Tiling catalogue

Generated page. Built by docs/tools/gen-tiling-catalogue.py from the program’s own tiling lists and data files; thumbnails are drawn by JETCARP (docs/tools/tiling-thumbs.rhai). Do not edit by hand.

Every tiling JETCARP can run, with its neighbour counts. That is 25 tiling families (some of them reshapeable), 81 isohedral types, and the 1248 k-uniform tilings of Galebach’s catalogue, which include the regular and Archimedean tilings again under their catalogue ids.

How to read the degrees. A tile’s edge neighbours share a side with it; its touch neighbours share at least one corner (see Tilings as neighbour graphs). The lists give every value that occurs in the interior of the tiling. Rules that count neighbours (Life-like, Generations, JACA and others) see these numbers, which is why the same rule behaves so differently from one tiling to the next.

Thumbnails show a small patch with each tile in a random colour so the shapes stand out. Reshapeable families and isohedral types are drawn with their default shapes.

Tiling families

These are the families you pick by name (tiling: {"family": "cairo"} in a scene). Symmetry is the wallpaper group of the default shape (none for the aperiodic tilings). Storage is how JETCARP holds the cells: dense for a plain lattice, periodic for a repeating unit cell, explicit for a finite aperiodic patch.

	Id	Name	Symmetry	Storage	Edge degrees	Touch degrees
	lizard	Three-fold hexagon	p3	periodic	6	6
	hex	Hexagon	p1	dense	6	6
	brick	Brick	p1	dense	6	6
	square	Square (parallelogram)	p1	dense	4	8
	quad	Any quadrilateral	p2	periodic	4	8
	triangle	Any triangle	p2	periodic	3	12
	glide	Glide rectangle	pg	periodic	4	8
	fourfold	Four-fold square	p4	periodic	4	8
	cairo	Cairo pentagon	p4	periodic	5	7
	spectre	Spectre	none	explicit	4, 5, 6, 7	6, 7
	hat	Hat	none	explicit	4, 5, 6	6, 7
	penrose-kd	Penrose kites and darts (P2)	none	explicit	4	8, 9, 10
	penrose-rhomb	Penrose rhombs (P3)	none	explicit	4	7, 8, 9, 10, 11
	hex_flat	Hexagon, flat top	p6m	periodic	6	6
	oct	4.8.8 truncated square	p4m	periodic	4, 8	4, 8
	trihex	3.6.3.6 trihexagonal	p6m	periodic	3, 6	6, 12
	elongtri	3.3.3.4.4 elongated triangular	cmm	periodic	3, 4	9, 12
	snubsq	3.3.4.3.4 snub square	p4g	periodic	3, 4	9, 12
	snubhex	3.3.3.3.6 snub hexagonal	p6	periodic	3, 6	9, 18
	rhombitrihex	3.4.6.4 rhombitrihexagonal	p6m	periodic	3, 4, 6	6, 8, 12
	trunchex	3.12.12 truncated hexagonal	p6m	periodic	3, 12	3, 12
	trunctrihex	4.6.12 truncated trihexagonal	p6m	periodic	4, 6, 12	4, 6, 12
	periodic	Periodic (custom or catalog)	any	periodic	3, 4, 6	6, 8, 12
	herring	Herringbone	pgg	periodic	6	6
	pyth	Pythagorean squares	p4	periodic	4, 8	4, 8

Pages: Regular and Archimedean tilings, Editable tile families, Penrose tilings, The hat and the spectre.

Isohedral types

The 81 isohedral types that Craig Kaplan’s Tactile library supports (IH01 to IH93 in Grünbaum and Shephard’s numbering, less the 12 types Tactile does not support). Parameters is the number of shape sliders; aspects is the number of differently oriented copies of the tile in one unit cell. See Isohedral tilings.

