Talk:Tessellation

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. At present this is largely a random collection of a few specific topics. It needs going through with reference to Tilings and Patterns to make sure that all the main topics and terminology are covered in a coherent arrangement; in the process, most of the "See also" links should become links at appropriate points in the body of the article instead. Joseph Myers 02:20, 20 May 2007 (UTC)

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Given the Latin, and the spelling given in the Concise Oxford Dictionary, I'd say tesselation isn't correct.

Charles Matthews 07:18, 30 Nov 2003 (UTC)

So what'd you say is correct? I ignore Latin and have not the C.O.D. ... --euyyn 22:03, 7 June 2006 (UTC)

[edit] Tiling

What is the difference between tiling and tessellation? --Henrygb 00:35, 25 Mar 2005 (UTC)

According to Mathworld, a tessellation is formally a periodic tiling, and, as such, this entry is factually inaccurate. (See: http://mathworld.wolfram.com/Tiling.html, http://mathworld.wolfram.com/Tessellation.html)

The definition of a tiling is "a collection of disjoint open sets, the closures of which cover the plane" which makes no claims about periodicity or lack thereof. I think this should be corrected and another entry written for 'Tiling'. Also in such a case the graphic on this page would be wholly inappropriate, being an aperiodic tiling. Perhaps a mosaic would be more fitting. --Caliprincess 18:17, 18 August 2005 (UTC)

According to Grünbaum & Shepherd (which I trust more than MathWorld), tiling and tessellation are used synonymously or with similar meanings in the mathematical literature. -- Jitse Niesen (talk) 16:01, 31 August 2005 (UTC)

[edit] "average number of sides meeting at a vertex"

This phrase, in the section Tessellation#Number of sides of a polygon vs. number of sides at a vertex, needs clarification - it is easy to assume this means that you should count the lines meeting at each of the 5 vertices of the bathroom tiling, and divide by 5, to get an average of 3.4.

Is it sufficient just to add "over the entire plane"? Or should it tell the reader how to determine what the average is? I have only a vague idea how to do the latter, and am not at all sure that it generalises to more complex tessellations. Hv 10:44, 6 September 2005 (UTC)

I think you need to take the harmonic mean of the number of lines meeting at each vertex. The statement seems to be true with some caveats (I'm not sure whether the average always exist or what happens if an edge of the tessellation consists of more than one edge of the polygon). A reference would be nice to settle the matter. -- Jitse Niesen (talk) 14:08, 6 September 2005 (UTC)

For the bathroom tiling we have 2x4+4x3, divided by 6, the ordinary mean value.--Patrick 15:47, 6 September 2005 (UTC)

I added some details and a proviso for extreme cases.--Patrick 16:28, 6 September 2005 (UTC)

This section gives a version of Euler's Theorem for Tilings (Grünbaum and Shephard, 3.3.3) but not expressed very precisely. The theorem only applies for normal tilings (3.2), which excludes the case of "tiles getting smaller and smaller outwardly", while limits may "depend on how the region is expanded to infinity" whenever the tiling is not metrically balanced (3.6); note the theorem can still be expressed (with more care) for tilings which are not metrically balanced. More discussion of these concepts and the precise form of the theorem might be good. Joseph Myers 17:47, 6 September 2005 (UTC)

You're right, I found the statement in the form 1/a + 1/b = 1/2 (notation as in the article) in (G&S, 3.5.13). There they say it's true for each "strongly balanced" tiling, which means that the averages are well defined. -- Jitse Niesen (talk) 18:06, 6 September 2005 (UTC)

[edit] Number of sides of a polygon versus number of sides at a vertex

In my opinion this section needs to be completely rewritten. From the perspective of someone unfamiliar with the subject, they're given a list of numbers with no explination for how they were chosen, and then told that bricks are hexagons and squares are pentagons.

Meekohi 15:45, 5 March 2006 (UTC)

I completely agree. I was so puzzled by the captions of the images that I read the section just to find out who and for what counted those rectangular bricks as hexagons and those rectangular tiles as pentagons. Of course I refered to the section by chance, as the title (N1 versus N2!!) is surrealist to me. In the end, I only found a formula which stands for those bricks and tiles when you treat the word side in an odd way... --euyyn 22:01, 7 June 2006 (UTC)

Definitely, this section needs a rewrite. I added to a caption (for the "hexagonal" bricks) what I thought was the the correct answer but frankly a BS addition and I'm not terribly happy with it. If you interpret where the the lines where bricks join as edges, the corners as vertexes (as in a planar graph), and the areas enclosed as "shapes" the "count" makes sense, though.

To make this section more understandable, alot more background information, like some basic definitions need to be added. Root⁴(one) 19:23, 3 February 2007 (UTC)

I rewrote the captions, linked to the hexagonal tiling and Cairo pentagonal tiling of topologically identical tilings. I admit more explanation could help still. Tom Ruen 21:02, 3 February 2007 (UTC)

[edit] Seven Colours?

