Archimedean Tessellations

(iii) There are infinitely many precise colorings of
(3,3,4,3,4). None is perfect. However, two of them are chirally perfect,
and each reflects into the other. See Figure 5 for one of these.
(iv) There is only one precise coloring of (3,3,3,3,6) for which the hexagons are not all the same color. It is chirally perfect. See Figure 6. There are exactly two precise chirally perfect colorings of (3,3,3,3,6) in which the hexagons are all the same color. In one of these (Figure 7) the pairs of triangles at opposite edges of a hexagon are the same color. In the other (Figure 8) the three triangles at alternate edges of a hexagon are the same color.
(v) There are infinitely many precise colorings of (3,3,3,4,4). None
are perfect. However, there are exactly eight precise chirally perfect
colorings, of which four reflect into the other four. These four are shown
in Figures 9, 10, 11, 12. It is remarkable that in the last of these all
five colors appear in each strip of squares, whereas in the first three two
colors alternate in each strip.
Some indications of the proof of the theorem follow:
In (3,4,6,4) the assignment of four colors at some vertex determines all other colors. The reason for this is that the three squares around a triangle must have three different colors. This determines the colors of the hexagons at the vertices of the triangle, etc. The eventual result is that all triangles are the same color, yellow in Figure 3. That this coloring is perfect can be verified visually (with some patience!). Similarly, in (3,6,3,6) the three hexagons around a triangle must have three different colors, which in turn determines the colors of the triangles at the vertices of this triangle, etc. The result is shown in Figure 4, which can be visually verified to be a perfect coloring. Note, however, that in contrast to Figure 3 all colors are congruent; in fact any two of them can be interchanged by a translation.
(iii) We propose to show that when the colors
For this purpose denote the midpoint of the edge between triangles
Note that in Figure 5 there is an exceptional color, gray (color
(iv) First, consider the case where not all hexagons are the same color.
Let
Next, suppose that all hexagons are the same color, say
If
If
(v) The tessellation (3,3,3,4,4) is naturally viewed as horizontal rows
of squares separated by rows of triangles. If the three triangles and
two squares at a vertex on the upper edge of a row of squares are
colored
It is easily seen, since the two colors in the rows of squares never occur in the rows of triangles in Figures 9, 10, 11, that uncountably many precise (but not chirally perfect) colorings of (3,3,3,4,4) result by arbitrarily sliding rows of squares to the right or left.
This completes the enumeration of all precise perfect (or chirally
perfect) colorings of the Archimedean tessellations.
Received: 27 November 1998 |