2. Escher Counts

Consider the rotations-only case for a moment (case 1) and consider the question of how many truly different signatures there are. Two signatures are considered equivalent when they yield congruent tilings for all motifs. Naively, there are , or 256 signatures, since each position can hold one of 4 instructions. But of course any signature is equivalent to one with a “1” in the upper-left corner. So there are at most 64 distinct tilings. But many of the 64 signatures yield congruent tilings. For example,   is equivalent to via a 90° rotation.

Figure 5. The signature yields a tiling that is congruent (by a 90° rotation) to that obtained from , shown in Figure 3.

It turns out that there are exactly 23 nonequivalent signatures. Escher knew this and made a complete compilation of the 23 tilings. A detailed explanation of the counting techniques that can be used to prove that 23 is correct can be found in [Schattschneider, 1997]. Table 1 shows the complete list (available in the package as EscherList).

Table 1. Escher’s 23 signatures that yield, up to congruence, all tilings using only rotations.

EXERCISE.  How many of these 23 signatures are unique in that they admit no other equivalent signature (with a 1 in the upper-left corner)?

When including flips (case 2) it is important to note that Escher found some restrictions useful in generating aesthetically pleasing patterns. He always used the restriction to the case of two rotations and two flips. Moreover, he either used signatures so that the two rotations are the same, as are the two flips (case 2A; e.g., ), or he insisted that the the two rotations not be the same and the two flips not be the same (case 2B; e.g.,  ). See Table 2, below.

		23	23
			—
		154	—
		67	—
		10	10
		39	36

Table 2. This summarizes what is known about the number of different patterns.

It is a dizzying task to examine by hand all the possibilities and determine which are equivalent. Thus it is not surprising that not all Escher’s counts are perfect. He missed three in case 2B (see Figure 6).

Figure 6. Escher’s count of a certain type of pattern was 36 when there are really 39. These three are the ones he missed (see [Schattschneider 1997]).