User:Double sharp/List of uniform tilings by Schwarz triangle

Yes, I'm fully aware that this is a misnomer: Schwarz triangles are spherical. I should really be referring to Coxeter polytopes (since we're referring to tilings of E²/H², I could say Coxeter polygons.)

This is the Euclidean and hyperbolic version of List of uniform polyhedra by Schwarz triangle.

What exactly sabotages {5,10/3}, {5/2,10}, {7,14/5}, {7/2,14/3}, {7/3,14} etc. and makes them have infinite density though they'd fit in the plane? – ah, I see, from Regular Polytopes (3rd ed.), p.108: they would have rotational symmetries that are not 2-, 3-, 4-, or 6-fold, which we know to be impossible.

Basically what we need is:

Add and subtract interior angles of {3} (60°), {4} (90°), {6} (120°), {8} (135°), {8/3} (45°), {12} (150°), {12/5} (30°), {∞} (180°) to yield some multiple of 360°. This will determine a candidate vertex figure. We may reject everything containing consecutive positive and negative terms of the same angle as degenerate. Some candidates may also be excluded by symmetric concerns, such as {8/3, 8} above. (Though we need to prove that only the above tiles are possible.) However such a bald listing ends up creating lots of junk as well...

Summary table

The eight forms for the Wythoff constructions from a general triangle (p q r).

There are seven generator points with each set of p,q,r (and a few special forms):

General				Right triangle (r=2)
Description	Wythoff symbol	Vertex configuration	Coxeter diagram	Wythoff symbol	Vertex configuration	Schläfli symbol	Coxeter diagram
regular and quasiregular	q | p r	(p.r)q		q | p 2	pq	{p,q}
	p | q r	(q.r)p		p | q 2	qp	{q,p}
	r | p q	(q.p)r		2 | p q	(q.p)²	t1{p,q}
truncated and expanded	q r | p	q.2p.r.2p		q 2 | p	q.2p.2p	t0,1{p,q}
	p r | q	p.2q.r.2q		p 2 | q	p. 2q.2q	t0,1{q,p}
	p q | r	2r.q.2r.p		p q | 2	4.q.4.p	t0,2{p,q}
even-faced	p q r |	2r.2q.2p		p q 2 |	4.2q.2p	t0,1,2{p,q}
	p q (r s) |	2p.2q.-2p.-2q	-	p 2 (r s) |	2p.4.-2p.4/3		-
snub	| p q r	3.r.3.q.3.p		| p q 2	3.3.q.3.p	s{p,q}
	| p q r s	(4.p.4.q.4.r.4.s)/2	-	-	-		-

There are three special cases:

• p q (r s) | – This is a mixture of p q r | and p q s |.
• | p q r – Snub forms (alternated) are give this otherwise unused symbol.
• | p q r s – A unique snub form for U75 that isn't Wythoff-constructible.

Euclidean tilings

The only plane triangles that tile the plane once over are (3 3 3), (4 2 4), and (3 2 6): they are respectively the equilateral triangle, the 45-45-90 right isosceles triangle, and the 30-60-90 right triangle. It follows that any plane triangle tiling the plane multiple times must be built up from multiple copies of one of these. The only possibility is the 30-120-120 isosceles triangle (3/2 6 6) = (6 2 3) + (2 6 3) tiling the plane twice over. Each triangle counts twice with opposite orientations, with a branch point at the 120° vertices.

The tiling {∞,2} made from two apeirogons is not accepted, because its faces meet at more than one edge. Here ∞' denotes the retrograde counterpart to ∞.

