Regular heptagons, of course, can’t tile a plane by themselves. Of all tessellations of the plane which include regular heptagons, I think this is the one which minimizes between-heptagon gap-size (the parts of the plane outside any heptagon). However, I do not have a proof of this. The shape of each of the polygons which fill the “heptagon-only gaps” is a biconcave, equilateral octagon. With these octagons, this is a tessellation, but without them, it wouldn’t fit the definition of that term.

[Later edit: on Facebook, a friend showed me two others with smaller gap-sizes. In other words, the conjecture above has now been shown to be wrong.]

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Very interesting but not good enough.

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Good enough for . . . ?

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Can you provide links to the other heptagon tilings with smaller gaps?

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I would if I could, but that information has been lost since I posted this.

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OK. Thanks for the quick reply!

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