Un- / interesting Labyrinths

In my last post I have referred to the interesting labyrinths by Tony Phillips. But what does this mean: interesting? Tony shows by means of combinatorics that there exist a vast number of one-arm labyrinths. And he subdivides them in uninteresting, interesting and very interesting labyrinths.

  • Accordingly, uninteresting are labyrinths that can be generated by simply attaching additional trivial circuits to smaller labyrinths at the inside or outside.
  • Interesting are labyrinths that cannot be generated this way. This implies that the path of such labyrinths may not enter on the first circuit nor reach the center from the last circuit.
  • Very interesting are the self-dual among the interesting labyrinths.

Now let us have a closer look at what this actually implies. With Arnol’d’s figures in mind, we are already familiar with all labyrinths with one arm and five circuits. So let us classify them according to the above-mentioned criteria.

Illustration 1 shows the uninteresting labyrinths. These correspond with Arnol’d’s figures 1 – 4.

Illustration 1: Uninteresting Labyrinths

Illustration 1: Uninteresting Labyrinths

Figure 1 (Labyrinth Näpfchenstein) consists of a mere series of circuits aligned to each other. Figures 2 to 4 are composed of a smaller Knossos-type labyirnth (black) and additional circuits (grey). In figure 2 (Löwenstein 5b) these circuits are attached at the outside, in figure 4 (Löwenstein 5a) at the inside to the smaller labyrinth. Figur 3 has one additional circuit attached to the out- and inside. Among the uninteresting labyrinths there are also self-dual examples, i.e. figures 1 und 3.

Illustration 2 shows the interesting and very interesting labyrinths. These correspond with Arnol’d’s figures 5 – 8.

Illustration 2: Interesting Labyrinths

Illustration 2: Interesting Labyrinths

Figure 5 and the dual in figure 7 are interesting labyrinths. Figure 6 and figure 8 are self-dual each and thus very interesting labyrinths.

Tony’s classification is very plausible. It provides a qualitative ranking of the labyrinths.

Labyrinths with a pattern of a serpentine from the outside in are uninteresting accoring to this classification. This applies to some historical labyrinths: Tholos, Löwenstein 3, Näpfchenstein, Casale Monferrato. Also uninteresting are labyrinths where the path enters on the first circuit or reaches the center from the last circuit. There are also some historical labyrinth types with this property: Temple Cowley, Löwenstein 5a und 5b, von Xanten, Zikkaron und Cakra-vyuh.

Uninteresting labyrinths are composed of a smaller interesting labyrinth and additional circuits. However, which is the intersting labyrinth that forms part of figure 1 (illustration 1)? Illustration 3 shows three options a – c to address this question.

Illustration 3: The Smallest Labyrinth?

Illustration 3: The Smallest Labyrinth?

Figure 1 is made up of a series of five circuits. A first option would be to consider figure a with one circuit (black) as the simplest labyrinth. However, this contradicts with the definition of a labyrinth provided by Kern (Kern H. Through the Labyrinth. Munich: Prestel 2000, p. 23), as the path does not repeatedly change its direction. The second option is figure b with two consecutive circuits. Here the path does not repeatedly but only once change direction. The third option then, figure c, is made up of three circuits with the path changing its drection twice. This is in full accordance with the defintion by Kern.

My answer to this question is that I consider figure b as the smallest / simplest labyrinth. There exists a historical labyrinth of this type, that lies in the Tholos of Epidauros. Also for option c, a historical labyrinth exists: Löwenstein 3.

The smallest / simplest interesting labyrinth, however, is the historical Knossos-type labyrinth. This has three circuits and a pattern of a single meander in my words, or, in Erwin’s terminology, a labyrinth-suited meander type 4. As this labyrinth is self-dual it is even a very interesting labyrinth.

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