The degenerate named forms are:

• chatit: compound of 3 hexagonal tilings + triangular tiling
• chata: compound of 3 hexagonal tilings + triangular tiling + double covers of apeirogons along all edge sequences
• cha: compound of 3 hexagonal tilings + double covers of apeirogons along all edge sequences
• cosa: square tiling + double covers of apeirogons along all edge sequences

(p q r)	q | p r (p.r)q	p | q r (q.r)p	r | p q (q.p)r	q r | p q.2p.r.2p	p r | q p.2q.r.2q	p q | r 2r.q.2r.p	p q r | 2r.2q.2p	| p q r 3.r.3.q.3.p
(6 3 2)	6.6.6 hexat	3.3.3.3.3.3 trat	3.6.3.6 that	3.12.12 toxat	6.6.6 hexat	4.3.4.6 srothat	4.6.12 grothat	3.3.3.3.6 snathat
(4 4 2)	4.4.4.4 squat	4.4.4.4 squat	4.4.4.4 squat	4.8.8 tosquat	4.8.8 tosquat	4.4.4.4 squat	4.8.8 tosquat	3.3.4.3.4 snasquat
(3 3 3)	3.3.3.3.3.3 trat	3.3.3.3.3.3 trat	3.3.3.3.3.3 trat	3.6.3.6 that	3.6.3.6 that	3.6.3.6 that	6.6.6 hexat	3.3.3.3.3.3 trat
(∞ 2 2)	—	—	—	—	4.4.∞ azip	4.4.∞ azip	4.4.∞ azip	3.3.3.∞ azap
(3/2 3/2 3)	3.3.3.3.3.3 trat	3.3.3.3.3.3 trat	3.3.3.3.3.3 trat	∞-covered {3}	∞-covered {3}	3.6.3.6 that	[degenerate]	?
(4 4/3 2)	4.4.4.4 squat	4.4.4.4 squat	4.4.4.4 squat	4.8.8 tosquat	4.8/5.8/5 quitsquat	∞-covered {4}	4.8/3.8/7 qrasquit	?
(4/3 4/3 2)	4.4.4.4 squat	4.4.4.4 squat	4.4.4.4 squat	4.8/5.8/5 quitsquat	4.8/5.8/5 quitsquat	4.4.4.4 squat	4.8/5.8/5 quitsquat	3.3.4/3.3.4/3 rasisquat
(3/2 6 2)	3.3.3.3.3.3 trat	6.6.6 hexat	3.6.3.6 that	[degenerate]	3.12.12 toxat	3/2.4.6/5.4 qrothat	[degenerate]	?
(3 6/5 2)	3.3.3.3.3.3 trat	6.6.6 hexat	3.6.3.6 that	6.6.6 hexat	3/2.12/5.12/5 quothat	3/2.4.6/5.4 qrothat	4.6/5.12/5 quitothit	?
