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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the Square tiling is a regular tiling of the Euclidean plane. It has Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 of .

Conway
John Horton Conway

John Horton Conway is a prolific mathematician active in the theory of finite group , knot theory, number theory, combinatorial game theory and coding theory....
 calls it a quadrille.

The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling
Triangular tiling

In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....
 and the hexagonal tiling
Hexagonal tiling

In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....
.

e are 9 distinct uniform coloring
Uniform coloring

In geometry, a uniform coloring is a property of a uniform figure that is colored to be vertex-transitive. Different Symmetry can be expressed on the same geometric figure with the Face following different uniform color patterns....
s of a square tiling.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the Square tiling is a regular tiling of the Euclidean plane. It has Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 of .

Conway
John Horton Conway

John Horton Conway is a prolific mathematician active in the theory of finite group , knot theory, number theory, combinatorial game theory and coding theory....
 calls it a quadrille.

The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling
Triangular tiling

In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....
 and the hexagonal tiling
Hexagonal tiling

In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....
.

Uniform colorings

There are 9 distinct uniform coloring
Uniform coloring

In geometry, a uniform coloring is a property of a uniform figure that is colored to be vertex-transitive. Different Symmetry can be expressed on the same geometric figure with the Face following different uniform color patterns....
s of a square tiling. (Naming the colors by indices on the 4 squares around a vertex: 1111, 1112(i), 1112(ii), 1122, 1123(i), 1123(ii), 1212, 1213, 1234. (i) cases have simple reflection symmetry, and (ii) glide reflection symmetry.)

Wythoff constructions from square tiling


Like the uniform polyhedra
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
 there are eight uniform tilings that can be based from the regular square tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three unique topologically forms: square tiling, truncated square tiling
Truncated square tiling

In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....
, snub square tiling
Snub square tiling

In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex . It has Schl?fli symbol of s....
.

OperationSchläfli
symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
Wythoff
Symbol
Wythoff construction

File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....
Vertex figure
Vertex configuration

In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
Image
Parentt0 4 | 2 444
Truncation
Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
t0,1 2 4 | 44.8.8
Truncated square tiling

In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....
Rectification
Rectification (geometry)

In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....
t1 2 | 4 4(4.4)2
Bitruncation
Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
t1,2 2 4 | 44.8.8
Dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
t2 4 | 2 444
Cantellation
Cantellation (geometry)

In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex....
t0,2 4 4 | 24.4.4.4
Omnitruncation
Omnitruncation (geometry)

In geometry, an omnitruncation is an operation applied to a regular polytope in a Wythoff construction that creates a maximum number of facets....
t0,1,2 2 4 4 |4.8.8
Truncated square tiling

In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....
Snubbing
Snub (geometry)

In geometry, an alternation is an operation on a polyhedron or tessellation that fully truncates alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedron....
s | 2 4 43.3.4.3.4
Snub square tiling

In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex . It has Schl?fli symbol of s....


See also

  • List of uniform tilings
  • List of regular polytopes
    List of regular polytopes

    This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....
  • Tilings of regular polygons
  • Square lattice
    Square lattice

    In mathematics, the square lattice is one of the five two-dimensional Lattice types. It is the two-dimensional version of the integer lattice....
  • Checkerboard
    Checkerboard

    A checkerboard is a board on which English draughts is played. It is an 8×8 board and the 64 squares are of alternating dark and light color, often red and black....

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