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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]
Dual semi-regular Article Face configuration Schläfli symbol Image Apeirogonal deltohedron: V3 3.∞ : dsr{2,∞} Apeirogonal bipyramid: V4 2.∞ : dt{2,∞} Cairo pentagonal 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).
Although a cube is the only regular polyhedron that admits of tessellation, many non-regular 3-dimensional shapes can tessellate, such as the truncated octahedron. The second part of Hilbert's eighteenth problem asked for a single polyhedron tiling Euclidean 3-space, such that no tiling by it is isohedral (an anisohedral tile).
The regular pentagon cannot form Cairo tilings, as it does not tile the plane without gaps. There is a unique equilateral pentagon that can form a type 4 Cairo tiling; it has five equal sides but its angles are unequal, and its tiling is bilaterally symmetric. [4] [13] Infinitely many other equilateral pentagons can form type 2 Cairo tilings. [4]
A kaleidoscope whose mirrors are arranged in the shape of one of these tiles will produce the appearance of an edge tessellation. However, in the tessellations generated by kaleidoscopes, it does not work to have vertices of odd degree, because when the image within a single tile is asymmetric there would be no way to reflect that image ...
Like the cube, it can tessellate (or "pack") 3-dimensional space, as a permutohedron. The truncated octahedron was called the "mecon" by Buckminster Fuller. [1] Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths 9 / 8 √ 2 and 3 / ...
The rhombic dodecahedron is a space-filling polyhedron, meaning it can be applied to tessellate three-dimensional space: it can be stacked to fill a space, much like hexagons fill a plane. It is a parallelohedron because it can be space-filling a honeycomb in which all of its copies meet face-to-face. [7]