(3/2 6/5 2)	3.3.3.3.3.3 trat	6.6.6 hexat	3.6.3.6 that	[degenerate]	3/2.12/5.12/5 quothat	3.4.6.4 srothat	[degenerate]	?
(3/2 6 6)	(3/2.6)6 chatit	(6.6.6.6.6.6)/2 2hexat	(3/2.6)6 chatit	[degenerate]	3/2.12.6.12 shothat	3/2.12.6.12 shothat	[degenerate]	?
(3 6 6/5)	(3/2.6)6 chatit	(6.6.6.6.6.6)/2 2hexat	(3/2.6)6 chatit	∞-covered {6}	3/2.12.6.12 shothat	3.12/5.6/5.12/5 ghothat	6.12/5.12/11 thotithit	?
(3/2 6/5 6/5)	(3/2.6)6 chatit	(6.6.6.6.6.6)/2 2hexat	(3/2.6)6 chatit	[degenerate]	3.12/5.6/5.12/5 ghothat	3.12/5.6/5.12/5 ghothat	[degenerate]	?
(3 3/2 ∞)	(3.∞)3/2 = (3/2.∞)3 ditatha	(3.∞)3/2 = (3/2.∞)3 ditatha	—	6.3/2.6.∞ chata	[degenerate]	3.∞.3/2.∞ tha	[degenerate]	?
(3 3 ∞')	(3.∞)3/2 = (3/2.∞)3 ditatha	(3.∞)3/2 = (3/2.∞)3 ditatha	—	6.3/2.6.∞ chata	6.3/2.6.∞ chata	[degenerate]	[degenerate]	?
(3/2 3/2 ∞')	(3.∞)3/2 = (3/2.∞)3 ditatha	(3.∞)3/2 = (3/2.∞)3 ditatha	—	[degenerate]	[degenerate]	[degenerate]	[degenerate]	?
(4 4/3 ∞)	(4.∞)4/3 cosa	(4.∞)4/3 cosa	—	8.4/3.8.∞ gossa	8/3.4.8/3.∞ sossa	4.∞.4/3.∞ sha	8.8/3.∞ satsa	3.4.3.4/3.3.∞ snassa
(4 4 ∞')	(4.∞)4/3 cosa	(4.∞)4/3 cosa	—	8.4/3.8.∞ gossa	8.4/3.8.∞ gossa	[degenerate]	[degenerate]	?
(4/3 4/3 ∞')	(4.∞)4/3 cosa	(4.∞)4/3 cosa	—	8/3.4.8/3.∞ sossa	8/3.4.8/3.∞ sossa	[degenerate]	[degenerate]	?
(6 6/5 ∞)	(6.∞)6/5 cha	(6.∞)6/5 cha	—	6/5.12.∞.12 ghaha	6.12/5.∞.12/5 shaha	6.∞.6/5.∞ 2hoha	12.12/5.∞ hatha	?
(6 6 ∞')	(6.∞)6/5 cha	(6.∞)6/5 cha	—	6/5.12.∞.12 ghaha	6/5.12.∞.12 ghaha	[degenerate]	[degenerate]	?
(6/5 6/5 ∞')	(6.∞)6/5 cha	(6.∞)6/5 cha	—	6.12/5.∞.12/5 shaha	6.12/5.∞.12/5 shaha	[degenerate]	[degenerate]	?

