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If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane. The Conway criterion is a sufficient, but not necessary, set of rules for deciding whether a given shape tiles the plane periodically without reflections: some tiles fail the criterion, but still tile the plane. [ 19 ]
Example tessellation based on a Type 1 hexagonal tile. In its simplest form, the criterion simply states that any hexagon with a pair of opposite sides that are parallel and congruent will tessellate the plane. [8] In Gardner's article, this is called a type 1 hexagon. [7] This is also true of parallelograms.
The tesseract can make a regular tessellation of 4-dimensional hyperbolic space, with 5 tesseracts around each face, with Schläfli symbol {4,3,3,5}, called an order-5 tesseractic honeycomb. The Ammann–Beenker tiling is an aperiodic tiling in 2 dimensions obtained by cut-and-project on the tesseractic honeycomb along an eightfold rotational ...
All five can create isohedral tilings, meaning that the symmetries of the tiling can take any tile to any other tile (more formally, the automorphism group acts transitively on the tiles). B. Grünbaum and G. C. Shephard have shown that there are exactly twenty-four distinct "types" of isohedral tilings of the plane by pentagons according to ...
Create account; Log in; Personal tools. Donate; Create account; ... This is a list of tessellations. This list is incomplete; you can help by adding missing items.
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}.
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names include half cubic honeycomb, half cubic cellulation, or tetragonal disphenoidal cellulation.
A Pythagorean tiling Street Musicians at the Door, Jacob Ochtervelt, 1665.As observed by Nelsen [1] the floor tiles in this painting are set in the Pythagorean tiling. A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides.