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:

N=even



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:

N=odd



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.

_________________

It isn't trivial!

First edition: Oct 28th, 2003 - Last revision: Nov 6th, 2003

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