[math-fun] Tiling the Plane with 5-, 6- and 7-gons only
A colleague wants a computer to define a 2D-graph, a tiling of the plane if you will, which only has pentagons, hexagons and heptagons. He's a chemist, which may give someone a clue what this is about, but that someone is not me. I don't suppose the n-gons can all be convex (but don't have a proof of that) but concave n-gons are not necessarily an issue. Anyone got an algorithm or even better, a program. My colleague did have the suggestion of just tiling the plane with triangles and then knocking sides out, but the problem then becomes one of inspecting what is left to see that it meets the requirements. Thanks - Guy
You're obviously not looking for the answer "tile the plane with regular hexagons", so what are the additional requirements you're making? Are you requiring all three types of polygons to appear (at least once? infinitely often?)? Are you looking for some notion of a randomly-generated tiling, with some sampling distribution properties over the space of all tilings? Etc. --Michael Kleber On Wed, Jun 18, 2008 at 9:41 AM, Guy Haworth <g.haworth@reading.ac.uk> wrote:
A colleague wants a computer to define a 2D-graph, a tiling of the plane if you will, which only has pentagons, hexagons and heptagons.
He's a chemist, which may give someone a clue what this is about, but that someone is not me.
I don't suppose the n-gons can all be convex (but don't have a proof of that) but concave n-gons are not necessarily an issue.
Anyone got an algorithm or even better, a program. My colleague did have the suggestion of just tiling the plane with triangles and then knocking sides out, but the problem then becomes one of inspecting what is left to see that it meets the requirements.
Thanks - Guy
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This problem surely needs to be made more specific. There are many ways to tile the plane with identical convex pentagons. Similarly for convex hexagons. The plane can be tiled with identical non-convex heptagons. It would be surprising if there were not infinitely many possibilities if all 3 must occur, but I don't know either if they could all be convex. I would suggest your friend have a look at the book "Tilings and Patterns" by Grünbaum and Shephard. Jim Buddenhagen
Doodling over lunch, I can say that the convexity constraint isn't a big deal. Starting with the regular tiling of the plane with hexagons, it's easy to convert the four hexagons touching a single edge into two pentagons and two heptagons, by "rotating the edge 90 degrees". Hmm, I'll try an ascii art rendition; apologies if this comes through muddled... _______ / \ / \ / \ _______/ \_______ / \ / \ / \ / \ / \ / \ _______/ \_______/ \_______ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \_______/ \_______/ \ \ / \__ __/ \ / \ / \__ __/ \ / \ / \ / \ / \_______/ | \_______/ / \ | / \ / \ __/ \__ / \ / \ __/ \__ / \ / \_______/ \_______/ \ \ / \ / \ / \ / \ / \ / \ / \ / \ / \_______/ \_______/ \_______/ \ / \ / \ / \ / \ / \ / \_______/ \_______/ \ / \ / \ / \_______/ Anyway, that doesn't help us with the question of what your friend really wants. --Michael Kleber On Wed, Jun 18, 2008 at 10:04 AM, James Buddenhagen <jbuddenh@gmail.com> wrote:
This problem surely needs to be made more specific. There are many ways to tile the plane with identical convex pentagons. Similarly for convex hexagons. The plane can be tiled with identical non-convex heptagons. It would be surprising if there were not infinitely many possibilities if all 3 must occur, but I don't know either if they could all be convex.
I would suggest your friend have a look at the book "Tilings and Patterns" by Grünbaum and Shephard.
Jim Buddenhagen
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participants (3)
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Guy Haworth -
James Buddenhagen -
Michael Kleber