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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.
The SI has special names for 22 of these coherent derived units (for example, hertz, the SI unit of measurement of frequency), but the rest merely reflect their derivation: for example, the square metre (m 2), the SI derived unit of area; and the kilogram per cubic metre (kg/m 3 or kg⋅m −3), the SI derived unit of density.
(Increasing any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.) Point groups: (p 2 2) dihedral symmetry, =,, … (order ) (3 3 2) tetrahedral symmetry (order 24) (4 3 2) octahedral symmetry (order 48) (5 3 2) icosahedral symmetry (order 120)
(4 4/3 ∞), (3/2 3 ∞), (6 6/5 ∞), (3 3/2 ∞). Branko Grünbaum and G. C. Shephard , in the 1987 book Tilings and patterns , section 12.3, enumerate a list of 25 uniform tilings, including the 11 convex forms, and add 14 more they call hollow tilings , using the first two expansions above: star polygon faces and generalized vertex figures.
For example: 3 6; 3 6; 3 4.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling. Broken down, 3 6 ; 3 6 (both of different transitivity class), or (3 6 ) 2 , tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided ...
The nit (nt) is a unit of luminance equal to one candela per metre squared (1 cd⋅m −2). The lambert (L) is a unit of luminance equal to 10 4 /π cd⋅m −2 . The lumerg is a unit of luminous energy equal to 10 −7 lumen-seconds (100 nlm s).
The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7) Schwarz triangle.This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a triangulation ...
Comparison of 1 square metre with some Imperial and metric units of area. The square metre (international spelling as used by the International Bureau of Weights and Measures) or square meter (American spelling) is the unit of area in the International System of Units (SI) with symbol m 2. [1] It is the area of a square with sides one metre in ...