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

edit
 
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

edit

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.[1]

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

edit

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

edit

(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

edit

Klitzing:

McNeill:

  1. ^ Coxeter, Regular Polytopes, p. 114