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In the list of triangles for the icosahedral symmetry, 4 triangles are missing: (3/2 5 5/2), (3/2 5/4, 5/3), (3, 5/4, 5/2) and (3, 5, 5/3) [ contact ivan.deman@hotmail.com ] - 05:57, October 16, 2008 85.27.18.29
Why is (3/2 4 4) listed as density 2? That's the triangle you get when you dissect (2 2 2) into three equal triangles meeting at the centroid of (2 2 2). So shouldn't it be density 1? -- Vaughan Pratt ( talk) 01:07, 3 October 2009 (UTC)
In fact ANY triangle, or any non-self-intersecting quadrilateral, can be used to tile the plane with no overlaps (simply start with a triangle or non-self-intersecting quadrilateral and add further congruent shapes by rotating existing shapes through 180 degrees about the midpoint of any side; in the case of a general triangle, adding another triangle in this fashion causes the two together to form a parallelogram, and it is easy to tile parallelograms in the plane). 86.4.253.180 ( talk) 17:57, 11 April 2013 (UTC) 86.4.253.180 ( talk) 17:57, 11 April 2013 (UTC)
I was not able to find a paper with that title by Schwarz. Perhaps the paper "Ueber diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt" is meant? My German is not very good, sorry. Sam nead ( talk) 12:59, 24 August 2013 (UTC)
I notice that where (p q r) are all integers they are listed in ascending order, but I haven't found a pattern where they are fractional. Any objecting to changing
from | to |
---|---|
(2 3/2 3) | (3/2 2 3) |
(3 4/3 4) | (4/3 3 4) |
(3 5/3 5) | (5/3 3 5) |
(2 3/2 3/2) | (3/2 3/2 2) |
(2 3/2 4) | (3/2 2 4) |
(2 3 4/3) | (4/3 2 3) |
(2 3 5/2) | (2 5/2 3) |
(2 5/3 5) | (5/3 2 5) |
(3 5/3 5/2) | (5/3 5/2 3) |
(3 5/4 5) | (5/4 3 5) |
(2 3/2 4/3) | (4/3 3/2 2) |
(2 3/2 5) | (3/2 2 5) |
(2 3 5/3) | (5/3 2 3) |
(3/2 4/3 4/3) | (4/3 4/3 3/2) |
(3 3 5/4) | (5/4 3 3) |
(3 5/4 5/2) | (5/4 5/2 3) |
(2 3/2 5/2) | (3/2 3/2 2) |
(3/2 3 5/3) | (3/2 5/3 3) |
(2 3 5/4) | (5/4 2 3) |
(2 5/4 5/2) | (5/4 2 5/2) |
(2 3/2 5/3) | (3/2 5/3 2) |
(2 5/4 5/3) | (5/4 5/3 2) |
(2 3/2 5/4) | (5/4 3/2 2) |
(3/2 5/4 5/3) | (5/4 3/2 5/3) |
(3/2 3/2 5/4) | (5/4 3/2 3/2) |
(3/2 5/4 5/4) | (5/4 5/4 3/2) |
? |
— Tamfang ( talk) 00:21, 14 March 2014 (UTC)
I censored the first of these because it is not generated by reflections in a Schwartz triangle. Compare the second pic, which is. — Tamfang ( talk) 06:15, 15 August 2023 (UTC)
I'm a little confused by this - I thought tilings were non-overlapping by definition. Can anyone clarify? Fbm23.4 ( talk) 16:14, 8 February 2024 (UTC)