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The origin of this type of tessellated pavement remains uncertain. The size and shape of these polygons appears to be dependent to a large extent on the grain size, texture, and coherence of the rock. This polygonal tessellation is best developed in relatively fine-grained, uniform, and siliceous or silicified sandstones. [1]
Dual semi-regular Article Face configuration Schläfli symbol Image Apeirogonal deltohedron: V3 3.∞ : dsr{2,∞} Apeirogonal bipyramid: V4 2.∞ : dt{2,∞} Cairo pentagonal tiling
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]
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]
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).
Architectonic tessellation Catoptric tessellation Name Coxeter diagram Image Vertex figure Image Cells Name Cell Vertex figures; J 11,15 A 1 W 1 G 22 δ 4 nc [4,3,4] Cubille (Cubic honeycomb) Octahedron, Cubille: Cube, J 12,32 A 15 W 14 G 7 t 1 δ 4 nc [4,3,4] Cuboctahedrille (Rectified cubic honeycomb) Cuboid, Oblate octahedrille: Isosceles ...
The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal spheres in ordinary space (see Kepler conjecture ).
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 ...