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・ Pentagonal gyrocupolarotunda
・ Pentagonal hexecontahedron
・ Pentagonal icositetrahedron
・ Pentagonal number
・ Pentagonal number theorem
・ Pentagonal orthobicupola
・ Pentagonal orthobirotunda
・ Pentagonal orthocupolarotunda
・ Pentagonal planar molecular geometry
・ Pentagonal polytope
・ Pentagonal prism
・ Pentagonal pyramid
・ Pentagonal pyramidal molecular geometry
・ Pentagonal pyramidal number
・ Pentagonal rotunda
Pentagonal tiling
・ Pentagonal trapezohedron
・ Pentagonia
・ Pentagonia (disambiguation)
・ Pentagonia (plant)
・ Pentagonia involucrata
・ Pentagonia orthoneura
・ Pentagonia peruviana
・ Pentagonia rubiflora
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Pentagonal tiling : ウィキペディア英語版
Pentagonal tiling
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.
A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn.
== Monohedral convex pentagonal tilings ==

Fifteen types of convex pentagons are known to tile the plane monohedrally (i.e. with one type of tile).〔Tilings and Patterns, Sec. 9.3 Other Monohedral tilings by convex polygons〕 The most recent one was discovered in 2015. It is not known whether this list is complete.〔 showed that there are only eight edge-to-edge convex types, a result obtained independently by .
Each enumerated tiling family contains pentagons that belong to no other type; however, some individual pentagons may belong to multiple types. In addition, some of the pentagons in the known tiling types also permit alternative tiling patterns beyond the standard tiling exhibited by all members of its type.
The sides of length a, b, c, d, e are directly clockwise from the angles at vertices A, B, C, D, E respectively. (Thus,
A, B, C, D, E are opposite to d, e, a, b, c respectively.)
Many of these monohedral tile types have degrees of freedom. These freedoms include variations of internal angles and edge lengths. In the limit, edges may have lengths that approach zero or angles that approach 180°. Types 1, 2, 4, 5, 6, 7, 8, 9, and 13 allow parametric possibilities with nonconvex prototiles.
Periodic tilings are characterised by their wallpaper group symmetry, for example p2 (2222) is defined by four 2-fold gyration points. This nomenclature is used in the diagrams below, where the tiles are also colored by their ''k''-isohedral positions within the symmetry.
A ''primitive unit'' is a section of the tiling that generates the whole tiling using only translations, and is as small as possible.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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