	Id	Symmetry	Parameters	Aspects	Edge degrees	Touch degrees
	ih01	p1	4	1	6	6
	ih02	pg	4	2	6	6
	ih03	pg	4	2	6	6
	ih04	p2	6	2	6	6
	ih05	pgg	5	4	6	6
	ih06	pgg	5	4	6	6
	ih07	p3	2	3	6	6
	ih08	p2	4	1	6	6
	ih09	pgg	3	2	6	6
	ih10	p3	0	1	6	6
	ih11	p6	0	1	6	6
	ih12	cm	2	1	6	6
	ih13	pmg	3	2	6	6
	ih14	cm	2	1	6	6
	ih15	pmg	3	2	6	6
	ih16	p31m	1	3	6	6
	ih17	cmm	2	1	6	6
	ih18	p31m	0	1	6	6
	ih20	p6m	0	1	6	6
	ih21	p6	2	6	5	8
	ih22	cm	3	2	5	7
	ih23	p2	4	2	5	7
	ih24	pmg	4	4	5	7
	ih25	pgg	3	4	5	7
	ih26	cmm	2	2	5	7
	ih27	pgg	3	4	5	7
	ih28	p4	2	4	5	7
	ih29	p4g	1	4	5	7
	ih30	p31m	1	6	4	9
	ih31	p6	0	6	4	9
	ih32	p6m	0	6	4	9
	ih33	p3	0	3	4	10
	ih34	p6	0	3	4	10
	ih36	p31m	0	3	4	10
	ih37	p6m	0	3	4	10
	ih38	p31m	0	6	3	21
	ih39	p6	0	6	3	21
	ih40	p6m	0	6	3	21
	ih41	p1	2	1	4	8
	ih42	pm	2	2	4	8
	ih43	pg	2	2	4	8
	ih44	pg	2	2	4	8
	ih45	cm	2	2	4	8
	ih46	p2	4	2	4	8
	ih47	p2	2	2	4	8
	ih49	pmg	3	4	4	8
	ih50	pmg	2	4	4	8
	ih51	pgg	3	4	4	8
	ih52	pgg	1	4	4	8
	ih53	pgg	3	4	4	8
	ih54	cmm	2	4	4	8
	ih55	p4	0	4	4	8
	ih56	p4g	1	8	4	8
	ih57	p2	2	1	4	8
	ih58	pmg	2	2	4	8
	ih59	pgg	1	2	4	8
	ih61	p4	0	2	4	8
	ih62	p4	0	1	4	8
	ih64	pm	1	1	4	8
	ih66	pmg	1	2	4	8
	ih67	cmm	2	2	4	8
	ih68	cm	1	1	4	8
	ih69	pmg	2	2	4	8
	ih71	p4g	0	4	4	8
	ih72	pmm	1	1	4	8
	ih73	p4g	0	2	4	8
	ih74	cmm	1	1	4	8
	ih76	p4m	0	1	4	8
	ih77	p6m	0	12	3	16
	ih78	cmm	1	4	3	14
	ih79	p4	0	4	3	14
	ih81	p4g	0	8	3	14
	ih82	p4m	0	4	3	14
	ih83	cm	1	2	3	12
	ih84	p2	2	2	3	12
	ih85	pmg	2	4	3	12
	ih86	pgg	1	4	3	12
	ih88	p6	0	6	3	12
	ih90	p6	0	2	3	12
	ih91	cmm	1	2	3	12
	ih93	p6m	0	2	3	12

k-uniform tilings, k = 1 to 3

Edge-to-edge tilings by regular polygons with k kinds of vertex (orbits under the tiling’s symmetries), from Brian Galebach’s catalogue in José Ezequiel Soto Sánchez’s encoding (CC BY 4.0). The name lists the vertex type of each orbit. Tiles is the number of tiles in one unit cell; mix is the share of each polygon; mean touch is the average touch degree over the tiles. k = 4 to 6 are on their own page. See k-uniform tilings.

