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In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle, (p q r), defined by internal angles as π/p, π/q, and π/r. Special cases are right triangles (p q 2).
The scaling is uniform if and only if the scaling factors are equal (v x = v y = v z). If all except one of the scale factors are equal to 1, we have directional scaling. In the case where v x = v y = v z = k, scaling increases the area of any surface by a factor of k 2 and the volume of any solid object by a factor of k 3.
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 ...
For example 4.8.8 means one square and two octagons on a vertex. These 11 uniform tilings have 32 different uniform colorings. A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices.
A fundamental domain triangle is denoted (p q r), where p, q, r are whole numbers > 1, i.e. ≥ 2; a fundamental domain right triangle is denoted (p q 2). The triangle may exist as a spherical triangle , a Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of p , q , and r .
Namely, the bijection X × X → Y × Y sends (x 1,x 2) to (f(x 1),f(x 2)); the bijection P(X) → P(Y) sends a subset A of X into its image f(A) in Y; and so on, recursively: a scale set being either product of scale sets or power set of a scale set, one of the two constructions applies. Let (X,U) and (Y,V) be two structures of the same signature.
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