Heptagon 48 -2012.08.28Up I designed new puzzle and name it "Heptagon 48". Will a regular heptagon tile a plane? The answer is NO. A “tetrahept” is a figure made of 4 regular heptagons joined edge to edge. |
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This puzzle uses the following 6 kinds of tetrahepts. |
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[Goal] |
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[Mechanical Version] Acctually, I have made the mechanical version of this puzzle out of two type of materials. The first one is made out of wood. The other one is made out of marble, and this was submited to the Puzzle Design Competition 2012. |
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[Further Information] As mentioned above, regular heptagon can tile the plane if allowing for the pentagonal space. There are two types of this tiling, shown in Fig.1. Puzzle “Heptagon 48” uses the type 1 tiling in Fig 1. In this tiling, Six tetrahepts are compatible, although only five tetrahepts are compatible in the type 2. |
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As for the basic of tiling, the following John J. G. Savard's site is very good 2). Here is the Basic Tilings Page. I found this two type of tiling in the process of my making idea for this puzzle. Surprisingly, however, this type of tiling occur in nature. It exists in the Boron-Chemistry and Carbon Nano-tube Chemistry. This figure is called "Pentaheptite". In the paper of "Pentaheptite Modifications of the Graphite Sheet" (J Chem Inf Comput Sci. 2000 Nov-Dec;40(6):1325-32.), it is described that pentaheptites (three-coordinate tilings of the plane by pentagons and heptagons only) are classified under the chemically motivated restriction that all pentagons occur in isolated pairs and all heptagons have three heptagonal neighbors 3). The example of pentaheptite nets in YCrB4 and ThMoB4 are also shown, and That is the same in Figure 1. My puzzle was based on the findings that there can be heptagonal tiling with pentagonal space and six tetrahepts is compatible in the pentaheptite network (this heptagonal tiling). The number of solutions are not known yet, but I found 57 solutions manually until now. I think these solutions are rich in diversity than originally expected. [Solution] |
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lf you'd like to see the other solutions I have found, crick here. | |||||||||||||||||||||
--Koshi Arai (2012.09.06) [Refference]
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[雑記メニューへ]
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