Counts: k = 1: 11, k = 2: 20, k = 3: 61, k = 4: 151, k = 5: 332, k = 6: 673.

k = 1 (11 tilings)

	Id	Vertex types	Symmetry	Tiles	Mix	Edge degrees	Touch degrees	Mean touch
	k1-001	6^3 hexagonal	p6m	1	hexagons 100%	6	6	6.00
	k1-002	4.8^2 truncated square	p4m	2	squares 50%, octagons 50%	4, 8	4, 8	6.00
	k1-003	4.6.12 truncated trihexagonal	p6m	6	squares 50%, hexagons 33%, dodecagons 17%	4, 6, 12	4, 6, 12	6.00
	k1-004	3.12^2 truncated hexagonal	p6m	3	triangles 67%, dodecagons 33%	3, 12	3, 12	6.00
	k1-005	4^4 square	p4m	1	squares 100%	4	8	8.00
	k1-006	3.4.6.4 rhombitrihexagonal	p6m	6	triangles 33%, squares 50%, hexagons 17%	3, 4, 6	6, 8, 12	8.00
	k1-007	3.6.3.6 trihexagonal	p6m	3	triangles 67%, hexagons 33%	3, 6	6, 12	8.00
	k1-008	3^3.4^2 elongated triangular	cmm	3	triangles 67%, squares 33%	3, 4	9, 12	10.00
	k1-009	3^2.4.3.4 snub square	p4g	6	triangles 67%, squares 33%	3, 4	9, 12	10.00
	k1-010	3^4.6 snub hexagonal	p6	9	triangles 89%, hexagons 11%	3, 6	9, 18	10.00
	k1-011	3^6 triangular	p6m	2	triangles 100%	3	12	12.00

k = 2 (20 tilings)

	Id	Vertex types	Symmetry	Tiles	Mix	Edge degrees	Touch degrees	Mean touch
	k2-001	[3.4.6.4; 4.6.12]	p6m	12	triangles 17%, squares 50%, hexagons 25%, dodecagons 8%	3, 4, 6, 12	6, 8, 12	7.00
	k2-002	[3.4.3.12; 3.12^2]	p4m	6	triangles 67%, squares 17%, dodecagons 17%	3, 4, 12	5, 8, 16	7.33
	k2-003	[3^3.4^2; 4^4] #1	cmm	5	triangles 40%, squares 60%	3, 4	8, 9, 10	9.20
	k2-004	[3^3.4^2; 4^4] #2	cmm	4	triangles 50%, squares 50%	3, 4	9, 10	9.50
	k2-005	[3.4^2.6; 3.4.6.4]	p6m	18	triangles 33%, squares 50%, hexagons 17%	3, 4, 6	6, 8, 12	8.00
	k2-006	[3.4^2.6; 3.6.3.6] #1	cmm	5	triangles 40%, squares 40%, hexagons 20%	3, 4, 6	6, 8, 12	8.00
	k2-007	[3.4^2.6; 3.6.3.6] #2	pmm	5	triangles 40%, squares 40%, hexagons 20%	3, 4, 6	6, 8, 12	8.00
	k2-008	[3^3.4^2; 3.4.6.4]	p6m	15	triangles 53%, squares 40%, hexagons 7%	3, 4, 6	8, 9, 10, 12	9.20
	k2-009	[3^2.4.3.4; 3.4.6.4]	p6m	15	triangles 53%, squares 40%, hexagons 7%	3, 4, 6	8, 9, 10, 12	9.20
	k2-010	[3^6; 3^2.6^2]	p6m	8	triangles 75%, hexagons 25%	3, 6	8, 12	9.00
	k2-011	[3^2.6^2; 3.6.3.6]	pmm	3	triangles 67%, hexagons 33%	3, 6	6, 12	8.00
	k2-012	[3^4.6; 3^2.6^2]	cmm	5	triangles 80%, hexagons 20%	3, 6	8, 14	9.20
	k2-013	[3^6; 3^2.4.12]	p6m	16	triangles 75%, squares 19%, dodecagons 6%	3, 4, 12	8, 24	9.00
	k2-014	[3^6; 3^3.4^2] #1	pmm	5	triangles 80%, squares 20%	3, 4	10, 11, 12	10.80
	k2-015	[3^6; 3^3.4^2] #2	cmm	7	triangles 86%, squares 14%	3, 4	10, 11, 12	11.14
	k2-016	[3^3.4^2; 3^2.4.3.4] #1	p4g	18	triangles 67%, squares 33%	3, 4	9, 12	10.00
	k2-017	[3^3.4^2; 3^2.4.3.4] #2	pgg	12	triangles 67%, squares 33%	3, 4	9, 12	10.00
	k2-018	[3^6; 3^2.4.3.4]	p6m	11	triangles 73%, squares 27%	3, 4	9, 10, 12	10.36
	k2-019	[3^6; 3^4.6] #1	p6m	13	triangles 92%, hexagons 8%	3, 6	10, 18	10.62
	k2-020	[3^6; 3^4.6] #2	p6	21	triangles 95%, hexagons 5%	3, 6	10, 11, 12, 18	11.14

