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This is a list of tessellations.
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This is a list of tessellations.
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| Regular Spherical (n=1, 2, 3, ...) | |||
|---|---|---|---|
| Article | Vertex configuration | Schläfli symbol | Image |
| Dihedron | n2 | {n,2} |  {6,2} |
| Hosohedron | 2n | {2,n} |  {2,6} |
| Spherical tetrahedron | 33 | {3,3} |  |
| Spherical octahedron | 34 | {3,4} |  |
| Spherical cube | 43 | {4,3} |  |
| Spherical icosahedron | 35 | {3,5} |  |
| Spherical dodecahedron | 53 | {5,3} |  |
| Semi-regular Spherical (n=2, 3, ...) | |||
|---|---|---|---|
| Article | Vertex configuration | Schläfli symbol | Image |
| Prism | 4.4.n | t{2, n} = { }×{n} |  { }×{6} |
| Antiprism | 33.n | sr{2,n} = { }⊗{n} |  { }⊗{6} |
| Dual semi-regular Spherical (n=2, 3, ...) | |||
|---|---|---|---|
| Article | Vertex configuration | Schläfli symbol | Image |
| Bipyramid | V42.n | dt{2, n} = { }+{n} |  { }+{6} |
| Trapezohedron | V33.n | dsr{2,n} = { }⨁{n} |  { }⨁{6} |
| Regular | |||
|---|---|---|---|
| Article | Vertex configuration | Schläfli symbol | Image |
| Apeirogonal hosohedron | 2∞ | {2,∞} |  |
| Order-2 apeirogonal tiling | ∞2 | {∞,2} |  |
| Square tiling | 44 | {4,4} |  |
| Triangular tiling | 36 | {3,6} |  |
| Hexagonal tiling | 63 | {6,3} |  |
| Semi-regular | |||
|---|---|---|---|
| Article | Vertex configuration | Schläfli symbol | Image |
| Apeirogonal prism | 42.∞ | t{2,∞} |  |
| Apeirogonal antiprism | 33.∞ | sr{2,∞} |  |
| Snub square tiling | 32.4.3.4 | s{4,4} |  |
| Elongated triangular tiling | 33.42 | {3,6}:e |  |
| Snub trihexagonal tiling | 34.6 | sr{6,3} |  |
| Rhombitrihexagonal tiling | 3.4.6.4 | rr{6,3} |  |
| Trihexagonal tiling | 3.6.3.6 | r{6,3} |  |
| Truncated hexagonal tiling | 3.122 | t{6,3} |  |
| Truncated trihexagonal tiling | 4.6.12 | tr{6,3} |  |
| Truncated square tiling | 4.82 | tr{4,4} |  |
| Dual semi-regular | |||
|---|---|---|---|
| Article | Face configuration | Schläfli symbol | Image |
| Apeirogonal deltohedron | V33.∞ | dsr{2,∞} |  |
| Apeirogonal bipyramid | V42.∞ | dt{2,∞} |  |
| Cairo pentagonal tiling | V32.4.3.4 | ds{4,4} |  |
| Prismatic pentagonal tiling | V33.42 | d{3,6}:e |  |
| Floret pentagonal tiling | V34.6 | dsr{6,3} |  |
| Deltoidal trihexagonal tiling | V3.4.6.4 | drr{6,3} |  |
| Rhombille tiling | V3.6.3.6 | dr{6,3} |  |
| Triakis triangular tiling | V3.122 | dt{6,3} |  |
| Kisrhombille tiling | V4.6.12 | dtr{6,3} |  |
| Tetrakis square tiling | 4.82 | tr{4,4} |  |
In geometry, a polytope or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi.
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} .
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex. Conway called it a quadrille.
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.
In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. Later the Coxeter diagram was developed to mark uniform polytopes and honeycombs in n-dimensional space within a fundamental simplex.
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex. Its dimension can be clarified as n-honeycomb for an n-dimensional honeycomb.
In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}.
In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.
In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.
In geometry, the truncated order-3 apeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{∞,3}.
In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}.
In geometry, the rhombtriapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of rr{∞,3}.
In geometry, the rhombitetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{∞,4}.
In geometry, the snub tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{∞,4}.
In geometry, the pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{6,5} or t1{6,5}.
In geometry, the rhombipentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,2{6,5}.
In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane.
