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A note about the graphics.

The hexagonal grid of sections 2 through 5 was calculated using the zsi system, with underlying coordinates as in figure X1.

Figure X1 Figure 2B1
repeated for comparison

Any combination of three integers totaling zero denotes a cell in this system.

The triangular grid of section 6 was figured by using an adaptation of the zsi system. Only a few of the hexagonal cells were selected, and those are drawn in red in figure X2 below:

The yellow shaded lines indicate the boundaries of the ultimate triangles. Triangular boundaries intersect at the points where all three coordinates are multiples of 3, with the corresponding hexagons shaded in blue.

figure X2A

In the final images, the scale was substantially reduced so that each triangle in section 6 has the same area as each hexagon in sections 2 through 5.

figure X2B

The hexagon-and-square grid of section 7 starts with the grid of figure X3A. Then any cell all of whose coordinates are even numbers remains a hexagon; all others become squares. Because the squares are smaller than the hexagons, the cells must be moved closer to one another to prevent gaps between hexagons and squares. Alternatively, the hexagons and squares can be proportionally enlarged jointly.

figure X3A
figure X3B

An important feature is that the coordinates of any square are the average of the coordinates of the two adjoining hexagons. Similarly, the coordinates of a hexagon are the average of the coordinates of any two opposite adjoining squares; that is, two adjoining squares of the same color.

The underlying grid of section 8 is shown in figure X4 below. The grid of section 7 would not work here because the coordinates of the triangles would not have been integers. To fix that problem, coordinates of corresponding cells have been multiplied by three.

figure X4

The averaging considerations of the section-7 grid apply here, with the triangles becoming full participants.