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Pick's Theorem is a well-known way to calculate the area of a plane figure whose vertices all lie on grid points. Usually the grid is orthogonal - but that is far too square for us here in the Hall of Hexagons. So we'll look at an isometric grid, which is (naturally) much better for drawing hexagons. Pick's Theorem states that the area of a simple plane figure whose vertices all lie on grid points is
area = i + e/2 -1
where
The
Now:
Having proved Pick's theorem on an isometric grid, it is easy to extend it to an orthogonal grid. We observe that shearing the grid relative to a horizontal axis will not change the area of any figure on it, nor will it change the number of vertices inside the figure or on its edges. Furthermore stretching the grid vertically will not change the number of vertices inside the figure or on its edges, but will change the area of the figure, and of the smallest parallegram (which is our unit of measurement) by the same fraction. So, with a shear and a stretch of the vertical axis we have Pick's Theorem on an orthogonal grid. On an orthogonal grid, the small parallelogram has become a square; if the grid points are spaced one unit apart, the area of this square is one. Thus Pick's Theorem is:
area = i + e/2 -1On an isometric grid where the grid points are spaced one unit apart, the area of the small parallelogram is sqrt(3)/2. Pick's Theorem expressed numerically is therefore:
area = (i + e/2 -1) * sqrt(3)/2It is impossible to draw an equilateral triangle with its vertices on orthogonal grid points. Pick's Theorem tells us why: any figure with vertices on orthogonal grid points will have a rational area, whereas an equilateral triangle has an irrational area. Of course the same applies to a regular hexagon. But both equilateral triangles and hexagons can be drawn in abundance on an isometric gird. | ||||

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