k = 3 (61 tilings)

	Id	Vertex types	Symmetry	Tiles	Mix	Edge degrees	Touch degrees	Mean touch
	k3-001	[3^2.6^2; 3.6.3.6; 6^3] #1	p6m	15	triangles 53%, hexagons 47%	3, 6	6, 10	7.60
	k3-002	[3^2.6^2; 3.6.3.6; 6^3] #2	cmm	4	triangles 50%, hexagons 50%	3, 6	6, 9	7.50
	k3-003	[3^4.6; 3^2.6^2; 6^3]	pmg	6	triangles 67%, hexagons 33%	3, 6	8, 10	8.67
	k3-004	[3^6; 3^2.6^2; 6^3]	p6m	9	triangles 67%, hexagons 33%	3, 6	8, 10	8.67
	k3-005	[3.4^2.6; 3.6.3.6; 4.6.12]	p6m	24	triangles 33%, squares 38%, hexagons 25%, dodecagons 4%	3, 4, 6, 12	6, 8, 10, 12	7.50
	k3-006	[3^6; 3^2.4.12; 4.6.12]	p3m1	11	triangles 55%, squares 27%, hexagons 9%, dodecagons 9%	3, 4, 6, 12	6, 8, 18	8.18
	k3-007	[3^2.4.12; 3.4.6.4; 3.12^2]	p6m	21	triangles 57%, squares 29%, hexagons 5%, dodecagons 10%	3, 4, 6, 12	5, 6, 8, 12, 18	7.71
	k3-008	[3.4.3.12; 3.4.6.4; 3.12^2]	p6m	21	triangles 57%, squares 29%, hexagons 5%, dodecagons 10%	3, 4, 6, 12	5, 6, 8, 12, 18	7.71
	k3-009	[3^2.4.12; 3.4.3.12; 3.12^2]	cmm	9	triangles 67%, squares 22%, dodecagons 11%	3, 4, 12	5, 6, 8, 20	7.78
	k3-010	[3^3.4^2; 4^4; 4^4] #1	cmm	7	triangles 29%, squares 71%	3, 4	8, 9, 10	8.86
	k3-011	[3.4^2.6; 3.6.3.6; 4^4] #1	cmm	9	triangles 22%, squares 67%, hexagons 11%	3, 4, 6	6, 8, 12	8.00
	k3-012	[3.4^2.6; 3.6.3.6; 4^4] #2	pmm	9	triangles 22%, squares 67%, hexagons 11%	3, 4, 6	6, 8, 12	8.00
	k3-013	[3^3.4^2; 3^3.4^2; 4^4] #1	pmm	8	triangles 50%, squares 50%	3, 4	8, 9, 10, 12	9.50
	k3-014	[3^6; 3^3.4^2; 4^4] #1	pmm	7	triangles 57%, squares 43%	3, 4	8, 10, 11	10.00
	k3-015	[3^6; 3^3.4^2; 4^4] #2	cmm	9	triangles 67%, squares 33%	3, 4	8, 10, 11, 12	10.44
	k3-016	[3^3.4^2; 4^4; 4^4] #2	cmm	6	triangles 33%, squares 67%	3, 4	8, 9, 10	9.00
	k3-017	[3.4^2.6; 3.6.3.6; 4^4] #3	cmm	7	triangles 29%, squares 57%, hexagons 14%	3, 4, 6	6, 8, 12	8.00
	k3-018	[3.4^2.6; 3.6.3.6; 4^4] #4	pmm	7	triangles 29%, squares 57%, hexagons 14%	3, 4, 6	6, 8, 12	8.00
	k3-019	[3^3.4^2; 3^3.4^2; 4^4] #2	pmm	7	triangles 57%, squares 43%	3, 4	9, 10, 12	9.71
