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

Image:Wythoff construction-pqr.png

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

class="wikitable" !colspan=4\|General !colspan=4\|Right triangle (r=2)
Description !Wythoff symbol !Vertex configuration !Coxeter diagram {{CDD\|pqr}} !Wythoff symbol !Vertex configuration !Schläfli symbol !Coxeter diagram {{CDD\|node\|p\|node\|q\|node}}
align=center \|rowspan=3\|regular and quasiregular \| q \| p r \| (p.r)^q \|{{CDD\|3\|node_1\|p\|node\|q\|node\|r}} \| q \| p 2 \| p^q \| {p,q} \|{{CDD\|node_1\|p\|node\|q\|node}}
align=center \| p \| q r \| (q.r)^p \|{{CDD\|3\|node\|p\|node\|q\|node_1\|r}} \| p \| q 2 \| q^p \| {q,p} \|{{CDD\|node\|p\|node\|q\|node_1}}
align=center \| r \| p q \|(q.p)^r \|{{CDD\|3\|node\|p\|node_1\|q\|node\|r}} \| 2 \| p q \|''(q.p)² \| t₁{p,q} \|{{CDD\|node\|p\|node_1\|q\|node}}
align=center \|rowspan=3\|truncated and expanded \| q r \| p \|q.2p.r.2p \|{{CDD\|3\|node_1\|p\|node_1\|q\|node\|r}} \| q 2 \| p \|''q.2p.2p \| t_0,1{p,q} \|{{CDD\|node_1\|p\|node_1\|q\|node}}
align=center \| p r \| q \| p.2q.r.2q \|{{CDD\|3\|node\|p\|node_1\|q\|node_1\|r}} \| p 2 \| q \| p. 2q.2q \| t_0,1{q,p} \|{{CDD\|node\|p\|node_1\|q\|node_1}}
align=center \| p q \| r \|2r.q.2r.p \|{{CDD\|3\|node_1\|p\|node\|q\|node_1\|r}} \| p q \| 2 \|4.q.4.p \| t_0,2{p,q} \|{{CDD\|node_1\|p\|node\|q\|node_1}}
align=center \|rowspan=2\| even-faced \| p q r \| \| 2r.2q.2p \|{{CDD\|3\|node_1\|p\|node_1\|q\|node_1\|r}} \| p q 2 \| \| 4.2q.2p \| t_0,1,2{p,q} \|{{CDD\|node_1\|p\|node_1\|q\|node_1}}
align=center \| p q (r s) \| \| 2p.2q.-2p.-2q \| - \| p 2 (r s) \| \| 2p.4.-2p.⁴/₃ \| \| -
align=center \|rowspan=2\| snub \| \| p q r \| 3.r.3.q.3.p \|{{CDD\|3\|node_h\|p\|node_h\|q\|node_h\|r}} \| \| p q 2 \| 3.3.q.3.p \| s{p,q} \|{{CDD\|node_h\|p\|node_h\|q\|node_h}}
align=center \| \| p q r s \| (4.p.4.q.4.r.4.s)/2 \| - \| - \| - \| \| -

class="wikitable"

!colspan=4|General

!colspan=4|Right triangle (r=2)

Description

!Wythoff
symbol

!Vertex
configuration

!Coxeter
diagram
{{CDD|pqr}}

!Wythoff
symbol

!Vertex
configuration

!Schläfli
symbol

!Coxeter
diagram
{{CDD|node|p|node|q|node}}

align=center

|rowspan=3|regular and
quasiregular

| q | p r

| (p.r)^q

|{{CDD|3|node_1|p|node|q|node|r}}

| q | p 2

| p^q

| {p,q}

|{{CDD|node_1|p|node|q|node}}

align=center

| p | q r

| (q.r)^p

|{{CDD|3|node|p|node|q|node_1|r}}

| p | q 2

| q^p

| {q,p}

|{{CDD|node|p|node|q|node_1}}

align=center

| r | p q

|(q.p)^r

|{{CDD|3|node|p|node_1|q|node|r}}

| 2 | p q

|''(q.p)²

| t₁{p,q}

|{{CDD|node|p|node_1|q|node}}

align=center

|rowspan=3|truncated and
expanded

| q r | p

|q.2p.r.2p

|{{CDD|3|node_1|p|node_1|q|node|r}}

| q 2 | p

|''q.2p.2p

| t_0,1{p,q}

|{{CDD|node_1|p|node_1|q|node}}

align=center

| p r | q

| p.2q.r.2q

|{{CDD|3|node|p|node_1|q|node_1|r}}

| p 2 | q

| p. 2q.2q

| t_0,1{q,p}

|{{CDD|node|p|node_1|q|node_1}}

align=center

| p q | r

|2r.q.2r.p

|{{CDD|3|node_1|p|node|q|node_1|r}}

| p q | 2

|4.q.4.p

| t_0,2{p,q}

|{{CDD|node_1|p|node|q|node_1}}

align=center

|rowspan=2| even-faced

| p q r |

| 2r.2q.2p

|{{CDD|3|node_1|p|node_1|q|node_1|r}}

| p q 2 |

| 4.2q.2p

| t_0,1,2{p,q}

|{{CDD|node_1|p|node_1|q|node_1}}

align=center

| p q (r s) |

| 2p.2q.-2p.-2q

| -

| p 2 (r s) |

| 2p.4.-2p.⁴/₃

| -

align=center

|rowspan=2| snub

| | p q r

| 3.r.3.q.3.p

|{{CDD|3|node_h|p|node_h|q|node_h|r}}

| | p q 2

| 3.3.q.3.p

| s{p,q}

|{{CDD|node_h|p|node_h|q|node_h}}

align=center

| | 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.Coxeter, Regular Polytopes, p. 114

