Talk:Tessellation

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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)

Contents 1 Tiling 2 "average number of sides meeting at a vertex" 3 Number of sides of a polygon versus number of sides at a vertex 4 Seven Colours?

[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)