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        3. The Perfect Precise Colorings of
             Archimedean Tessellations


Theorem:
       (i)    There are no precise colorings of (3,12,12).
       (ii)   There is only one precise coloring of each of (4,6,12), (4,8,8), (3,4,6,4) and (3,6,3,6). Each is perfect. See Figures 1-4.

3Kb        2Kb

6Kb        4Kb

Figures 1-4

       (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.

5Kb

Figure 5

       (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:

       (i) Lemma: If a 3-valent planar graph (infinite or finite) has a region with an odd number of edges then it admits no precise coloring. This lemma is proved by a simple parity argument on the (two) colors of the regions surrounding the "odd" region. This lemma also rules out any precise coloring of the regular tetrahedron or regular dodecahedron. (For the same reason there are no precise colorings of four of the Archimedean polyhedra: the truncations of the tetrahedron, cube, dodecahedron, and icosahedron.)

       (ii) In (4,6,12) and (4,8,8) the three colors at any vertex immediately force all the colors at all other vertices. Then, in Figure 1, any isometry takes squares to squares, hexagons to hexagons and dodecagons to dodecagons. Hence it is the identity permutation of their colors, so the coloring is perfect. In Figure 2 every rotation about the centers of the octagons is the identity permutation of the colors. Every quarter-turn about the center of a square interchanges the two octagon colors; similarly, every rotation about a mid-edge point interchanges the two octagon colors and fixes the square color. Likewise, every translation is the identity permutation or interchanges two colors. Every reflection or glide-reflection is the identity permutation or interchanges two colors. Hence the coloring is perfect.

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 a, b, c, d, e are assigned to the two triangles, square, triangle, square around one vertex (in that cyclic order) then there are exactly two ways that assignment can be completed to yield a chirally perfect precise coloring of the entire tessellation.

For this purpose denote the midpoint of the edge between triangles a and b by P; denote the center of square c by Q. Denote by S the square having an edge in common with triangle b and a vertex in common with square c. Then S must be colored d or e.

       First case: S is colored e. Then a half-turn about P, rot P, fixes color e and interchanges a and b. Hence rot P is the permutation (ab) or (ab)(cd). If rot P = (ab) then the quarter-turn around Q, rot Q (which has period 2 or 4 and fixes colors c and e) must be (bd). Using rot P and rot Q repeatedly, the "precise" condition yields a unique precise coloring. However, the resulting coloring is not consistent with translations, and hence is not chirally perfect. On the other hand, if rot P = (ab)(cd) then again rot Q = (bd). Then rot P, rot Q and translations determine a unique precize coloring, which is shown in Figure 5. It is chirally perfect, but not perfect.

       Second case: S is colored d. Then rot P = (ab)(de). It follows that rot Q (taken to be in the direction that takes b to d) is (abde). Then rot P, rot Q and the "precise" condition determine a unique coloring, which is chirally perfect, but not perfect.

Note that in Figure 5 there is an exceptional color, gray (color e in the first case above, and color c in the second case), which occurs only in squares, whereas the other four colors appear in both squares and triangles. If Figure 5 is reflected in a line then it is equivalent to the coloring obtained in the second case above. The gray squares are the ones colored c, and the other colors are a permutation of the colors of the first case.

       (iv) First, consider the case where not all hexagons are the same color. Let T be a triangle with no edge in common with a hexagon. Suppose this triangle is colored t, and the three triangles at its edges are colored b, c, d. Since not all hexagons are the same color we can choose T so that the three hexagons having edges in common with these three triangles do not all have the same color. If one of these hexagons is colored e, then the "precise" condition at the vertices of T eventually implies that all three have color e, contradicting our original assumption. Then, given that none of these hexagons is colored e (and none is t, since each shares a vertex with T), the "precise" condition determines each of their colors. With this begining (and resolving occasional apparent dichotomies) a unique precise coloring, shown in Figure 6, is obtained. Remarkably, the local "precise" condition has been enough (as it was in the four simpler cases treated in (ii)) to guarantee that this coloring is chirally perfect.

Next, suppose that all hexagons are the same color, say e. Assign colors a, b, c, d to the four triangles at a vertex, E, of one of these hexagons, in cyclic order (so that a and d are the colors of the triangles sharing an edge with E). Let rot E be the permutation of colors induced by rotation by a one-sixth turn around the center of E (in the direction from color a to color d). Let rot P be the rotation about the midpoint, P, of the edge between triangles colored a and b. Then the permutation of colors induced by rot P is (ab)(cd) or (ab).

If rot P is (ab)(cd) then the only apparent possibilities for rot E are (ad), (adb), or (adc). Filling in the colors of triangles around E shows that the first two lead to contradictions. The case of rot E = (adc) yields Figure 7, which is chirally perfect.

If rot P is (ab) then there are the same three possibilities for rot E. In this case, only rot E = (ad) succeeds. The result is shown in Figure 8, which is chirally perfect.

7Kb

5Kb        5Kb

Figures 6-8

       (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 a, b, c, d, e in that cyclic order (triangles a, b, c; squares d, e) then triangle next to c must be colored either a or e.

First case: The triangle next to c is colored a. Then the square (in the row above) sharing an edge with this triangle must be colored d or e. These two possibilities turn out to lead to colorings which reflect into each other; hence we assume this square is colored d. Then the permutation induced by translation V (up and to the right) fixes d and e. Consistency of the coloring with translations only requires that V permute a, b, c arbitrarily. However, for consistency with half-turns about centers of squares only the three even permutations of a, b, c are permitted. These three even permutations yield Figures 9, 10 and (the reflection of) Figure 11. (This reflection of Figure 11 is the one obtained when V is the identity permutation of colors.)

Second case: The triangle next to c is colored e. Then the colors of three of the squares in the next higher row are immediately determined. From this, and the "precise" condition, the permutation of colors induced by V is (edacb). It is easy to see that the entire coloring is thereby determined. The result is shown as (the reflection of) Figure 12.

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.

4Kb        7Kb

4Kb        4Kb

Figures 9-12



Acknowlegments: The author is indebted to John Rigby for the initial ideas as well as for correcting an omission in an earlier version, and to Bob Wilson for assistance with the computerization.

Received: 27 November 1998