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  2. Euclidean tilings by convex regular polygons - Wikipedia

    en.wikipedia.org/wiki/Euclidean_tilings_by...

    Following Grünbaum and Shephard (section 1.3), a tiling is said to be regular if the symmetry group of the tiling acts transitively on the flags of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first ...

  3. List of Euclidean uniform tilings - Wikipedia

    en.wikipedia.org/wiki/List_of_euclidean_uniform...

    An example of uniform tiling in the Archeological Museum of Seville, Sevilla, Spain: rhombitrihexagonal tiling Regular tilings and their duals drawn by Max Brückner in Vielecke und Vielflache (1900) This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane , and their dual tilings.

  4. Lists of uniform tilings on the sphere, plane, and hyperbolic ...

    en.wikipedia.org/wiki/Lists_of_uniform_tilings...

    There are 4 symmetry classes of reflection on the sphere, and three in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are also listed. (Increasing any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.) Point groups:

  5. Uniform tiling - Wikipedia

    en.wikipedia.org/wiki/Uniform_tiling

    The elongated triangular tiling, the only non-Wythoffian convex uniform tiling There are symmetry groups on the Euclidean plane constructed from fundamental triangles: (4 4 2), (6 3 2), and (3 3 3). Each is represented by a set of lines of reflection that divide the plane into fundamental triangles.

  6. List of regular polytope compounds - Wikipedia

    en.wikipedia.org/wiki/List_of_regular_polytope...

    The only regular Euclidean compound honeycombs are an infinite family of compounds of cubic honeycombs, all sharing vertices and faces with another cubic honeycomb. This compound can have any number of cubic honeycombs. The Coxeter notation is {4,3,4}[d{4,3,4}]{4,3,4}.

  7. List of regular polytopes - Wikipedia

    en.wikipedia.org/wiki/List_of_regular_polytopes

    One could go further (as is done in the table above) and find tilings with ultra-ideal vertices, outside the Poincaré disc, which are dual to tiles inscribed in hypercycles; in what is symbolised {p, iπ/λ} above, infinitely many tiles still fit around each ultra-ideal vertex. [16]

  8. Uniform tiling symmetry mutations - Wikipedia

    en.wikipedia.org/wiki/Uniform_tiling_symmetry...

    In geometry, a symmetry mutation is a mapping of fundamental domains between two symmetry groups. [1] They are compactly expressed in orbifold notation.These mutations can occur from spherical tilings to Euclidean tilings to hyperbolic tilings.

  9. Hexagonal tiling - Wikipedia

    en.wikipedia.org/wiki/Hexagonal_tiling

    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} (as a truncated triangular tiling).