (3.4.6.4; 3.4$^2$.6) | p6m, *632 (4.6.12; 3.4.6.4) | p6m, *632 (3$^6$; 3$^2$.4.12) |

## Ghyka's list (1946)

Ghyka lists 10 of them with 2 or 3 vertex types, calling them semiregular polymorph partitions.
| Plate XXVII No. 12 4.6.12 3.4.6.4 | No. 13 3.4.6.4 3.3.3.4.4 | No. 13 bis. 3.4.4.6 3.3.4.3.4 | No. 13 ter. 3.4.4.6 3.3.3.4.4 | Plate XXIV No. 13 quatuor. 3.4.6.4 3.3.4.3.4 |
| No. 14 3$^3$.4$^2$ 3$^6$ | Plate XXVI No. 14 bis. 3.3.4.3.4 3.3.3.4.4 3$^6$ | No. 14 ter. 3$^3$.4$^2$ 3$^6$ | No. 15 3.3.4.12 3$^6$ | Plate XXV No. 16 3.3.4.12 3.3.4.3.4 3$^6$ |

## Steinhaus's list (1969)

Steinhaus gives 5 examples of non-homogeneous tessellations of regular polygons beyond the 11 regular and semiregular ones. (All of them have 2 types of vertices, while one is 3-uniform.)
| 2-uniform | 2-uniform | 2-uniform | 2-uniform | 3-uniform |
| Image 85 3$^3$.4$^2$ 3.4.6.4 | Image 86 3$^2$.4.3.4 3.4.6.4 | Image 87 3.3.4.12 3$^6$ | Image 89 3$^3$.4$^2$ 3$^2$.4.3.4 | Image 88 3.12.12 3.3.4.12 |

## Critchlow's list (1970)

Critchlow identifies 14 demi-regular tessellations, with 7 being 2-uniform, and 7 being 3-uniform.
He codes letter names for the vertex types, with superscripts to distinguish face orders. He recognizes A, B, C, D, F, and J can't be a part of continuous coverings of the whole plane.
| A (none) | B (none) | C (none) | D (none) | E (semi) | F (none) | G (semi) | H (semi) | J (none) | K (2) (reg) |
| 3.7.42 | 3.8.24 | 3.9.18 | 3.10.15 | 3.12.12 | 4.5.20 | 4.6.12 | 4.8.8 | 5.5.10 | 6$^3$ |
| L1 (demi) | L2 (demi) | M1 (demi) | M2 (semi) | N1 (demi) | N2 (semi) | P (3) (reg) | Q1 (semi) | Q2 (semi) | R (semi) | S (1) (reg) |
| 3.3.4.12 | 3.4.3.12 | 3.3.6.6 | 3.6.3.6 | 3.4.4.6 | 3.4.6.4 | 4$^4$ | 3.3.4.3.4 | 3.3.3.4.4 | 3.3.3.3.6 | 3$^6$ |
2-uniforms :
| 1 | 2 | 4 | 6 | 7 | 10 | 14 |
| (3.12.12; 3.4.3.12) | (3$^6$; 3$^2$.4.12) | (4.6.12; 3.4.6.4) | ((3.6)$^2$; 3$^2$.6$^2$) | (3.4.6.4; 3$^2$.4.3.4) | (3$^6$; 3$^2$.4.3.4) | (3.4.6.4; 3.4$^2$.6) |
| E+L2 | L1+(1) | N1+G | M1+M2 | N2+Q1 | Q1+(1) | N1+Q2 |
3-uniforms :
| 3 | 5 | 8 | 9 | 11 | 12 | 13 |
| (3.3.4.3.4; 3.3.4.12, 3.4.3.12) | (3$^6$; 3.3.4.12; 3.3.4.3.4) | (3.3.4.3.4; 3.3.3.4.4, 4.3.4.6) | (3$^6$, 3.3.4.3.4) | (3$^6$; 3.3.4.3.4, 3.3.3.4.4) | (3$^6$; 3.3.4.3.4; 3.3.3.4.4) | (3.4.6.4; 3.4$^2$.6) |
| L1+L2+Q1 | L1+Q1+(1) | N1+Q1+Q2 | Q1+(1) | Q1+Q2+(1) | Q1+Q2+(1) | N1+N2 |
| Claimed Tilings and Duals |||||||