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

class="wikitable"
(p q r) ! q {{pipe}} p r (p.r)^q ! p {{pipe}} q r (q.r)^p ! r {{pipe}} p q (q.p)^r ! q r {{pipe}} p q.2p.r.2p ! p r {{pipe}} q p.2q.r.2q ! p q {{pipe}} r 2r.q.2r.p ! p q r {{pipe}} 2r.2q.2p ! {{pipe}} p q r 3.r.3.q.3.p
(6 3 2) \| bgcolor=#99ff99 \| 108px 6.6.6 hexat \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3 trat \| bgcolor=#99ff99 \| 108px 3.6.3.6 that \| bgcolor=#99ff99 \| 108px 3.12.12 toxat \| bgcolor=#99ff99 \| 108px 6.6.6 hexat \| bgcolor=#99ff99 \| 108px 4.3.4.6 srothat \| bgcolor=#99ff99 \| 108px 4.6.12 grothat \| bgcolor=#99ff99 \| 108px 3.3.3.3.6 snathat
(4 4 2) \| bgcolor=#99ff99 \| 108px 4.4.4.4 squat \| bgcolor=#99ff99 \| 108px 4.4.4.4 squat \| bgcolor=#99ff99 \| 108px 4.4.4.4 squat \| bgcolor=#99ff99 \| 108px 4.8.8 tosquat \| bgcolor=#99ff99 \| 108px 4.8.8 tosquat \| bgcolor=#99ff99 \| 108px 4.4.4.4 squat \| bgcolor=#99ff99 \| 108px 4.8.8 tosquat \| bgcolor=#99ff99 \| 108px 3.3.4.3.4 snasquat
(3 3 3) \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3 trat \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3 trat \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3 trat \| bgcolor=#99ff99 \| 108px 3.6.3.6 that \| bgcolor=#99ff99 \| 108px 3.6.3.6 that \| bgcolor=#99ff99 \| 108px 3.6.3.6 that \| bgcolor=#99ff99 \| 108px 6.6.6 hexat \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3 trat
(∞ 2 2) \| bgcolor=#ff9999 \| — \| bgcolor=#ff9999 \| — \| bgcolor=#ff9999 \| — \| bgcolor=#ff9999 \| — \| bgcolor=#99ff99 \| 108px 4.4.∞ azip \| bgcolor=#99ff99 \| 108px 4.4.∞ azip \| bgcolor=#99ff99 \| 108px 4.4.∞ azip \| bgcolor=#99ff99 \| 108px 3.3.3.∞ azap
(3/2 3/2 3) \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3 trat \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3 trat \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3 trat \| bgcolor=#ff9999 \| ∞-covered {3} \| bgcolor=#ff9999 \| ∞-covered {3} \| bgcolor=#99ff99 \| 108px 3.6.3.6 that \| bgcolor=#ff9999 \| [degenerate] \| ?
(4 4/3 2) \| bgcolor=#99ff99 \| 108px 4.4.4.4 squat \| bgcolor=#99ff99 \| 108px 4.4.4.4 squat \| bgcolor=#99ff99 \| 108px 4.4.4.4 squat \| bgcolor=#99ff99 \| 108px 4.8.8 tosquat \| 108px 4.8/5.8/5 quitsquat \| bgcolor=#ff9999 \| ∞-covered {4} \| 108px 4.8/3.8/7 qrasquit \| ?
(4/3 4/3 2) \| bgcolor=#99ff99 \| 108px 4.4.4.4 squat \| bgcolor=#99ff99 \| 108px 4.4.4.4 squat \| bgcolor=#99ff99 \| 108px 4.4.4.4 squat \| 108px 4.8/5.8/5 quitsquat \| 108px 4.8/5.8/5 quitsquat \| bgcolor=#99ff99 \| 108px 4.4.4.4 squat \| 108px 4.8/5.8/5 quitsquat \| 108px 3.3.4/3.3.4/3 rasisquat
(3/2 6 2) \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3 trat \| bgcolor=#99ff99 \| 108px 6.6.6 hexat \| bgcolor=#99ff99 \| 108px 3.6.3.6 that \| bgcolor=#ff9999 \| [degenerate] \| bgcolor=#99ff99 \| 108px 3.12.12 toxat \| 108px 3/2.4.6/5.4 qrothat \| bgcolor=#ff9999 \| [degenerate] \| ?
(3 6/5 2) \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3 trat \| bgcolor=#99ff99 \| 108px 6.6.6 hexat \| bgcolor=#99ff99 \| 108px 3.6.3.6 that \| bgcolor=#99ff99 \| 108px 6.6.6 hexat \| 108px 3/2.12/5.12/5 quothat \| 108px 3/2.4.6/5.4 qrothat \| 108px 4.6/5.12/5 quitothit \| ?
