User:Tamfang

hyperbolic tilings

Shown in the conformal ('stereographic') disc model. In each, the origin is equidistant from the three defining mirrors.

Made by crude little Python programs. Full size is 2520 pixels (least common multiple of 1,2,3,4,5,6,7,8,9,10).

Ranked by the area of the fundamental triangle.

You will notice that many of the duals are missing; because, where an odd number of facets meet at a vertex, I have not thought of an algorithm to color them. (The black-and-white figures are made by counting mirror-flips from the pixel to the interior of the triangle that contains the centre.)

p q r	xxx	xox	oox	oxx	oxo	xxo	xoo	snub
2 3 7 area π/42
2 4 5 area π/20
3 3 4 area π/12
2 3 ∞ area π/6
2 ∞ ∞ area π/2
∞ ∞ ∞ area π

The difference between the two snubs in each row is whether or not the central triangle contains a vertex.

In my opinion the above six rows abundantly illustrate the principles; but, by popular demand, I made a hundred more. (And it appears that each row now has at least one article in Wikipedia. I lament my role as enabler.)

uniform tilings of Euclidean 3-space and their vertex figures

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stereographic projections of uniform tilings of the 3-sphere

made with Jenn

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other

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  Orbits of two idealized asteroids of the Hilda family, in the rotating reference frame of Jupiter (blue).
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  an example of T_h symmetry

See also

• My page on English Wikipedia.
• My personal site.