Hyperbolic Planar Tesselations

by Don Hatch

Each tesselation is represented by a Schlafli symbol of the form {p,q}, which means that q regular p-gons surround each vertex. There exists a hyperbolic tesselation {p,q} for every p,q such that (p-2)*(q-2) > 4.

Each tesselation is shown in various stages of truncation.

The dual of each tesselation or truncated tesselation is shown in blue. At the final stage of truncation (4.0) the object becomes its dual so those images are identical to the untruncated images except that the colors are reversed.

You may want to make your browser window wide so you can see them all at once. Click on an image to see a bigger version of it.

Note that the ones based on a regular {p,q} are the same as the ones based on a regular {q,p}, but shown in a different orientation.

The Omnitruncated {3,7} is the "most nearly planar" of all semiregular or regular hyperbolic tesselations, in the sense that if you tried to construct it from Euclidean planar polygons, the sum of the angles at each vertex would be as small as possible while exceeding 360 degrees.

Enumeration of Uniform Tesselations

Fundamental Tilings (p 0 |p 1 |...|p n-1 |) (note this is NOT a standard Wythoff symbol, see above)

0

1

n-1

i

0

1

n-1

1

n-1

0

0

1

n-1

0

1

n-1

0

1

n-1

The dual of the fundamental tiling is composed of "Schwarz polygons" denoted (p 0 p 1 ... p n-1 ), with interior angles pi/p 0 , pi/p 1 , ..., pi/p n-1 , and reflected about their edges.

We can color each vertex of the uniform tiling (or Schwarz polygon of the dual) "even" or "odd" depending on whether it is generated as an even or odd number of reflections of a fixed initial vertex (or Schwarz polygon of the dual, respectively), i.e. whether it is an even or odd distance (number of edges) from the initial vertex.

The resulting tiling is spherical, planar, or hyperbolic, depending on whether the sum of (pi - pi/p i ) is 2pi respectively. We can enumerate all possible spherical and planar Schwarz polygons, with corresponding fundamental uniform tilings:

Schwarz Fundamental Vertex Familiar Polygon Tiling Configuration Name Spherical: (2 3 3) (2|3|3|) (4,6,6) truncated octahedron (2 3 4) (2|3|4|) (4,6,8) truncated cuboctahedron (2 3 5) (2|3|5|) (4,6,10) truncated icosidodecahedron (2 2 p) (2<=p<=inf) (2|2|p|) (4,4,2*p) 2p-gonal prism Planar: (2 3 6) (2|3|6|) (4,6,12) omnitruncated triangle tiling (2 4 4) (2|4|4|) (4,8,8) truncated square tiling (2 2 inf) (2|2|inf|) (4,4,inf) infinity-gonal prism (3 3 3) (3|3|3|) (6,6,6) hexagonal tiling (2 2 2 2) (2|2|2|2|) (4,4,4,4) square tiling Hyperbolic: All other (p 0 p 1 ... p n-1 ), n>=3, p i >=2.

0

1

n-1

0

1

n-1

Every uniform tiling I know of (with the sole exception of the planar tiling with vertex configuration (3,3,3,4,4)) can be constructed from a fundamental tiling in one of the following ways:

The fundamental tiling itself

The fundamental tiling with one of its edge types (i.e. one edge and all its reflections) shrunk to a point

The fundamental tiling with one of its face types p i (i.e. two consecutive edge types) shrunk to a point

(i.e. two consecutive edge types) shrunk to a point The "snub" of the fundamental tiling (this will be described below).

A fundamental tiling (as described above) is denoted by the Coxeter-Dynkin symbol (p 0 |p 1 |...|p n-1 |). The vertical bar between p i and p i+1 represents the edge type between a 2p i -gon and a 2p i+1 -gon in the tiling. When we shrink an edge type to zero size, we remove the corresponding bar in the symbol.

Fundamental Tilings with one edge type removed, (p 0 |p 1 |...|p n-1 )

n-1

0

0

1

n-1

1

n-1

0

0

1

2

n-2

n-1

n-2

2

1

0

n-1

Fundamental Tilings with one face type removed, (p 0 p 1 |...|p n-1 )

0

0

1

n-1

1

n-1

0

0

1

2

3

n-2

n-1

n-2

3

2

0

1

n-1

Snub Tilings, (p 0 ||p 1 ||...||p n-1 ||)