(3/2 6/5 2) \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3 trat \| bgcolor=#99ff99 \| 108px 6.6.6 hexat \| bgcolor=#99ff99 \| 108px 3.6.3.6 that \| bgcolor=#ff9999 \| [degenerate] \| 108px 3/2.12/5.12/5 quothat \| bgcolor=#99ff99 \| 108px 3.4.6.4 srothat \| bgcolor=#ff9999 \| [degenerate] \| ?
(3/2 6 6) \| bgcolor=#ff9999 \| 108px (3/2.6)⁶ chatit \| bgcolor=#ff9999 \| 108px (6.6.6.6.6.6)/2 2hexat \| bgcolor=#ff9999 \| 108px (3/2.6)⁶ chatit \| bgcolor=#ff9999 \| [degenerate] \| 108px 3/2.12.6.12 shothat \| 108px 3/2.12.6.12 shothat \| bgcolor=#ff9999 \| [degenerate] \| ?
(3 6 6/5) \| bgcolor=#ff9999 \| 108px (3/2.6)⁶ chatit \| bgcolor=#ff9999 \| 108px (6.6.6.6.6.6)/2 2hexat \| bgcolor=#ff9999 \| 108px (3/2.6)⁶ chatit \| bgcolor=#ff9999 \| ∞-covered {6} \| 108px 3/2.12.6.12 shothat \| 108px 3.12/5.6/5.12/5 ghothat \| 108px 6.12/5.12/11 thotithit \| ?
(3/2 6/5 6/5) \| bgcolor=#ff9999 \| 108px (3/2.6)⁶ chatit \| bgcolor=#ff9999 \| 108px (6.6.6.6.6.6)/2 2hexat \| bgcolor=#ff9999 \| 108px (3/2.6)⁶ chatit \| bgcolor=#ff9999 \| [degenerate] \| 108px 3.12/5.6/5.12/5 ghothat \| 108px 3.12/5.6/5.12/5 ghothat \| bgcolor=#ff9999 \| [degenerate] \| ?
(3 3/2 ∞) \| 108px (3.∞)³/2 = (3/2.∞)³ ditatha \| 108px (3.∞)³/2 = (3/2.∞)³ ditatha \| bgcolor=#ff9999 \| — \| bgcolor=#ff9999 \| 108px 6.3/2.6.∞ chata \| bgcolor=#ff9999 \| [degenerate] \| 108px 3.∞.3/2.∞ tha \| bgcolor=#ff9999 \| [degenerate] \| ?
(3 3 ∞') \| 108px (3.∞)³/2 = (3/2.∞)³ ditatha \| 108px (3.∞)³/2 = (3/2.∞)³ ditatha \| bgcolor=#ff9999 \| — \| bgcolor=#ff9999 \| 108px 6.3/2.6.∞ chata \| bgcolor=#ff9999 \| 108px 6.3/2.6.∞ chata \| bgcolor=#ff9999 \| [degenerate] \| bgcolor=#ff9999 \| [degenerate] \| ?
(3/2 3/2 ∞') \| 108px (3.∞)³/2 = (3/2.∞)³ ditatha \| 108px (3.∞)³/2 = (3/2.∞)³ ditatha \| bgcolor=#ff9999 \| — \| bgcolor=#ff9999 \| [degenerate] \| bgcolor=#ff9999 \| [degenerate] \| bgcolor=#ff9999 \| [degenerate] \| bgcolor=#ff9999 \| [degenerate] \| ?
(4 4/3 ∞) \| bgcolor=#ff9999 \| 108px (4.∞)⁴/3 cosa \| bgcolor=#ff9999 \| 108px (4.∞)⁴/3 cosa \| bgcolor=#ff9999 \| — \| 108px 8.4/3.8.∞ gossa \| 108px 8/3.4.8/3.∞ sossa \| 108px 4.∞.4/3.∞ sha \| 108px 8.8/3.∞ satsa \| 108px 3.4.3.4/3.3.∞ snassa
(4 4 ∞') \| bgcolor=#ff9999 \| 108px (4.∞)⁴/3 cosa \| bgcolor=#ff9999 \| 108px (4.∞)⁴/3 cosa \| bgcolor=#ff9999 \| — \| 108px 8.4/3.8.∞ gossa \| 108px 8.4/3.8.∞ gossa \| bgcolor=#ff9999 \| [degenerate] \| bgcolor=#ff9999 \| [degenerate] \| ?
(4/3 4/3 ∞') \| bgcolor=#ff9999 \| 108px (4.∞)⁴/3 cosa \| bgcolor=#ff9999 \| 108px (4.∞)⁴/3 cosa \| bgcolor=#ff9999 \| — \| 108px 8/3.4.8/3.∞ sossa \| 108px 8/3.4.8/3.∞ sossa \| bgcolor=#ff9999 \| [degenerate] \| bgcolor=#ff9999 \| [degenerate] \| ?
(6 6/5 ∞) \| bgcolor=#ff9999 \| 108px (6.∞)⁶/5 cha \| bgcolor=#ff9999 \| 108px (6.∞)⁶/5 cha \| bgcolor=#ff9999 \| — \| 108px 6/5.12.∞.12 ghaha \| 108px 6.12/5.∞.12/5 shaha \| bgcolor=#ff9999 \| 108px 6.∞.6/5.∞ 2hoha \| 108px 12.12/5.∞ hatha \| ?
(6 6 ∞') \| bgcolor=#ff9999 \| 108px (6.∞)⁶/5 cha \| bgcolor=#ff9999 \| 108px (6.∞)⁶/5 cha \| bgcolor=#ff9999 \| — \| 108px 6/5.12.∞.12 ghaha \| 108px 6/5.12.∞.12 ghaha \| bgcolor=#ff9999 \| [degenerate] \| bgcolor=#ff9999 \| [degenerate] \| ?
(6/5 6/5 ∞') \| bgcolor=#ff9999 \| 108px (6.∞)⁶/5 cha \| bgcolor=#ff9999 \| 108px (6.∞)⁶/5 cha \| bgcolor=#ff9999 \| — \| 108px 6.12/5.∞.12/5 shaha \| 108px 6.12/5.∞.12/5 shaha \| bgcolor=#ff9999 \| [degenerate] \| bgcolor=#ff9999 \| [degenerate] \| ?

