![]() In addition, one could require them to be polygons. ![]() In addition, one could require the tiles to be compact. In particular, one could require the tiling to be invariant under some discrete group of isometries. One could draw the line in a variety of ways and degrees of generality. Is there a clear separation between these two groups somehow? 16 that the word tesselation is used "synonymously or with similar meaning" to "tiling" English Wikipedia agrees. It appears that the terms "tiling" and "tessellation" can be interchanged Grünbaum and Shephard, the authors of the book Patterns and Tilings, at first intended to make it just the first chapter of a book on geometry, but ended up with a 700-pages book just on this subject. I would like to come up with a final list of "tilings"Īn ambitious goal. Would you mind clarifying how I can grab ahold better conceptually of Wythoff, Vertex figures, and the "group theory symbols"? Is the coxeter diagram stuff important? I don't understand that one yet, but i think knowing this much should get me further along. And I'm not sure what the "group theory symbols" mean. Vertex figures with their numbers like q.2p.2p and their p's and q's tell us about the polygons around a vertex somehow. Wythoff tells us something about the triangle generator. They are all focusing on different pieces of the puzzle. Sometimes they will list several Wythoff constructions, like here, what does that mean?:īasically in summary, we have 4 (or 5?) notations: For example on that last like (k uniform tilings), it has stuff like. It shows a Coxeter diagram, Wythoff symbol, Vertex figure, and then p4m,, (*442), where does that come from, what is all that? I think that may be a group theory thing? More of those here. Then we have this on the Euclidean uniform tilings page: Where did the (p 3 2) and (p = 3, 4, 5) come from, what does that mean? Also, in the Vertex figure, why do they use p and q and do q.2p.2p, instead of like the example of 3.5.3.5? Finally for this, why does the Wythoff symbol use a number 2 in there? The sequence of triangles (p 3 2) change from spherical (p = 3, 4, 5), to Euclidean (p = 6), to hyperbolic (p ≥ 7). ![]() | p q r is designated for the case where all mirrors are active, but odd-numbered reflected images are ignored.p q r | indicates that the generator lies in the interior of the triangle.p q | r indicates that the generator lies on the edge between p and q,.p | q r indicates that the generator lies on the corner p,.The three numbers in Wythoff's symbol, p, q, and r, represent the corners of the Schwarz triangle used in the construction. The Wythoff construction begins by choosing a generator point on a fundamental triangle. The notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. The notation "a.b.c" describes a vertex that has 3 faces around it, faces with a, b, and c sides.įor example, "3.5.3.5" indicates a vertex belonging to 4 faces, alternating triangles and pentagons. So the Wythoff symbol is up above each image, the Vertex figure down below.Ī vertex configuration/figure is given as a sequence of numbers representing the number of sides of the faces going around the vertex. I have a few scattered questions revolving around this that I am hoping to clarify. Escher's works), and others have much higher symmetry. Sidenote, it appears that the terms "tiling" and "tessellation" can be interchanged, where some are arbitrary repeated patterns (like M.C. I would like to come up with a final list of "tilings", but am having hard determining what the name or even a standard representation of the tiling is. ![]()
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