Hyperbolic Planar Tesselations

by Don Hatch

Here are pictures of some regular tesselations of the hyperbolic plane.

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.

Truncation: 0 .5 1 1.5 2 2.5 3 3.5 4 (dual)
{3,7}
{4,5}
{5,5}
{5,4}
{7,3}
{8,3}
{9,3}
{10,3}
{20,3}
{infinity,3}
{infinity,3}
another
view
Truncation: 0 .5 1 1.5 2 2.5 3 3.5 4 (dual)
Here are some more semiregular hyperbolic tesselations, based on regular hyperbolic tesselations.

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.
Omnitruncated Runcinated Snub
{3,7}
{4,5}
{5,5}
{5,4}
{7,3}
{8,3}
{9,3}
{10,3}
{20,3}
{infinity,3}
{infinity,3}
another
view
Omnitruncated Runcinated Snub

Enumeration of Uniform Tesselations

A tesselation (a.k.a. tiling) is uniform if its faces are regular and its symmetry group (including reflections) is transitive on the vertices. A uniform tesselation whose dual is also uniform is called regular. A uniform tesselation that is not regular is called semiregular. We'll consider spherical, planar, and hyperbolic tilings all at once, using "Schwarz polygons" (a generalization of Schwarz triangles) to generate the symmetry groups, and using a generalized Coxeter-Dynkin symbol to name the resulting tesselations. (The Wythoff symbol could probably be used instead, except that I find the Wythoff symbol to be horribly unintuitive and I don't see how to extend it to Schwarz polygons of more than 3 sides). We will only consider tilings composed of convex (non-star) polygons.

Fundamental Tilings (p0|p1|...|pn-1|) (note this is NOT a standard Wythoff symbol, see above)

Start with any cyclic ordered list of integers p0,p1,...,pn-1 (n>=3, pi>=2). (By "cyclic", I mean that the list p0,p1,...,pn-1 can also be written as p1,...,pn-1,p0 or any other cyclic permutation.) The fundamental uniform tiling corresponding to p0,p1,...,pn-1 has vertex configuration (2p0,2p1,...,2pn-1) (that is, each vertex is surrounded by n regular polygons whose numbers of sides are 2p0,2p1,...,2pn-1 in CCW or CW order), with symmetry group generated by reflections about the perpendicular bisectors of the edges of the tiling.

The dual of the fundamental tiling is composed of "Schwarz polygons" denoted (p0 p1 ... pn-1), with interior angles pi/p0, pi/p1, ..., pi/pn-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/pi) is <, =, or > 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 (p0 p1 ... pn-1), n>=3, pi>=2.
(Note, I haven't proved the above construction works for all cyclic lists p0,p1,...,pn-1, but it seems to work for all lists I have tried. Thus there seems to be a (at least one) uniform tiling having every conceivable all-even vertex configuration (2p0,2p1,...,2pn-1). Note that this construction does not work in general for vertex configurations having one or more odd elements; in fact, for example, there is no semiregular tiling with vertex configuration (3,4,5) (by this construction or any other).

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:

(shrinking out more than two edge types, or two non-consecutive edge types, would result in problematic over-collapsing, disconnection, and changing of the symmetry group).

A fundamental tiling (as described above) is denoted by the Coxeter-Dynkin symbol (p0|p1|...|pn-1|). The vertical bar between pi and pi+1 represents the edge type between a 2pi-gon and a 2pi+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, (p0|p1|...|pn-1)

If we remove (i.e. shrink to zero size) the edge type between the 2pn-1-gons and 2p0-gons of the fundamental tiling, we denote the result by (p0|p1|...|pn-1) (which is the same as (p1|...|pn-1 p0|), etc.) This has the effect of merging each "even" vertex with an adjacent "odd" vertex. The resulting merged vertex configuration is (p0,2p1,2p2,...,2pn-2,pn-1,2pn-2,...,2p2,2p1). Note that it is symmetric about p0 and pn-1.

Fundamental Tilings with one face type removed, (p0 p1|...|pn-1)

If we remove (i.e. shrink to zero size) the face type 2p0 of the fundamental tiling, we denote the result by (p0 p1|...|pn-1) (which is the same as (p1|...|pn-1 p0), etc.) This has the effect of merging together groups of 2p0 vertices. The resulting merged vertex configuration is (p1,2p2,2p3,...,2pn-2,pn-1,2pn-2,...,2p3,2p2)p0. Note that it is symmetric about each p1 and pn-1.

Snub Tilings, (p0||p1||...||pn-1||)

Starting with the fundamental tiling, add edges connecting each pair of "even" vertices that share an "odd" neighbor vertex, then remove all "odd" vertices along with all the original edges. (The edge lengths and angles will then need to be adjusted so that the resulting tiles are regular polygons again.) The resulting vertex configuration is (p0,n,p1,n,...,pn-1,n). We denote this by the symbol (p0||p1||...||pn-1||) (which is the same as (p1||...||pn-1||p0||), etc.)

Examples

In all cases, 2-gons of zero area are removed from the result. 
                       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!

This is a somewhat surprising fact, since there are no spherical or planar examples of it. But here are some examples where two different (non-isomorphic) uniform hyperbolic tilings have the same vertex configuration:

XXX TBD


Last modified: Thu Apr 25 00:04:14 PDT 2002
Don Hatch
hatch@plunk.org
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