The tiling 6 6/5 | ∞ is generated as a double cover by Wythoff's construction:

6.∞.6/5.∞ hoha hemi(6 6/5 | ∞)

Also there are a few tilings with the mixed symbol p q ^r
_s |:

4.12.4/3.12/11 sraht 2 6 3/2 3 |

4.12/5.4/3.12/7 graht 2 6/5 3/2 3 |

8/3.8.8/5.8/7 sost 4/3 4 2 ∞ |

12/5.12.12/7.12/11 huht 6/5 6 3 ∞ |

There are also some non-Wythoffian tilings:

3.3.3.4.4 etrat

3.3.3.4/3.4/3 retrat

4.8.8/3.4/3.∞ rorisassa

4.8/3.8.4/3.∞ rosassa

4.8.4/3.8.4/3.∞ rarsisresa

4.8/3.4.8/3.4/3.∞ rassersa

Hyperbolic

OK, apparently the hyperbolic fundamental domains are called Lannér triangles (compact) per Coxeter–Dynkin diagram#Hyperbolic Coxeter groups, Koszul triangles (paracompact) and Vinberg triangles (noncompact). But these are only right for simplices, no? So in general I'd write "Coxeter polygons" again.

(p q r)	q | p r (p.r)q	p | q r (q.r)p	r | p q (q.p)r	q r | p q.2p.r.2p	p r | q p.2q.r.2q	p q | r 2r.q.2r.p	p q r | 2r.2q.2p	| p q r 3.r.3.q.3.p
(7 3 2)	7.7.7 heat	3.3.3.3.3.3.3 hetrat	3.7.3.7 thet	3.14.14 theat	6.6.7 thetrat	4.3.4.7 srothet	4.6.14 grothet	3.3.3.3.7 snathet
(8 3 2)	8.8.8 ocat	3.3.3.3.3.3.3.3 otrat	3.8.3.8 toct	3.16.16 tocat	6.6.8 totrat	4.3.4.8 srotoct	4.6.16 grotoct	3.3.3.3.8 snatoct
(5 4 2)	5.5.5.5 peat	4.4.4.4.4 pesquat	4.5.4.5 tepet	4.10.10 topeat	5.8.8 topesquat	4.4.4.5 srotepet	4.8.10 grotepet	3.3.4.3.5 stepet
(6 4 2)	6.6.6.6 shexat	4.4.4.4.4.4 hisquat	4.6.4.6 tehat	4.12.12 toshexat	6.8.8 thisquat	4.4.4.6 srotehat	4.8.12 grotehat	3.3.4.3.6 snatehat
(5 5 2)	5.5.5.5.5 pepat	5.5.5.5.5 pepat	5.5.5.5 peat	5.10.10 topepat	5.10.10 topepat	4.5.4.5 tepet	4.10.10 topeat	3.3.5.3.5 spepat
(6 6 2)	6.6.6.6.6.6 hihat	6.6.6.6.6.6 hihat	6.6.6.6 shexat	6.12.12 thihat	6.12.12 thihat	4.6.4.6 tehat	4.12.12 toshexat	3.3.6.3.6 shihat
(4 3 3)	3.4.3.4.3.4 dittitecat	3.3.3.3.3.3.3.3 otrat	3.4.3.4.3.4 dittitecat	3.8.3.8 toct	6.3.6.4 sittitetrat	6.3.6.4 sittitetrat	6.6.8 totrat	3.3.3.3.3.4 stititet
(4 4 3)	3.4.3.4.3.4.3.4 ditetetrat	3.4.3.4.3.4.3.4 ditetetrat	4.4.4.4.4.4 hisquat	4.8.3.8 sittiteteat	4.8.3.8 sittiteteat	6.4.6.4 tehat	6.8.8 thisquat	3.3.3.4.3.4 stitetet
(4 4 4)	4.4.4.4.4.4.4.4 osquat	3.4.3.4.3.4.3.4 osquat	4.4.4.4.4.4 osquat	4.8.4.8 teoct	4.8.4.8 teoct	4.8.4.8 teoct	8.8.8 ocat	3.4.3.4.3.4 dittitecat

Symmetry mutations

(should really also add *333, but this is a start, from Mandara: The World of Uniform Tessellations

Families which contain only degenerate members (e.g. the quasitruncated {3,n}) are not shown; neither are those Wythoff symbols that already contain reducible fractions. Those that turn out to be degenerate anyway but do not satisfy either criterion are still shown. In some cases I have naughtily silently corrected the "doubled" constructions of the hemipolyhedra. Some of the Euclidean families involving {∞} correspond quite nicely to the hemipolyhedra, taking {∞} as an equator of r{4,4} or r{3,6}. However, some others do not have clear spherical analogues.

