**Maximum PolygonsLivio 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