class="wikitable"

(p q r)

! q {{pipe}} p r
(p.r)^q

! p {{pipe}} q r
(q.r)^p

! r {{pipe}} p q
(q.p)^r

! q r {{pipe}} p
q.2p.r.2p

! p r {{pipe}} q
p.2q.r.2q

! p q {{pipe}} r
2r.q.2r.p

! p q r {{pipe}}
2r.2q.2p

! {{pipe}} p q r
3.r.3.q.3.p

(6 3 2)

| bgcolor=#99ff99 | 108px
6.6.6
hexat

| bgcolor=#99ff99 | 108px
3.3.3.3.3.3
trat

| bgcolor=#99ff99 | 108px
3.6.3.6
that

| bgcolor=#99ff99 | 108px
3.12.12
toxat

| bgcolor=#99ff99 | 108px
6.6.6
hexat

| bgcolor=#99ff99 | 108px
4.3.4.6
srothat

| bgcolor=#99ff99 | 108px
4.6.12
grothat

| bgcolor=#99ff99 | 108px
3.3.3.3.6
snathat

(4 4 2)

| bgcolor=#99ff99 | 108px
4.4.4.4
squat

| bgcolor=#99ff99 | 108px
4.8.8
tosquat

| bgcolor=#99ff99 | 108px
4.4.4.4
squat

| bgcolor=#99ff99 | 108px
4.8.8
tosquat

| bgcolor=#99ff99 | 108px
3.3.4.3.4
snasquat

(3 3 3)

| bgcolor=#99ff99 | 108px
3.3.3.3.3.3
trat

| bgcolor=#99ff99 | 108px
3.6.3.6
that

| bgcolor=#99ff99 | 108px
6.6.6
hexat

| bgcolor=#99ff99 | 108px
3.3.3.3.3.3
trat

(∞ 2 2)

| bgcolor=#ff9999 | —

| bgcolor=#99ff99 | 108px
4.4.∞
azip

| bgcolor=#99ff99 | 108px
3.3.3.∞
azap

(3/2 3/2 3)

| bgcolor=#99ff99 | 108px
3.3.3.3.3.3
trat

| bgcolor=#ff9999 | ∞-covered {3}

| bgcolor=#99ff99 | 108px
3.6.3.6
that

| bgcolor=#ff9999 | [degenerate]

(4 4/3 2)

| bgcolor=#99ff99 | 108px
4.4.4.4
squat

| bgcolor=#99ff99 | 108px
4.8.8
tosquat

| 108px
4.8/5.8/5
quitsquat

| bgcolor=#ff9999 | ∞-covered {4}

| 108px
4.8/3.8/7
qrasquit

(4/3 4/3 2)

| bgcolor=#99ff99 | 108px
4.4.4.4
squat

| 108px
4.8/5.8/5
quitsquat

| bgcolor=#99ff99 | 108px
4.4.4.4
squat

| 108px
4.8/5.8/5
quitsquat

| 108px
3.3.4/3.3.4/3
rasisquat

(3/2 6 2)

| bgcolor=#99ff99 | 108px
3.3.3.3.3.3
trat

| bgcolor=#99ff99 | 108px
6.6.6
hexat

| bgcolor=#99ff99 | 108px
3.6.3.6
that

| bgcolor=#ff9999 | [degenerate]

| bgcolor=#99ff99 | 108px
3.12.12
toxat

| 108px
3/2.4.6/5.4
qrothat

| bgcolor=#ff9999 | [degenerate]

(3 6/5 2)

| bgcolor=#99ff99 | 108px
3.3.3.3.3.3
trat

| bgcolor=#99ff99 | 108px
6.6.6
hexat

| bgcolor=#99ff99 | 108px
3.6.3.6
that

| bgcolor=#99ff99 | 108px
6.6.6
hexat

| 108px
3/2.12/5.12/5
quothat

| 108px
3/2.4.6/5.4
qrothat

| 108px
4.6/5.12/5
quitothit

(3/2 6/5 2)

| bgcolor=#99ff99 | 108px
3.3.3.3.3.3
trat

| bgcolor=#99ff99 | 108px
6.6.6
hexat

| bgcolor=#99ff99 | 108px
3.6.3.6
that

| bgcolor=#ff9999 | [degenerate]

| 108px
3/2.12/5.12/5
quothat

| bgcolor=#99ff99 | 108px
3.4.6.4
srothat

| bgcolor=#ff9999 | [degenerate]

(3/2 6 6)

| bgcolor=#ff9999 | 108px
(3/2.6)⁶
chatit

| bgcolor=#ff9999 | 108px
(6.6.6.6.6.6)/2
2hexat

| bgcolor=#ff9999 | 108px
(3/2.6)⁶
chatit

| bgcolor=#ff9999 | [degenerate]

| 108px
3/2.12.6.12
shothat

| bgcolor=#ff9999 | [degenerate]

(3 6 6/5)

| bgcolor=#ff9999 | 108px
(3/2.6)⁶
chatit

| bgcolor=#ff9999 | 108px
(6.6.6.6.6.6)/2
2hexat

| bgcolor=#ff9999 | 108px
(3/2.6)⁶
chatit

| bgcolor=#ff9999 | ∞-covered {6}

| 108px
3/2.12.6.12
shothat

| 108px
3.12/5.6/5.12/5
ghothat

| 108px
6.12/5.12/11
thotithit

(3/2 6/5 6/5)

| bgcolor=#ff9999 | 108px
(3/2.6)⁶
chatit

| bgcolor=#ff9999 | 108px
(6.6.6.6.6.6)/2
2hexat

| bgcolor=#ff9999 | 108px
(3/2.6)⁶
chatit

| bgcolor=#ff9999 | [degenerate]

| 108px
3.12/5.6/5.12/5
ghothat

| bgcolor=#ff9999 | [degenerate]

(3 3/2 ∞)

| 108px
(3.∞)³/2 = (3/2.∞)³
ditatha

| bgcolor=#ff9999 | —

| bgcolor=#ff9999 | 108px
6.3/2.6.∞
chata

| bgcolor=#ff9999 | [degenerate]

| 108px
3.∞.3/2.∞
tha

| bgcolor=#ff9999 | [degenerate]

(3 3 ∞')

| 108px
(3.∞)³/2 = (3/2.∞)³
ditatha

| bgcolor=#ff9999 | —

| bgcolor=#ff9999 | 108px
6.3/2.6.∞
chata

| bgcolor=#ff9999 | [degenerate]

(3/2 3/2 ∞')

| 108px
(3.∞)³/2 = (3/2.∞)³
ditatha

| bgcolor=#ff9999 | —

| bgcolor=#ff9999 | [degenerate]

