It is known that there are 8 alternating labyrinths with 1 arm and 5 circuits (see “Considering Meanders and Labyrinths”, related posts, below). Of these, four are not self-dual. These four all are in a relationship to each other via the duality and complementarity (see “The Complementary versus the Dual Labyrinth”, related posts, below). The other four labyrinths are self-dual.
I had already pointed to the relationship between complementary and self-dual labyrinths (see “The Complementary Labyrinth”, related posts, below). Here I want to elaborate on it further. For this purpose I use the same form of diagram I had already used in my previous post (see “The Complementary versus the Dual Labyrinth”). I also use the same numbers of the labyrinths according to the numbering of Arnol’d’s meanders (see “Considering Meanders and Labyrinths”), that underlie them.
Figure 1. Labyrinths 1 and 6
The first of the Arnol’d’s labyrinths, number 1, is self-dual. In the diagram, the dual is situated in the same row, the complementary in the same column with the original labyrinth. The dual of number 1 is again number 1 (what actually is the meaning of selfdual). The complementary of number 1 is number 6. And – of course – is the dual to the complementary again number 6. So in the case of self-dual labyrinths, we only captured two different labyrinths, whereas it were four in the case of not self-dual labyrinths.
Thus, two more labyrinths are still missing. We need another diagram to capture labyrinths number 3 and number 8 (fig. 2).
Figure 2. Labyrinths 3 and 8
And, indeed, these two are complementary to each other. So in self-dual labyrinths, only two different labyrinths are in a relationship to each other.
Here the question arises: Do there also exist self-complementary labyrinths? Up to now we have not yet found such a labyrinth. So let us remember, what self-dual imples. The patterns of the original and self-dual labyrinths are self-covering. In fig. 3 I show what that means. The two patterns in the same row are dual. If we shift them together, we can easily see, what I mean.
Figure 3. Self-dual patterns are self-covering
Thus, self-complementary would imply that the original and complementary pattern would also be self-covering.
Figure 4. Complementary patterns are not self-covering
Fig. 4 shows, that even though there is a certain similarity between these two patterns, they are not self-covering. In my opinion there are no self-complementary labyrinths. This is because vertical mirroring with uninterrupted connections to the entrance and center modifies the sequence of circuits. This, however, woult have to remain unaltered.
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