*332	*432	*532	*632	*442
3.3.3 tet 3 | 2 3	4.4.4 cube 4 | 2 3	5.5.5 doe 5 | 2 3	6.6.6 hexat 6 | 2 3	4.4.4.4 squat 4 | 2 4
3.3.3 tet 3 | 2 3	3.3.3.3 oct 3 | 2 4	3.3.3.3.3 ike 3 | 2 5	3.3.3.3.3.3 trat 3 | 2 6	4.4.4.4 squat 4 | 2 4
3.3.3.3 oct 2 | 3 3	3.4.3.4 co 2 | 3 4	3.5.3.5 id 2 | 3 5	3.6.3.6 that 2 | 3 6	4.4.4.4 squat 2 | 4 4
3.4.3/2.4 thah 3 3/2 | 2	3.6.3/2.6 oho 3 3/2 | 3	3.10.3/2.10 seihid 3 3/2 | 5	3.∞.3/2.∞ tha 3 3/2 | ∞	4.∞.4/3.∞ sha 4 4/3 | ∞
3.4.3/2.4 thah 3 3/2 | 2	4.6.4/3.6 cho 4 4/3 | 3	5.10.5/4.10 sidhid 5 5/4 | 5	6.∞.6/5.∞ hoha 6 6/5 | ∞	4.∞.4/3.∞ sha 4 4/3 | ∞
3.6.6 tut 2 3 | 3	3.8.8 tic 2 3 | 4	3.10.10 tid 2 3 | 5	3.12.12 toxat 2 3 | 6	4.8.8 tosquat 2 4 | 4
3.6/2.6/2 3tet 2 3 | 3/2	3.8/3.8/3 quith 2 3 | 4/3	3.10/4.10/4 2sissid+gike 2 3 | 5/4	3/2.12/5.12/5 quothat 2 3 | 6/5	4/3.8/3.8/3 quitsquat 2 4 | 4/3
3.6.6 tut 2 3 | 3	4.6.6 toe 2 4 | 3	5.6.6 ti 2 5 | 3	6.6.6 hexat 2 6 | 3	4.8.8 tosquat 2 4 | 4
3.4.3.4 co 3 3 | 2	3.4.4.4 sirco 3 4 | 2	3.4.5.4 srid 3 5 | 2	3.4.6.4 rothat 3 6 | 2	4.4.4.4 squat 4 4 | 2
3.6.3/2.6 oho 3/2 3 | 3	4.8.3/2.8 socco 3/2 4 | 4	5.10.3/2.10 saddid 3/2 5 | 5	6.12.3/2.12 shothat 3/2 6 | 6
4.6.4/3.6 cho 2 3 (3/2 3/2) |	4.8.4/3.8/7 sroh 2 3 (3/2 4/2) |	4.10.4/3.10/9 srid 2 3 (3/2 5/2) |	4.12.4/3.12/11 sraht 2 3 (3/2 6/2) |
3/2.4.3.4 thah 3/2 3 | 2	3/2.4.4.4 querco 3/2 4 | 2	3/2.4.5/4.4 (gicdatrid) 3/2 5 | 2	3/2.4.6/5.4 qrothat 3/2 6 | 2	∞-covered {4} 4/3.4.4.4 4/3 4 | 2
3.6/2.3.6/2 2oct 3 3 | 3/2	3.8/3.4.8/3 gocco 3 4 | 4/3	3.10/4.5.10/4 (sidtid+ditdid) 3 5 | 5/4	3.12/5.6/5.12/5 ghothat 3 6 | 6/5
	4.8/3.4/3.8/5 groh 2 4/3 (3/2 4/2) |		4.12/5.4/3.12/7 graht 2 6/5 (3/2 6/2) |
3.6.3/2.6 oho 3/2 3 | 3		3.6.5/2.6 siid 5/2 3 | 3
	3.8.4/3.8 socco 3 4/3 | 4	3.10.5/3.10 sidditdid 3 5/3 | 5
	6.8.8/3 cotco 3 4 4/3 |	6.10.10/3 idtid 3 5 5/3 |
4.6.6 toe 2 3 3 |	4.6.8 girco 2 3 4 |	4.6.10 grid 2 3 5 |	4.6.12 othat 2 3 6 |	4.8.8 tosquat 2 4 4 |
4.6.6/2 cho+4{6/2} 2 3 3/2 |	4.6/5.8/3 quitco 2 3 4/3 |	4.6.10/4 ri+12{10/4} 2 3 5/4 |	4.6/5.12/5 quitothit 2 3 6/5 |	4/3.8.8/5 qrasquit 2 4 4/3 |
6.6.6/2 2tut 3 3 3/2 |	6.8.8/3 cotco 3 4 4/3 |	6.10.10/4 siddy+12{10/4} 3 5 5/4 |	6.12/11.12/5 thotithit 3 6 6/5 |
			12/5.6.12/5.∞ shaha 6 ∞ | 6/5	8/3.4.8/3.∞ sossa 4 ∞ | 4/3
			12.6/5.12.∞ ghaha 6/5 ∞ | 6	8.4/3.8.∞ gossa 4/3 ∞ | 4
			12/5.12.12/7.12/11 huht 6/5 6 (6/2 ∞/2) |	8/3.8.8/5.8/7 sost 4/3 4 (4/2 ∞/2) |
			12.12/5.∞ hatha 6/5 6 ∞ |	8.8/3.∞ satsa 4/3 4 ∞ |
3.3.3.3.3 ike | 2 3 3	3.3.3.3.4 snic | 2 3 4	3.3.3.3.5 snid | 2 3 5	3.3.3.3.6 snathat | 2 3 6	3.3.4.3.4 snasquat | 2 3 4
(3.3.3.3.3)/2 gike | 2 3/2 3/2				3.3.4/3.3.4/3 rasisquat | 2 4/3 4/3
				3.4.3.4/3.3.∞ snassa | 4 4/3 ∞

References

Klitzing:

• Euclidean tessellations and honeycombs
• Hyperbolic tessellations and honeycombs

McNeill:

• Tessellations of the Plane