(4 4/3 ∞)

| bgcolor=#ff9999 | 108px
(4.∞)⁴/3
cosa

| bgcolor=#ff9999 | —

| 108px
8.4/3.8.∞
gossa

| 108px
8/3.4.8/3.∞
sossa

| 108px
4.∞.4/3.∞
sha

| 108px
8.8/3.∞
satsa

| 108px
3.4.3.4/3.3.∞
snassa

(4 4 ∞')

| bgcolor=#ff9999 | 108px
(4.∞)⁴/3
cosa

| bgcolor=#ff9999 | —

| 108px
8.4/3.8.∞
gossa

| bgcolor=#ff9999 | [degenerate]

(4/3 4/3 ∞')

| bgcolor=#ff9999 | 108px
(4.∞)⁴/3
cosa

| bgcolor=#ff9999 | —

| 108px
8/3.4.8/3.∞
sossa

| bgcolor=#ff9999 | [degenerate]

(6 6/5 ∞)

| bgcolor=#ff9999 | 108px
(6.∞)⁶/5
cha

| bgcolor=#ff9999 | —

| 108px
6/5.12.∞.12
ghaha

| 108px
6.12/5.∞.12/5
shaha

| bgcolor=#ff9999 | 108px
6.∞.6/5.∞
2hoha

| 108px
12.12/5.∞
hatha

(6 6 ∞')

| bgcolor=#ff9999 | 108px
(6.∞)⁶/5
cha

| bgcolor=#ff9999 | —

| 108px
6/5.12.∞.12
ghaha

| bgcolor=#ff9999 | [degenerate]

(6/5 6/5 ∞')

| bgcolor=#ff9999 | 108px
(6.∞)⁶/5
cha

| bgcolor=#ff9999 | —

| 108px
6.12/5.∞.12/5
shaha

| bgcolor=#ff9999 | [degenerate]

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

class="wikitable"

108px
6.∞.6/5.∞
hoha
hemi(6 6/5 {{pipe}} ∞)

Also there are a few tilings with the mixed symbol p q {{su|p=r|b=s}} |:

class=wikitable

108px
4.12.4/3.12/11
sraht
2 6 {{su|p=3/2|b=3}} {{pipe}}

|108px
4.12/5.4/3.12/7
graht
2 6/5 {{su|p=3/2|b=3}} {{pipe}}

|108px
8/3.8.8/5.8/7
sost
4/3 4 {{su|p=2|b=∞}} {{pipe}}

|108px
12/5.12.12/7.12/11
huht
6/5 6 {{su|p=3|b=∞}} {{pipe}}

There are also some non-Wythoffian tilings:

class=wikitable

bgcolor=#99ff99 | 108px
3.3.3.4.4
etrat

|108px
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.

class="wikitable"
(p q r) ! q {{pipe}} p r (p.r)^q ! p {{pipe}} q r (q.r)^p ! r {{pipe}} p q (q.p)^r ! q r {{pipe}} p q.2p.r.2p ! p r {{pipe}} q p.2q.r.2q ! p q {{pipe}} r 2r.q.2r.p ! p q r {{pipe}} 2r.2q.2p ! {{pipe}} p q r 3.r.3.q.3.p
(7 3 2) \| bgcolor=#99ff99 \| 108px 7.7.7 heat \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3.3 hetrat \| bgcolor=#99ff99 \| 108px 3.7.3.7 thet \| bgcolor=#99ff99 \| 108px 3.14.14 theat \| bgcolor=#99ff99 \| 108px 6.6.7 thetrat \| bgcolor=#99ff99 \| 108px 4.3.4.7 srothet \| bgcolor=#99ff99 \| 108px 4.6.14 grothet \| bgcolor=#99ff99 \| 108px 3.3.3.3.7 snathet
(8 3 2) \| bgcolor=#99ff99 \| 108px 8.8.8 ocat \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3.3.3 otrat \| bgcolor=#99ff99 \| 108px 3.8.3.8 toct \| bgcolor=#99ff99 \| 108px 3.16.16 tocat \| bgcolor=#99ff99 \| 108px 6.6.8 totrat \| bgcolor=#99ff99 \| 108px 4.3.4.8 srotoct \| bgcolor=#99ff99 \| 108px 4.6.16 grotoct \| bgcolor=#99ff99 \| 108px 3.3.3.3.8 snatoct
(5 4 2) \| bgcolor=#99ff99 \| 108px 5.5.5.5 peat \| bgcolor=#99ff99 \| 108px 4.4.4.4.4 pesquat \| bgcolor=#99ff99 \| 108px 4.5.4.5 tepet \| bgcolor=#99ff99 \| 108px 4.10.10 topeat \| bgcolor=#99ff99 \| 108px 5.8.8 topesquat \| bgcolor=#99ff99 \| 108px 4.4.4.5 srotepet \| bgcolor=#99ff99 \| 108px 4.8.10 grotepet \| bgcolor=#99ff99 \| 108px 3.3.4.3.5 stepet
(6 4 2) \| bgcolor=#99ff99 \| 108px 6.6.6.6 shexat \| bgcolor=#99ff99 \| 108px 4.4.4.4.4.4 hisquat \| bgcolor=#99ff99 \| 108px 4.6.4.6 tehat \| bgcolor=#99ff99 \| 108px 4.12.12 toshexat \| bgcolor=#99ff99 \| 108px 6.8.8 thisquat \| bgcolor=#99ff99 \| 108px 4.4.4.6 srotehat \| bgcolor=#99ff99 \| 108px 4.8.12 grotehat \| bgcolor=#99ff99 \| 108px 3.3.4.3.6 snatehat
(5 5 2) \| bgcolor=#99ff99 \| 108px 5.5.5.5.5 pepat \| bgcolor=#99ff99 \| 108px 5.5.5.5.5 pepat \| bgcolor=#99ff99 \| 108px 5.5.5.5 peat \| bgcolor=#99ff99 \| 108px 5.10.10 topepat \| bgcolor=#99ff99 \| 108px 5.10.10 topepat \| bgcolor=#99ff99 \| 108px 4.5.4.5 tepet \| bgcolor=#99ff99 \| 108px 4.10.10 topeat \| bgcolor=#99ff99 \| 108px 3.3.5.3.5 spepat
(6 6 2) \| bgcolor=#99ff99 \| 108px 6.6.6.6.6.6 hihat \| bgcolor=#99ff99 \| 108px 6.6.6.6.6.6 hihat \| bgcolor=#99ff99 \| 108px 6.6.6.6 shexat \| bgcolor=#99ff99 \| 108px 6.12.12 thihat \| bgcolor=#99ff99 \| 108px 6.12.12 thihat \| bgcolor=#99ff99 \| 108px 4.6.4.6 tehat \| bgcolor=#99ff99 \| 108px 4.12.12 toshexat \| bgcolor=#99ff99 \| 108px 3.3.6.3.6 shihat
(4 3 3) \| bgcolor=#99ff99 \| 108px 3.4.3.4.3.4 dittitecat \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.3.3.3 otrat \| bgcolor=#99ff99 \| 108px 3.4.3.4.3.4 dittitecat \| bgcolor=#99ff99 \| 108px 3.8.3.8 toct \| bgcolor=#99ff99 \| 108px 6.3.6.4 sittitetrat \| bgcolor=#99ff99 \| 108px 6.3.6.4 sittitetrat \| bgcolor=#99ff99 \| 108px 6.6.8 totrat \| bgcolor=#99ff99 \| 108px 3.3.3.3.3.4 stititet
(4 4 3) \| bgcolor=#99ff99 \| 108px 3.4.3.4.3.4.3.4 ditetetrat \| bgcolor=#99ff99 \| 108px 3.4.3.4.3.4.3.4 ditetetrat \| bgcolor=#99ff99 \| 108px 4.4.4.4.4.4 hisquat \| bgcolor=#99ff99 \| 108px 4.8.3.8 sittiteteat \| bgcolor=#99ff99 \| 108px 4.8.3.8 sittiteteat \| bgcolor=#99ff99 \| 108px 6.4.6.4 tehat \| bgcolor=#99ff99 \| 108px 6.8.8 thisquat \| bgcolor=#99ff99 \| 108px 3.3.3.4.3.4 stitetet
(4 4 4) \| bgcolor=#99ff99 \| 108px 4.4.4.4.4.4.4.4 osquat \| bgcolor=#99ff99 \| 108px 3.4.3.4.3.4.3.4 osquat \| bgcolor=#99ff99 \| 108px 4.4.4.4.4.4 osquat \| bgcolor=#99ff99 \| 108px 4.8.4.8 teoct \| bgcolor=#99ff99 \| 108px 4.8.4.8 teoct \| bgcolor=#99ff99 \| 108px 4.8.4.8 teoct \| bgcolor=#99ff99 \| 108px 8.8.8 ocat \| bgcolor=#99ff99 \| 108px 3.4.3.4.3.4 dittitecat

