Many Man-in-the-Maze (MiM) illustrations show the Cretan-type labyrinth. Many others show figures that look alike but are no true labyrinths. With the seed pattern and the MiM auxiliary figure we are now able to draw every type of a one-arm labyrinth in the MiM-style. This can be done by varying the seed pattern, placing it into the center of the auxiliary figure and completing it to the whole labyrinth. For this, however, we have to know exactly, how many spokes and rings the auxiliary figure for a particular type of labyrinth must have. Both, the number of spokes and rings are determined by the seed pattern of the labyrinth to be drawn in the MiM-style. These numbers can be found by trying out. But they can also be calculated with a formula. Figure 1 shows this with the example of my demonstration labyrinth (see also post / 5 of this series, related posts below).
S is the number of spokes of the auxiliary figure
E is the number of ends of the seed pattern
R is the number of rings of the auxiliary figure
V is the number of nestings of the element with the most nested levels
Fig. 1 shows the seed pattern for the walls (blue) with all turns of the pathway (red) drawn in. (By the way: these red lines are nothing else than the seed pattern for the Ariadne’s Thread.)
To determine the number of spokes we use the entire seed pattern for the walls. Just count the ends. The number of spokes is the same as the number of ends of the seed pattern: S = E.
The calculation of the number of rings is somewhat more complicated. For this, the number of spokes is needed and, in addition, the element of the seed pattern for the Ariadne’s Thread with the most nested levels. In figure 1, this is the element of the right half with three nested turns. The number of rings is equal to the number of spokes divided by two plus the number of nestings plus 1, thus: R = S/2 + V + 1.
The auxiliary figure for my demonstration labyrinth in the MiM-style therefore needs 12 spokes and 12/2 + 3 + 1 = 10 rings. Let us test this formula with some of the other labyrinths of earlier posts of this series. The auxiliary figure for
- the Cretan requires 16 spokes and 16/2 + 2 + 1 = 11 rings (see post / 1, related posts below)
- the Löwenstein 3 – type labyrinth requires 8 spokes and 8/2 + 1 + 1 = 6 rings (see post / 4, related posts below)
- the Otfrid-type labyirnth requires 24 spokes and 24/2 + 2 + 1 =15 rings (see post / 4, related posts below)
The number of spokes depends only from the number of circuits of the labyrinth. All auxiliary figures of labyrinths with the same number of circuits have the same number of spokes.
The number of rings depends from the number of circuits too, but in addition also from the depth of the nestings of the seed pattern. Auxiliary figures of labyrinths with the same number of circuits can have different numbers of rings (see post / 3, related posts below).
At this point a precision with respect to the number of nestings has to be made. The element with the most nestings of my demonstration labyrinth has three nested turns, i.e. three nestings. However, there exist also seed patterns in which several turns on the same level are nested by a turn on the next level.
Two such examples are shown in fig. 2. The left figure has 4 elements, but only 3 nestings, as the two turns aligned vertically to the right lie on the same level. The right figure has 5 elements but also only 3 nestings. So, what counts is the number of nested levels, not the number of turns.
Let’s show this with an example that has not yet been shown in the MiM-style. For this, the alternating labyrinth with five circuits that corresponds with Arnol’d’s figure 6 and has already been repeatedly presented on this blog, is well suited.
The seed pattern of this labyrinth has two elements with 2 nestings. Each element includes 3 turns, i.e. 2 turns on the first level nested by one turn on the second level. According to the formula, the auxiliary figure for this type of a labyrinth has 12 spokes, as is the same with all labyrinths with five circuits, and 12/2 + 2 + 1 = 9 rings. This is 1 ring less than in the auxiliary figure for my demonstration labyrinth although this also has five circuits.
With this formula we now have everything we need to be able to transform any type of a one-arm labyrinth into the MiM-style. This can be done with the following steps:
- Choice of the type of labyrinth
- Extraction of the seed pattern
- Calculation of the number of spokes and rings of the auxiliary figure
- Variation of the seed pattern into the MiM-style
- Placing the seed pattern in the center of the axiliary figure
- Determining the situation of the center of the labyrinth
- Completion of the seed pattern around the center from inside out to the whole labyrinth.