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Truncated square tiling

 

 
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Truncated 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 truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon. 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 t0,1.

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 truncated quadrille, constructed as a 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....
 operation applied to a square tiling
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
 (quadrille).

Other names used for this pattern include Mediterranean tiling and octagonal tiling.

There are 3 regular and 8 semiregular tilings in the plane.

Uniform colorings
There are two 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 truncated 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 truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon. 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 t0,1.

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 truncated quadrille, constructed as a 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....
 operation applied to a square tiling
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
 (quadrille).

Other names used for this pattern include Mediterranean tiling and octagonal tiling.

There are 3 regular and 8 semiregular tilings in the plane.

Uniform colorings


There are two 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 truncated square tiling. (Naming the colors by indices around a vertex (4.8.8): 122, 123.)

2 colors: 122
3 colors: 123


Related polyhedra and tilings


It is topologically related to the polyhedron truncated octahedron
Truncated octahedron

The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
, 4.6.6
Truncated Octahedron
This tiling which alternates large and small squares can be seen as topologically identical to the truncated square tiling. The squares are rotated 45 degrees and octagons are distorted into squares with mid-edge vertices.

This tiling (with two chiral
Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
 forms and two 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) is sometimes called the Pythagorean tiling because its geometry can be used in a proof for the Pythagorean theorem
Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
.


See also

  • Tilings of regular polygons
  • List of uniform tilings


External links