class="wikitable"

(p q r)

! q {{pipe}} p r
(p.r)^q

! p {{pipe}} q r
(q.r)^p

! r {{pipe}} p q
(q.p)^r

! q r {{pipe}} p
q.2p.r.2p

! p r {{pipe}} q
p.2q.r.2q

! p q {{pipe}} r
2r.q.2r.p

! p q r {{pipe}}
2r.2q.2p

! {{pipe}} p q r
3.r.3.q.3.p

(7 3 2)

| bgcolor=#99ff99 | 108px
7.7.7
heat

| bgcolor=#99ff99 | 108px
3.3.3.3.3.3.3
hetrat

| bgcolor=#99ff99 | 108px
3.7.3.7
thet

| bgcolor=#99ff99 | 108px
3.14.14
theat

| bgcolor=#99ff99 | 108px
6.6.7
thetrat

| bgcolor=#99ff99 | 108px
4.3.4.7
srothet

| bgcolor=#99ff99 | 108px
4.6.14
grothet

| bgcolor=#99ff99 | 108px
3.3.3.3.7
snathet

(8 3 2)

| bgcolor=#99ff99 | 108px
8.8.8
ocat

| bgcolor=#99ff99 | 108px
3.3.3.3.3.3.3.3
otrat

| bgcolor=#99ff99 | 108px
3.8.3.8
toct

| bgcolor=#99ff99 | 108px
3.16.16
tocat

| bgcolor=#99ff99 | 108px
6.6.8
totrat

| bgcolor=#99ff99 | 108px
4.3.4.8
srotoct

| bgcolor=#99ff99 | 108px
4.6.16
grotoct

| bgcolor=#99ff99 | 108px
3.3.3.3.8
snatoct

(5 4 2)

| bgcolor=#99ff99 | 108px
5.5.5.5
peat

| bgcolor=#99ff99 | 108px
4.4.4.4.4
pesquat

| bgcolor=#99ff99 | 108px
4.5.4.5
tepet

| bgcolor=#99ff99 | 108px
4.10.10
topeat

| bgcolor=#99ff99 | 108px
5.8.8
topesquat

| bgcolor=#99ff99 | 108px
4.4.4.5
srotepet

| bgcolor=#99ff99 | 108px
4.8.10
grotepet

| bgcolor=#99ff99 | 108px
3.3.4.3.5
stepet

(6 4 2)

| bgcolor=#99ff99 | 108px
6.6.6.6
shexat

| bgcolor=#99ff99 | 108px
4.4.4.4.4.4
hisquat

| bgcolor=#99ff99 | 108px
4.6.4.6
tehat

| bgcolor=#99ff99 | 108px
4.12.12
toshexat

| bgcolor=#99ff99 | 108px
6.8.8
thisquat

| bgcolor=#99ff99 | 108px
4.4.4.6
srotehat

| bgcolor=#99ff99 | 108px
4.8.12
grotehat

| bgcolor=#99ff99 | 108px
3.3.4.3.6
snatehat

(5 5 2)

