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hyperbolic tilings[edit]

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
H2 tiling 237-7.png H2checkers 237.png H2 tiling 237-5.png H2 tiling 237-1.png H2 tiling 237-3.png H2 tiling 237-2.png H2 tiling 237-6.png H2 tiling 237-4.png H2 snub 237a.png H2 snub 237b.png
2 4 5
area π/20
H2 tiling 245-7.png H2checkers 245.png H2 tiling 245-5.png H2 tiling 245-1.png H2 tiling 245-3.png H2chess 245f.png H2 tiling 245-2.png H2 tiling 245-6.png H2 tiling 245-4.png H2chess 245c.png H2 snub 245a.png H2 snub 245b.png
3 3 4
area π/12
H2 tiling 334-7.png H2checkers 334.png H2 tiling 334-5.png H2 tiling 334-1.png H2 tiling 334-3.png H2 tiling 334-2.png H2 tiling 334-6.png H2 tiling 334-4.png H2 snub 334a.png H2 snub 334b.png
2 3 ∞
area π/6
H2 tiling 23i-7.png H2checkers 23i.png H2 tiling 23i-5.png H2 tiling 23i-1.png H2chess 23ib.png H2 tiling 23i-3.png H2 tiling 23i-2.png H2 tiling 23i-6.png H2chess 23ie.png H2 tiling 23i-4.png H2 snub 23ia.png H2 snub 23ib.png
2 ∞ ∞
area π/2
H2 tiling 2ii-7.png H2checkers 2ii.png H2 tiling 2ii-5.png H2chess 2iid.png H2 tiling 2ii-1.png H2chess 2iib.png H2 tiling 2ii-3.png H2chess 2iif.png H2 tiling 2ii-2.png H2chess 2iia.png H2 tiling 2ii-6.png H2chess 2iie.png H2 tiling 2ii-4.png H2chess 2iic.png H2 snub 2iia.png H2 snub 2iib.png
∞ ∞ ∞
area π
H2 tiling iii-7.png H2checkers iii.png H2 tiling iii-5.png H2chess iiid.png H2 tiling iii-1.png H2chess iiib.png H2 tiling iii-3.png H2chess iiif.png H2 tiling iii-2.png H2chess iiia.png H2 tiling iii-6.png H2chess iiie.png H2 tiling iii-4.png H2chess iiic.png H2 snub iiia.png H2 snub iiib.png

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.)


Hilda asteroid as seen from Jupiter.png

Orbits of two idealized asteroids of the Hilda family, in the rotating reference frame of Jupiter.

See also[edit]