	k3-020	[3^6; 3^3.4^2; 4^4] #3	pmm	6	triangles 67%, squares 33%	3, 4	10, 11	10.33
	k3-021	[3^6; 3^3.4^2; 4^4] #4	cmm	8	triangles 75%, squares 25%	3, 4	10, 11, 12	10.75
	k3-022	[3.4^2.6; 3.4.6.4; 4^4]	p4g	14	triangles 29%, squares 57%, hexagons 14%	3, 4, 6	6, 8, 12	8.00
	k3-023	[3^3.4^2; 3^2.4.3.4; 4^4]	p4	13	triangles 62%, squares 38%	3, 4	9, 11, 12	9.85
	k3-024	[3^6; 3^2.4.3.4; 3.4^2.6]	p6m	23	triangles 52%, squares 39%, hexagons 9%	3, 4, 6	7, 8, 10, 12	9.13
	k3-025	[3^2.4.3.4; 3.4^2.6; 3.4.6.4]	cmm	9	triangles 44%, squares 44%, hexagons 11%	3, 4, 6	7, 8, 9, 12	8.67
	k3-026	[3.4^2.6; 3.4.6.4; 3.4.6.4]	p6m	24	triangles 33%, squares 50%, hexagons 17%	3, 4, 6	6, 8, 12	8.00
	k3-027	[3^3.4^2; 3^2.6^2; 3.4^2.6]	cmm	11	triangles 55%, squares 27%, hexagons 18%	3, 4, 6	7, 8, 10, 12	8.55
	k3-028	[3.4^2.6; 3.4^2.6; 3.6.3.6]	cmm	10	triangles 40%, squares 40%, hexagons 20%	3, 4, 6	6, 8, 12	8.00
	k3-029	[3^2.6^2; 3.4^2.6; 3.6.3.6] #1	cmm	8	triangles 50%, squares 25%, hexagons 25%	3, 4, 6	6, 8, 12	8.00
	k3-030	[3.4^2.6; 3.6.3.6; 3.6.3.6] #1	pmg	8	triangles 50%, squares 25%, hexagons 25%	3, 4, 6	6, 8, 12	8.00
	k3-031	[3^2.6^2; 3.4^2.6; 3.6.3.6] #2	pmm	8	triangles 50%, squares 25%, hexagons 25%	3, 4, 6	6, 8, 12	8.00
	k3-032	[3.4^2.6; 3.6.3.6; 3.6.3.6] #2	cmm	8	triangles 50%, squares 25%, hexagons 25%	3, 4, 6	6, 8, 12	8.00
	k3-033	[3^4.6; 3^3.4^2; 3.4^2.6]	pmm	10	triangles 60%, squares 30%, hexagons 10%	3, 4, 6	8, 9, 10, 14	9.20
	k3-034	[3^6; 3^2.4.3.4; 3.4.6.4] #1	p6m	23	triangles 52%, squares 39%, hexagons 9%	3, 4, 6	7, 8, 10, 12	9.13
	k3-035	[3^3.4^2; 3^2.4.12; 3.4.6.4]	p6m	33	triangles 55%, squares 36%, hexagons 6%, dodecagons 3%	3, 4, 6, 12	7, 8, 10, 12, 24	8.55
	k3-036	[3^6; 3^3.4^2; 3.4.6.4]	p6m	28	triangles 64%, squares 32%, hexagons 4%	3, 4, 6	8, 10, 12	9.86
	k3-037	[3^6; 3^2.4.3.4; 3.4.6.4] #2	p6m	28	triangles 64%, squares 32%, hexagons 4%	3, 4, 6	8, 10, 12	9.86
	k3-038	[3^6; 3^4.6; 3^2.6^2] #1	pmg	14	triangles 86%, hexagons 14%	3, 6	8, 9, 10, 14	10.00
	k3-039	[3^2.6^2; 3.6.3.6; 3.6.3.6]	cmm	6	triangles 67%, hexagons 33%	3, 6	6, 12	8.00