| bgcolor=#99ff99 | 108px
5.5.5.5.5
pepat

| bgcolor=#99ff99 | 108px
5.5.5.5
peat

| bgcolor=#99ff99 | 108px
5.10.10
topepat

| bgcolor=#99ff99 | 108px
4.5.4.5
tepet

| bgcolor=#99ff99 | 108px
4.10.10
topeat

| bgcolor=#99ff99 | 108px
3.3.5.3.5
spepat

(6 6 2)

| bgcolor=#99ff99 | 108px
6.6.6.6.6.6
hihat

| bgcolor=#99ff99 | 108px
6.6.6.6
shexat

| bgcolor=#99ff99 | 108px
6.12.12
thihat

| bgcolor=#99ff99 | 108px
4.6.4.6
tehat

| bgcolor=#99ff99 | 108px
4.12.12
toshexat

| bgcolor=#99ff99 | 108px
3.3.6.3.6
shihat

(4 3 3)

| bgcolor=#99ff99 | 108px
3.4.3.4.3.4
dittitecat

| bgcolor=#99ff99 | 108px
3.3.3.3.3.3.3.3
otrat

| bgcolor=#99ff99 | 108px
3.4.3.4.3.4
dittitecat

| bgcolor=#99ff99 | 108px
3.8.3.8
toct

| bgcolor=#99ff99 | 108px
6.3.6.4
sittitetrat

| bgcolor=#99ff99 | 108px
6.6.8
totrat

| bgcolor=#99ff99 | 108px
3.3.3.3.3.4
stititet

(4 4 3)

| bgcolor=#99ff99 | 108px
3.4.3.4.3.4.3.4
ditetetrat

| bgcolor=#99ff99 | 108px
4.4.4.4.4.4
hisquat

| bgcolor=#99ff99 | 108px
4.8.3.8
sittiteteat

| bgcolor=#99ff99 | 108px
6.4.6.4
tehat

| bgcolor=#99ff99 | 108px
6.8.8
thisquat

| bgcolor=#99ff99 | 108px
3.3.3.4.3.4
stitetet

(4 4 4)

| bgcolor=#99ff99 | 108px
4.4.4.4.4.4.4.4
osquat

| bgcolor=#99ff99 | 108px
3.4.3.4.3.4.3.4
osquat

| bgcolor=#99ff99 | 108px
4.4.4.4.4.4
osquat

| bgcolor=#99ff99 | 108px
4.8.4.8
teoct

| bgcolor=#99ff99 | 108px
8.8.8
ocat

| bgcolor=#99ff99 | 108px
3.4.3.4.3.4
dittitecat

Symmetry mutations

(should really also add *333, but this is a start, from [http://www2u.biglobe.ne.jp/~hsaka/mandara/index.html 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.