0

1

n-1

0

1

n-1

1

n-1

0

Examples

Vertex Familiar Tiling Configuration Name (p|q|r|) (2p,2q,2r) (p|q|r ) (p,2q,r,2q) (p|q r|) (r,2p,q,2p) (p q|r|) (q,2r,p,2r) (p|q r ) (p,q)r (p q|r ) (q,r)p (p q r|) (r,p)q (p||q||r||) (p,3,q,3,r,3) Spherical: (2|3|3|) (4,6,6) truncated octahedron (2|3|3 ) (2,6,3,6) = (6,3,6) truncated tetrahedron (2|3 3|) (3,4,3,4) cuboctahedron (2 3|3|) (same as (2|3|3)) (2|3 3 ) (2,3)3 = 33 tetrahedron (2 3|3 ) (3,3)2 = 34 octahedron (2 3 3|) (same as (2|3 3)) (2||3||3||) (2,3,3,3,3,3) = 35 icosahedron (2|3|4|) (4,6,8) truncated cuboctahedron (2|3|4 ) (2,6,4,6) = (6,4,6) truncated octahedron (2|3 4|) (4,4,3,4) rhombicuboctahedron (2 3|4|) (3,8,2,8) = (3,8,8) truncated cube (2|3 4 ) (2,3)4 = 34 octahedron (2 3|4 ) (3,4)2 cuboctahedron (2 3 4|) (4,2)3 = 43 cube (2||3||4||) (2,3,3,3,4,3) = (3,3,3,4,3) snub cuboctahedron (2|3|5|) (4,6,10) truncated icosidodecahedron (2|3|5 ) (2,6,5,6) = (6,5,6) truncated icosahedron (2|3 5|) (5,4,3,4) rhombicosidodecahedron (2 3|5|) (3,10,2,10) = (3,10,10) truncated dodecahedron (2|3 5 ) (2,3)5 = 35 icosahedron (2 3|5 ) (3,5)2 icosidodecahedron (2 3 5|) (5,2)3 = 53 dodecahedron (2||3||5||) (2,3,3,3,5,3) = (3,3,3,5,3) snub icosidodecahedron (2|2|p|) (4,4,2p) 2p-gonal prism (2|2|p ) (2,4,p,4) = (4,p,4) p-gonal prism (2|2 p|) (same as (2|2|p)) (2 2|p|) (2,2p,2,2p) = (2p,2p) 2p-gonal dihedron (2|2 p ) (2,2)p 2p lunes of ambiguous width (2 2|p ) (2,p)2 = p2 p-gonal dihedron (2 2 p|) (same as (2 2|p)) (2||2||p||) (2,3,2,3,p,3) = (3,3,p,3) p-gonal antiprism Planar: (2|3|6|) (4,6,12) omnitruncated hexagon or triangle tiling (2|3|6 ) (2,6,6,6) = 63 regular hexagon tiling (2|3 6|) (6,4,3,4) runcinated hexagon or triangle tiling (2 3|6|) (3,12,2,12) = (3,12,12) truncated hexagon tiling (2|3 6 ) (2,3)6 = 36 regular triangle tiling (2 3|6 ) (3,6)2 bitruncated hexagon or triangle tiling (2 3 6|) (6,2)3 = 63 regular hexagon tiling (2||3||6||) (2,3,3,3,6,3) = (3,3,3,6,3) snub hexagon tiling (2|4|4|) (4,8,8) truncated square tiling (2|4|4 ) (2,8,4,8) = (8,4,8) truncated square tiling (2|4 4|) (4,4,4,4) = 44 square tiling (2 4|4|) (same as (2|4|4)) (2|4 4 ) (2,4)4 = 44 square tiling (2 4|4 ) (4,4)2 = 44 square tiling (2 4 4|) (same as (2|4 4)) (2||4||4||) (2,3,4,3,4,3) = (3,4,3,4,3) snub square tiling (2|2|inf|) (4,4,inf) infinity-gonal prism (2|2|inf ) (2,4,inf,4) = (4,inf,4) infinity-gonal prism (2|2 inf|) (same as (2|2|inf)) (2 2|inf|) (2,inf,2,inf) = (inf,inf) infinity-gonal dihedron (2|2 inf ) (2,2)inf (not really meaningful) (2 2|inf ) (2,inf)2 = inf2 infinity-gonal dihedron (2 2 inf|) (same as (2 2|inf)) (2||2||inf||) (2,3,2,3,inf,3) = (3,3,inf,3) infinity-gonal antiprism (3|3|3|) (6,6,6) = 63 regular hexagon tiling (3|3|3 ) (3,6,3,6) bitruncated hexagon or triangle tiling (3|3 3 ) (3,3)3 = 36 regular triangle tiling (3||3||3||) (3,3,3,3,3,3) = 36 regular triangle tiling (2|2|2|2|) (4,4,4,4) = 44 square tiling (2|2|2|2 ) (2,4,4,2,4,4) = 44 square tiling (2|2|2 2 ) (2 4 2 4)2 = 44 square tiling (2||2||2||2||) (2,4,2,4,2,4,2,4) = 44 square tiling Hyperbolic: For (p-2)*(q-2) > 4: (2|p|q|) (4,2p,2q) omnitruncated {p,q} or {q,p} (2|p|q ) (2,2p,q,2p) = (2p,q,2p) truncated {p,q} (2|p q|) (q,4,p,4) runcinated {p,q} or {q,p} (2 p|q|) (p,2q,2,2q) = (p,2q,2q) truncated {q,p} (2|p q ) (2,p)q = pq {p,q} (2 p|q ) (p,q)2 bitruncated {p,q} or {q,p} (2 p q|) (q,2)p = qp {q,p} (2||p||q||) (2,3,p,3,q,3) = (3,p,3,q,3) snub {p,q} or {q,p} (3|4|5|) (6,8,10) (3|4|5 ) (3,8,5,8) (3|4 5|) (5,6,4,6) (3 4|5|) (4,10,3,10) (3|4 5 ) (3,4)5 (3 4|5 ) (4,5)3 (3 4 5|) (5,3)4 (3||4||5||) (3,3,4,3,5,3) (3|2|2|2|) (6,4,4,4) NOT the same as runcinated {6,4}, see below (3|2|2|2 ) (3,4,4,2,4,4) = (3,4,4,4,4) (3|2|2 2|) (2,6,4,2,4,6) = (6,4,4,6) (3|2|2 2 ) (3,4,2,4)2 = (3,4,4)2 (3 2|2|2 ) (2,4,2,4)3 = 46 (3|2 2 2|) (2,6,2,6)2 = 64 (3||2||2||2||) (3,4,2,4,2,4,2,4) = (3,4,4,4,4)

A tiling is not uniquely determined by its vertex configuration!

XXX TBD

Don Hatch

hatch@plunk.org