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April 12

New problem

If you draw 2 grids of lines that intersect at 90 degree angles, you'll get a square tiling.

If you draw 3 grids of lines that intersect at 60 degree angles, you'll get a triangular tiling.

What will you get if you draw 4 grids of lines that intersect at 45 degree angles?? Georgia guy (talk) 22:30, 12 April 2020 (UTC)[reply]

It will be a different tiling of triangles. The 60-degree one makes ~~isosceles~~ equilateral triangles, the 45-degree one makes right triangles. Bubba73 ^{You talkin' to me?} 23:07, 12 April 2020 (UTC)[reply]

The tetrakis square tiling? In this tiling the distances between the diagonal lines are larger (by a factor of √2) than those between the horizontal/vertical lines. Or if you make them the same, you get an irregular tiling with isosceles right triangles of infinitely many sizes, down to any arbitrarily small size. --Lambiam 23:17, 12 April 2020 (UTC)[reply]

Lambiam, what would such a tiling look like?? Georgia guy (talk) 23:22, 12 April 2020 (UTC)[reply]

See the image (artist impression). --Lambiam 23:46, 12 April 2020 (UTC)[reply]

Is this related to {8/3, 8} (or even the same thing)? Double sharp (talk) 14:19, 13 April 2020 (UTC)[reply]

If I'm not mistaken, the notation {8/3} denotes a regular star polygon. The beasts we have here are not regular (except possibly for one at most) because the √2 is a surd. --Lambiam 20:04, 13 April 2020 (UTC)[reply]

It might be interesting to compare the set of vertices in {8/3,8} with the set of vertices in the diagram above (assuming that in the diagram there is a common central vertex between the two square tilings}. In {8/3,8} you're allowed to move in any of the cardinal directions (N, S, E, W) or in any of the intercardinal directions (NE, SE, SW, NW) at any time, while in the diagram you can move in any of the eight directions from the center, but once you start off from the center in one direction, you have to stick with that same set of directions (cardinal or intercardinal) until you get back to the center again. As a subset of R², in the first case you have the sum of two copies of Z² and in the second you have the union of two copies of Z². If we call the second copy of Z² Z², then Z² ⊕ Z² contains both (1, 0) and (√2, 0), so its intersection with R×{0} is dense in the line. Meanwhile the intersection of Z² ∪ Z² with R×{0} is just Z×{0} ∪ √2Z×{0} which is not dense. In fact, Z² ⊕ Z² is dense in R² and while there are pairs of points in Z² ∪ Z² that are arbitrarily close together, each individual point is isolated. I'm not convinced that the polytopes article is completely correct on this; if we consider {8/3,8} to be a set of overlapping star octagons together with their edges and vertices, then vertices all look alike and so the "tiling" is regular in some sense. It seems to me that the reason it fails to be a tiling is that each point in the plane is covered by an infinite number of faces. --RDBury (talk) 01:42, 14 April 2020 (UTC)[reply]

@RDBury: Thank you for the explanation! Now I understand the difference.

Yes, it seems to me that the polytopes article is wrong: the problem is really infinite density. (Because if I understand this right, surely translation by a unit length in a cardinal or intercardinal direction must transform {8/3,8} into itself, so there is periodicity unlike what is claimed.) Double sharp (talk) 07:05, 14 April 2020 (UTC)[reply]

I assume the complaint is not about the Polytope article but about List of regular polytopes and compounds#Euclidean star-tilings. In such limiting cases, precise definitions become important. A "tile" is said to be a "geometric shape" (Tesselation). Geometric shape defines the concept as a "geometric object" modulo the similarity transformations (although not using that terminology; I think that in many contexts, such as tiling, this is too loose: there congruence is required, and often mirroring is excluded). In the article, geometric object is an Easter egg for "mathematical object", which is "anything that has been (or could be) formally defined". Shape appears to define the geometric concept, in the section Equivalence of shapes, as "[any] subset[s] of a Euclidean space" (modulo certain transformations) – too restrictive by requiring the space to be Euclidean. This leaves us (or me) with a burning question. Is a one-point set a geometric shape? Can the plane be tiled with point tiles ("monotopes")? The point tiling is very strongly periodic. Note that the intersection of the star polygons of {8/3,8} covering a given point is a point tile. --Lambiam 11:02, 14 April 2020 (UTC)[reply]

Yeah, it appears the definition of "tiling" gets a bit vague when it comes to edge cases. MathWorld says "a tiling is a collection of disjoint open sets, the closures of which cover the plane." This would rule out the tiling by points, but it also rules out spaces with dimension other than 2 as well as non-Euclidean planes. Presumably one could substitute something more general for "plane" to include these cases and get a fairly comprehensive definition which rules out the problematic examples. Are polygons and polyhedra then tilings in some sense of the circle and sphere? And if so, what about self-intersecting polygons and polyhedra, compounds, etc. Unless someone can find something a bit more authoritative than MathWorld which speaks to this I doubt these issues can be settled here. --RDBury (talk) 14:22, 14 April 2020 (UTC)[reply]

I don't see why the MathWorld definition should rule out tiling non-Euclidean planes or higher-dimensionsonal spaces (by higher-dimensional polytopes). --Lambiam 18:12, 14 April 2020 (UTC)[reply]

The article doesn't seem consistent even with itself, for example in one sentence it mentions generalizations to higher dimensions and in the next it gives a definition that specifies only the plane. I think the main takeaway is that each tile must contain an open set, ruling out points, lines etc. as tiles. --RDBury (talk) 12:05, 15 April 2020 (UTC)[reply]