class=wikitable

! *332

! *432

! *532

! *632

! *442

108px
3.3.3
tet
3 {{pipe}} 2 3

|108px
4.4.4
cube
4 {{pipe}} 2 3

|108px
5.5.5
doe
5 {{pipe}} 2 3

|108px
6.6.6
hexat
6 {{pipe}} 2 3

|108px
4.4.4.4
squat
4 {{pipe}} 2 4

108px
3.3.3
tet
3 {{pipe}} 2 3

|108px
3.3.3.3
oct
3 {{pipe}} 2 4

|108px
3.3.3.3.3
ike
3 {{pipe}} 2 5

|108px
3.3.3.3.3.3
trat
3 {{pipe}} 2 6

|108px
4.4.4.4
squat
4 {{pipe}} 2 4

108px
3.3.3.3
oct
2 {{pipe}} 3 3

|108px
3.4.3.4
co
2 {{pipe}} 3 4

|108px
3.5.3.5
id
2 {{pipe}} 3 5

|108px
3.6.3.6
that
2 {{pipe}} 3 6

|108px
4.4.4.4
squat
2 {{pipe}} 4 4

108px
3.4.3/2.4
thah
3 3/2 {{pipe}} 2

|108px
3.6.3/2.6
oho
3 3/2 {{pipe}} 3

|108px
3.10.3/2.10
seihid
3 3/2 {{pipe}} 5

|108px
3.∞.3/2.∞
tha
3 3/2 {{pipe}} ∞

|108px
4.∞.4/3.∞
sha
4 4/3 {{pipe}} ∞

108px
3.4.3/2.4
thah
3 3/2 {{pipe}} 2

|108px
4.6.4/3.6
cho
4 4/3 {{pipe}} 3

|108px
5.10.5/4.10
sidhid
5 5/4 {{pipe}} 5

|108px
6.∞.6/5.∞
hoha
6 6/5 {{pipe}} ∞

|108px
4.∞.4/3.∞
sha
4 4/3 {{pipe}} ∞

108px
3.6.6
tut
2 3 {{pipe}} 3

|108px
3.8.8
tic
2 3 {{pipe}} 4

|108px
3.10.10
tid
2 3 {{pipe}} 5

|108px
3.12.12
toxat
2 3 {{pipe}} 6

|108px
4.8.8
tosquat
2 4 {{pipe}} 4

108px
3.6/2.6/2
3tet
2 3 {{pipe}} 3/2

|108px
3.8/3.8/3
quith
2 3 {{pipe}} 4/3

|108px
3.10/4.10/4
2sissid+gike
2 3 {{pipe}} 5/4

|108px
3/2.12/5.12/5
quothat
2 3 {{pipe}} 6/5

|108px
4/3.8/3.8/3
quitsquat
2 4 {{pipe}} 4/3

108px
3.6.6
tut
2 3 {{pipe}} 3

|108px
4.6.6
toe
2 4 {{pipe}} 3

|108px
5.6.6
ti
2 5 {{pipe}} 3

|108px
6.6.6
hexat
2 6 {{pipe}} 3

|108px
4.8.8
tosquat
2 4 {{pipe}} 4

108px
3.4.3.4
co
3 3 {{pipe}} 2

|108px
3.4.4.4
sirco
3 4 {{pipe}} 2

|108px
3.4.5.4
srid
3 5 {{pipe}} 2

|108px
3.4.6.4
rothat
3 6 {{pipe}} 2

|108px
4.4.4.4
squat
4 4 {{pipe}} 2

108px
3.6.3/2.6
oho
3/2 3 {{pipe}} 3

|108px
4.8.3/2.8
socco
3/2 4 {{pipe}} 4

|108px
5.10.3/2.10
saddid
3/2 5 {{pipe}} 5

|108px
6.12.3/2.12
shothat
3/2 6 {{pipe}} 6

108px
4.6.4/3.6
cho
2 3 (3/2 3/2) {{pipe}}

|108px
4.8.4/3.8/7
sroh
2 3 (3/2 4/2) {{pipe}}

|108px
4.10.4/3.10/9
srid
2 3 (3/2 5/2) {{pipe}}

|108px
4.12.4/3.12/11
sraht
2 3 (3/2 6/2) {{pipe}}

108px
3/2.4.3.4
thah
3/2 3 {{pipe}} 2

|108px
3/2.4.4.4
querco
3/2 4 {{pipe}} 2

|108px
3/2.4.5/4.4
(gicdatrid)
3/2 5 {{pipe}} 2

|108px
3/2.4.6/5.4
qrothat
3/2 6 {{pipe}} 2

|∞-covered {4}
4/3.4.4.4
4/3 4 {{pipe}} 2

108px
3.6/2.3.6/2
2oct
3 3 {{pipe}} 3/2

|108px
3.8/3.4.8/3
gocco
3 4 {{pipe}} 4/3

|108px
3.10/4.5.10/4
(sidtid+ditdid)
3 5 {{pipe}} 5/4

|108px
3.12/5.6/5.12/5
ghothat
3 6 {{pipe}} 6/5

|108px
4.8/3.4/3.8/5
groh
2 4/3 (3/2 4/2) {{pipe}}

|108px
4.12/5.4/3.12/7
graht
2 6/5 (3/2 6/2) {{pipe}}

108px
3.6.3/2.6
oho
3/2 3 {{pipe}} 3

|108px
3.6.5/2.6
siid
5/2 3 {{pipe}} 3

|108px
3.8.4/3.8
socco
3 4/3 {{pipe}} 4

|108px
3.10.5/3.10
sidditdid
3 5/3 {{pipe}} 5

|108px
6.8.8/3
cotco
3 4 4/3 {{pipe}}

|108px
6.10.10/3
idtid
3 5 5/3 {{pipe}}

108px
4.6.6
toe
2 3 3 {{pipe}}

|108px
4.6.8
girco
2 3 4 {{pipe}}

|108px
4.6.10
grid
2 3 5 {{pipe}}

|108px
4.6.12
othat
2 3 6 {{pipe}}

|108px
4.8.8
tosquat
2 4 4 {{pipe}}

108px
4.6.6/2
cho+4{6/2}
2 3 3/2 {{pipe}}

|108px
4.6/5.8/3
quitco
2 3 4/3 {{pipe}}

|108px
4.6.10/4
ri+12{10/4}
2 3 5/4 {{pipe}}

|108px
4.6/5.12/5
quitothit
2 3 6/5 {{pipe}}

|108px
4/3.8.8/5
qrasquit
2 4 4/3 {{pipe}}

108px
6.6.6/2
2tut
3 3 3/2 {{pipe}}

|108px
6.8.8/3
cotco
3 4 4/3 {{pipe}}

|108px
6.10.10/4
siddy+12{10/4}
3 5 5/4 {{pipe}}

|108px
6.12/11.12/5
thotithit
3 6 6/5 {{pipe}}

|108px
12/5.6.12/5.∞
shaha
6 ∞ {{pipe}} 6/5

|108px
8/3.4.8/3.∞
sossa
4 ∞ {{pipe}} 4/3

|108px
12.6/5.12.∞
ghaha
6/5 ∞ {{pipe}} 6

|108px
8.4/3.8.∞
gossa
4/3 ∞ {{pipe}} 4

|108px
12/5.12.12/7.12/11
huht
6/5 6 (6/2 ∞/2) {{pipe}}

|108px
8/3.8.8/5.8/7
sost
4/3 4 (4/2 ∞/2) {{pipe}}

|108px
12.12/5.∞
hatha
6/5 6 ∞ {{pipe}}

|108px
8.8/3.∞
satsa
4/3 4 ∞ {{pipe}}

108px
3.3.3.3.3
ike
{{pipe}} 2 3 3

|108px
3.3.3.3.4
snic
{{pipe}} 2 3 4

|108px
3.3.3.3.5
snid
{{pipe}} 2 3 5

|108px
3.3.3.3.6
snathat
{{pipe}} 2 3 6

|108px
3.3.4.3.4
snasquat
{{pipe}} 2 3 4

108px
(3.3.3.3.3)/2
gike
{{pipe}} 2 3/2 3/2

|108px
3.3.4/3.3.4/3
rasisquat
{{pipe}} 2 4/3 4/3

|108px
3.4.3.4/3.3.∞
snassa
{{pipe}} 4 4/3 ∞

References

Klitzing:

[http://www.bendwavy.org/klitzing/dimensions/flat.htm Euclidean tessellations and honeycombs]
[http://www.bendwavy.org/klitzing/dimensions/hyperbolic.htm Hyperbolic tessellations and honeycombs]

McNeill:

[http://www.orchidpalms.com/polyhedra/tessellations/tessel.htm Tessellations of the Plane]