It appears that the caption for Image:Torus-with-seven-colours.png on the Tessellation page is wrong, it inadvertantly claims the ability to violate the Four color theorem. The statement about this pattern requiring 7 colours only holds if it is wrapped onto a torus, in an infinite tiling the pattern still only needs 4 colours. Even if I heave the wrong end of the stick here, perhaps the caption could be clearer. --Cs01ab 14:14, 4 April 2006 (UTC)

The four colour theorem only hold for the plane, or equivalently the sphere. Both there surfaces have Euler characteristic two. The Euler characteristic of the torus is zero so it is topologically distinct, and yield a different number for the minimal colouring. See Four color theorem#Generalizations. --Salix alba (talk) 14:33, 4 April 2006 (UTC)

But the caption doesn't talk about joining the arrows to form a torus (as is expected by the arrows, which are, so, unnecesary and misleading), it talks about "infinitely repeating the region [in the plane]". The article's section mentions the number 7 by saying that "to produce a colorization that respects the symmetries of the tessellation, you may need as many as seven colors". The caption says that for this concrete tiling, you have to use no less than seven colors. So... where does the 7 come from? Is 7 the maximum number of colors that can be required? How does the 7 relate to the fact that the tile in question is "formed" by 7 parallelograms? I feel the 7 must be related to the torus thing, but doesn't know how. --euyyn 22:18, 7 June 2006 (UTC)

Well, indeed I do know a bit about how this relates to the torus, as a rectangular region in which opposite sides can be considered adjacent (and that's what happens if you are restricted to repeat the tile ad infinitum) isn't but a torus unfolded. So the answers to my questions are "From the generalization of the theorem to toruses", "Yes", and "The tile pictured is formed by 7 parallelogram just to show the minimum number of colors needed". Someone add an explanation in the article, please? (I'm not confident of my own English) --euyyn 22:27, 7 June 2006 (UTC)

[edit] Hyperbolic Square?

I know very little about hyperbolic shapes or hyperbolic geometry, but a recent edit comment I found kinda interesting... There do appear to be such entities as hyperbolic squares as this online paper Chapter 5 (of what?): The Farey tessellation and circle packing. But what M. C. Escher drew, if it was a hyperbolic square, I wouldn't know (and frankly I don't care to know at the moment.) The current edit may better reflect what should be the intended meaning as I think I can reasonably guess that those entities drawn next to the triangles are quadrilaterals in hyperbolic space. But again, I know hardly anything... like practically Zilch, with respect to hyperbolic geometry. Root⁴(one) 05:30, 7 May 2007 (UTC)

[edit] Russian interwiki is put incorrect

Please, correct somebody who knows how to do it.

[edit] Seven colours reprise

Image:Torus-with-seven-colours.png on the Tessellation page.. claims if painted before tiling then only four colours are needed, could you please explain this? it clearly requires seven, or my eyes decieve me. Please help ! SkyInTheSea 12:57 9/12/07

When the caption says "If we tile before coloring ...", it means that if you tile the plane with an uncoloured pattern of paralleograms, and then colour in the parallelograms afterwards without requiring the colouring to respect the symmetry of the repeating rectangular lattice (this is the key point) then you can colour the pattern of uncoloured parallelograms with only 4 colours. However, if you require the parallelograms that are in the same place in the repeating rectangle to always have the same colour (which has to be the case if you are going to wrap the colouring around a torus) then you have to use at least 7 colours, as is shown in the diagram.

As a 1 dimensional analogy, think of colouring the integers so that no two consecutive integers have the same colour. Obviously you only need two colours - you can colour odd numbers Red and even numbers Blue, so your repeating pattern is RB. However, if you also require integers that leave the same remainder when divided by 5 to have the same colour (or, equivalently, wrap the number line around a circle) then you need at least 3 colours, with a repeating pattern such as RBRBG. Gandalf61 (talk) 15:28, 12 December 2007 (UTC)

By the way, the article is not quite accurate when it states: Note that the coloring guaranteed by the four-color theorem will not in general respect the symmetries of the tessellation. To produce a coloring which does, as many as seven colors may be needed, as in the picture at right. The most reasonable way to interpret "respect the symmetries of the tessellation" seems to be that any symmetry of the uncolored tessellation must also take like colors to like colors when applied to the colored tessellation. In general this is not compatible with the "no tiles of equal colors meet" citerion. For example, the tiling of the plane with regular hexagons has symmetries that take any tile to any other tile, so in order to "respect the symmetries of the tesselation" all tiles must have the same color. (What's worse, this is also the case for the uncolored version of the example in the figure!)

I think that the "as many as seven colors" claim is true if we restrict our attention to a symmetry group of translations such that a tile never shares an edge with its image under a nontrivial symmetry. But I'm not sure how to express this condition succinctly. (By the way, the proper reference for 7 being sufficient would be the Heawood conjecture, as applied to a torus). –Henning Makholm 00:32, 14 January 2008 (UTC)

[edit] etymology

tessela may be immediately from Latin, but it's obviously from a Greek word for a four-sided piece. I'm just sayin'. —Tamfang (talk) 05:59, 6 January 2008 (UTC)

[edit]  ?

How do they work? —Preceding unsigned comment added by 74.142.116.75 (talk) 20:41, 2 March 2008 (UTC)

How do what work? —Tamfang (talk) 08:00, 17 March 2008 (UTC)

[edit] Tiling, Space-filling, Fractal

I began with looking at the Sierpinksi Triangle, Sierpinski Carpet where it appears evident that any suitable tiling will lend itself to a Sierpinski space-filling pattern. The Hausdorf dimension page leads into the fractal dimension of such curves and shapes. Futher discussion required. JK-Salisbury —Preceding unsigned comment added by 86.160.138.236 (talk) 12:38, 2 June 2008 (UTC)