Maximum Polygons
Livio Zucca

We define "Order N Maximum Polygon" the equilateral polygon with N unitary edges and greatest area that tiles the plane.

 Order Polygon Area Tiling 3 0.433013- 4 1 5 1.661438- 6 2,598076+ 7 3.089725- 8 4.403+ 9 5.391941+ 10 7.0406+ Generalization

All the Order N Maximum Polygon, for N>=6 and N even, are in the form below: For N even and alpha -> 180°

For R=0:
Area = M^2*sqr(3)*3/2

For R=2:
Area = ...

For R=4:
Area = ...

All the Order N Maximum Polygon, for N>=5 and N odd, are in the form below: For N odd and alpha -> 180°

For R=0:
Area = Sqr(2*M^2-(Na/2)^2)*Na/2+M^2

For R=1:
Area = Sqr((M+1)^2+M^2-(Na/2)^2)Na/2+M*(M+1)

Notice: on this page there is only a my conjecture. Many data are unproved. You could find a bigger polygon than these that tiles the plane. But I think it is improbable.

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It isn't trivial!

First edition: Oct 28th, 2003 - Last revision: Nov 6th, 2003 Below you'll find a mirror of Livio Zucca pages. Click here for my home