	k3-040	[3^6; 3^4.6; 3^2.6^2] #2	pmm	7	triangles 86%, hexagons 14%	3, 6	9, 10, 14	10.00
	k3-041	[3^6; 3^4.6; 3^2.6^2] #3	cmm	9	triangles 89%, hexagons 11%	3, 6	9, 11, 14	10.44
	k3-042	[3^6; 3^4.6; 3.6.3.6] #1	p6m	17	triangles 82%, hexagons 18%	3, 6	6, 8, 10, 14	9.53
	k3-043	[3^6; 3^4.6; 3.6.3.6] #2	p6m	14	triangles 86%, hexagons 14%	3, 6	8, 10, 15	9.86
	k3-044	[3^6; 3^4.6; 3.6.3.6] #3	cmm	11	triangles 91%, hexagons 9%	3, 6	8, 10, 11, 16	10.55
	k3-045	[3^4.6; 3^4.6; 3.6.3.6] #1	pmg	14	triangles 86%, hexagons 14%	3, 6	8, 9, 16	9.71
	k3-046	[3^4.6; 3^4.6; 3.6.3.6] #2	p2	7	triangles 86%, hexagons 14%	3, 6	8, 9, 16	9.71
	k3-047	[3^6; 3^3.4^2; 3^2.4.12]	p6m	33	triangles 79%, squares 18%, dodecagons 3%	3, 4, 12	8, 9, 10, 11, 12, 24	10.00
	k3-048	[3^6; 3^2.4.3.4; 3^2.4.12]	p6m	27	triangles 74%, squares 22%, dodecagons 4%	3, 4, 12	8, 9, 10, 24	9.56
	k3-049	[3^6; 3^6; 3^3.4^2] #1	pmm	9	triangles 89%, squares 11%	3, 4	10, 11, 12	11.33
	k3-050	[3^6; 3^6; 3^3.4^2] #2	cmm	11	triangles 91%, squares 9%	3, 4	10, 11, 12	11.45
	k3-051	[3^6; 3^3.4^2; 3^3.4^2] #1	cmm	8	triangles 75%, squares 25%	3, 4	9, 10, 11, 12	10.50
	k3-052	[3^6; 3^3.4^2; 3^3.4^2] #2	pmg	10	triangles 80%, squares 20%	3, 4	9, 10, 11, 12	10.80
	k3-053	[3^3.4^2; 3^3.4^2; 3^2.4.3.4]	pgg	18	triangles 67%, squares 33%	3, 4	9, 12	10.00
	k3-054	[3^6; 3^3.4^2; 3^2.4.3.4]	p6m	20	triangles 70%, squares 30%	3, 4	9, 10, 12	10.20
	k3-055	[3^3.4^2; 3^2.4.3.4; 3^2.4.3.4]	p2	9	triangles 67%, squares 33%	3, 4	9, 12	10.00
	k3-056	[3^4.6; 3^3.4^2; 3^2.4.3.4]	p6m	27	triangles 74%, squares 22%, hexagons 4%	3, 4, 6	9, 12, 18	10.00
	k3-057	[3^6; 3^2.4.3.4; 3^2.4.3.4]	p6m	20	triangles 70%, squares 30%	3, 4	9, 10, 12	10.20
	k3-058	[3^6; 3^6; 3^4.6] #1	p6m	19	triangles 95%, hexagons 5%	3, 6	10, 11, 18	11.05
	k3-059	[3^6; 3^6; 3^4.6] #2	p6m	27	triangles 96%, hexagons 4%	3, 6	10, 11, 12, 18	11.33
	k3-060	[3^6; 3^6; 3^4.6] #3	p6	33	triangles 97%, hexagons 3%	3, 6	10, 11, 12, 18	11.45
	k3-061	[3^6; 3^4.6; 3^4.6]	cmm	11	triangles 91%, hexagons 9%	3, 6	9, 10, 18	10.36