Talk:Tiling

need to mention Roger Penrose's work on tilings

This has been done a while ago, see tiling page. Karlscherer3 01:35, 19 Jun 2005 (UTC)

" Alternating tilings A tiling {T} of a shape S is called alternating if {T} is the union of two disjoint sets {T1} and {T2} of tiles such that any tile T adjacent to a tile T1 in {T1} is in {T2} and, vice versa, any tile T adjacent to a tile T2 in {T2} is in {T1}. Example : If we want to tile the plane with squares and dominoes in an alternating way, then we must find a way that the plane is fully covered without gaps or overlaps (otherwise it is not a tiling at all) and such that no two squares have a side or a part of a side in common (but having a point in common is allowed) and such that no two dominoes have a side or a part of a side in common (but having a point in common is allowed)1,2. " Do all the squares have to be the same size? Do all the dominoes have to be the same size? If so, I don't see how this can be done - can someone expand please. -- SGBailey 23:06 Apr 3, 2003 (UTC)

Since this restriction is not requested, I cannot see why you assume that it is to be imposed. If additional restrictions are needed, they would be stated. There is no need to state that some restrictions are NOT given. Using differently sized squares, of course there are solutions to this problem. Karlscherer3 00:09, 19 Jun 2005 (UTC)

What about invariants? For instance, Dehn's invariant can be used to show that three dimensional Euclidean space cannot be tiled with regular tetrahedra. PML.

This article needs pictures and drawings! 213.236.117.2 11:14, 8 Sep 2004 (UTC)

They are coming! I have added already more than 100 images to the pages the "types of tilings" list points to. (see pages on "convex tiling" and "tetrad (tiling)". It all takes a while, since I had to write about 100 pages on puzzles as well. I would like to see more people suppling images! It is also so easy to ask for more... why don't you provide images? Karlscherer3 00:25, 19 Jun 2005 (UTC)

article meandering and off

this article is quite meandering and off. The main issues are not discussed:

definition of a tiling.
definition of vertexs, edges, and tiles.

k-hedral. A tiling is k-hedral if it has k prototiles.
k-gonal. A tiling is k-gonal if it has k types of vertices.
k-toxal. A tiling is k-toxal if it has k types of edges.

consider the number of types of tiles, vertices, edges, with respect to the tiling's symmetry, we have k-isogonal, k-isohedral, k-isotoxal tilings. That is, if the symmetry of the tiling maps the prototiles into k classes, then that tiling is k-isohedral.

definition of regular tilings.
definition of archimedean tilings.
the problem of a tiling of r prototiles admits m number of tilings. If r==m==1, it the prototile is said to be isomorphic. If r==m, it's r-morphic.
topological class of tilings.
duality
aperiodic tilings.

Xah Lee 02:53, 2004 Sep 1 (UTC)

Good suggestions, so why didn't you add them in? I will add your suggestions, but maybe you can help edit the tiling page, because you seem very knowledgable in this area. Any help is welcome and more constructive than suggestions. By the way, aperiodic tilings have their own page, and the link is there. Karlscherer3 00:25, 19 Jun 2005 (UTC)

All fixed now, apart from:

the problem of a tiling of r prototiles admits m number of tilings. If r==m==1, it the prototile is said to be isomorphic. If r==m, it's r-morphic.
topological class of tilings.
duality

First topic: unclear, please explain. All topics: Please give sources where to find these topics in wiki or otherwise. Can someone please fill in the gaps here? Thank you. Karlscherer3 01:31, 19 Jun 2005 (UTC)

Tessellation

What is the difference between tiling and tessellation? --Henrygb 00:36, 25 Mar 2005 (UTC)

This has been answered in the amended tiling page now. Karlscherer3 23:54, 18 Jun 2005 (UTC)

wallpaper groups

Latest comment: 19 years ago1 comment1 person in discussion

There exists an article on Wallpaper groups. The tiling article and Wallpaper groups should mention each other. Dmharvey 19:35, 30 May 2005 (UTC)Reply

Thanks for the reference. I will do this. Karlscherer3 23:55, 18 Jun 2005 (UTC)

Alhambra symmetries

The article says examples of all 17 wallpaper groups can be found at the Alhambra. Does this rely on coloring to break a higher geometric symmetry? Can we confirm this with photographs?

B. Grünbaum, and later B. Grünbaum, Z. Grünbaum, & G. C. Shephard, assert that p2 and p3m1 are not found at the Alhambra, but do appear nearby in Toleda from about the same period; however pg and pgg are completely missing from Islamic art of the time. Contradicting this is an article by R. Pérez-Gómez entitled "The four regular mosaics missing in The Alhambra", asserting that the missing groups are not missing.

Not having copies of these articles, nor photographs to support the claim of 17, unlike the claim of 13, I'd like to see something definitive.

Anyone? KSmrq 21:38, 10 Jun 2005 (UTC)

Good news, partly. I found online a comment by Coxeter saying he accepts that 16 of the 17 groups appear, based on the Pérez-Gómez article. The only copy of that article I have found has pictures which are impossible to decipher; but I'm willing to trust this expert opinion. And given that the final missing group is reported nearby in Toledo, I'll let the current claim stand. KSmrq 18:05, 25 Jun 2005 (UTC)

Terminology

With regard to the notice about "abnormal terminology", I think the proper source for choosing preferred terminology would be Grünbaum and Shephard, Tilings and Patterns, which still serves as the standard work in the area although largely written in the 1970s. For example, "Neat tilings" should be "edge-to-edge", "Nowhere-neat tilings" should probably be "unilateral", "Polysquares" should be "Polyominoes" without mention of the nonstandard term "Polysquares" (there is no tiling content in that section at present, but tiling with polyominoes is a legitimate topic to discuss here), "Pure tilings" should be "monohedral", and concepts described in the article which are not discussed in Tilings and Patterns may not be important enough to cover unless referenced to multiple articles reviewed in Math Reviews. Joseph Myers 19:09, 23 Jun 2005 (UTC)

Note

Karl Scherer's edits to Wikipedia appear to be predominantly original research, and have been removed in accordance with VfD at

~~~~ 3 July 2005 17:42 (UTC)

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