Talk:Tiling by regular polygons

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[edit] Title

Shouldn't this be "Tilings by regular polygons" or even "Plane tilings by regular polygons"? Melchoir 01:18, 9 October 2005 (UTC)

  • I agree. Feel free to move/rename this title - I don't know how! (besides copying contents and putting old under delete) Tom Ruen 01:46, 9 October 2005 (UTC)

[edit] Cleanup and topics not covered

I cleaned up the article and removed the cleanup tag. There are still a number of topics discussed in Grünbaum and Shephard which could perhaps be discussed on this page but aren't yet:

  • More on k-uniform (and k-isohedral, k-isotoxal) tilings, with additional examples and something on the Krötenheerdt tilings; equitransitive tilings.
  • Non-edge-to-edge tilings: equitransitive unilateral tilings by squares; the problem of tiling the plane with exactly one square of each integer edge length.
  • Star polygons, both in the style of Kepler (Grünbaum and Shephard section 2.5, the lists there being incomplete) and as hollow self-intersecting polygons (section 12.3); I shouldn't make the call as to notability of the former, having published regarding them.
  • The duals of the uniform tilings (Laves tilings).
  • Archimedean and uniform colourings of tilings.

Joseph Myers 00:41, 6 October 2005 (UTC)

[edit] This is wrong, but I'm not sure how to fix it

This part:

In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same size. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd.

is correct, but the tilings it makes impossible are uniform (semiregular) tilings.

However, immediately below is a section which says things like "cannot appear in *any* tiling of regular polygons". (Not just uniform tilings).

There is no justification for the claim that it can't appear in *any* tiling. I can't figure out if this is a true claim with no justification, or a false claim that happened because someone confused "any tiling" and "any uniform tiling". Ken Arromdee 18:31, 22 November 2006 (UTC)

I'm pretty sure it can all be justified. E.g. for a tiling involving 3.7.42: a triangle with a vertex of this type would have to have another vertex of this type at the other endpoint of the edge shared between the 3 and the 42; consequently the third vertex of the 3 would have to be 3.7.7, which doesn't work. The same reasoning shows 3.8.24, 3.9.18, and 3.10.15 impossible. 4.5.20 is the only vertex type involving 20-5 and 4-5, so if a vertex of this type existed, you'd have to have 4's and 20's alternating around the 5, impossible by the same reasoning as you've blockquoted above, and the same reasoning works for 5.5.10.
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