Heterogeneous Seed Patterns
In all my previous posts on the Man-in-the-Maze labyrinth I have shown labyrinths with homogeneous seed patterns (see: related posts, below). I term seed patterns as homogenous, that are composed of a series of similar elements.
I have shown seed patterns with single (not nested) turns as well as seed patterns with elements of solely two nested turns or four nested turns. The labyrinths of the Löwenstein 3, Näpfchenstein, and Casale Monferrato types and the smaller of the two labyrinths with 7 circuits shown in part / 3 of this series all have single turns (Ill. 1, left figure). The patterns of these labyrinths are made-up of a serpentine from the outside in. The seed patterns of the Knossos, Cretan and Otfrid type labyrinths are composed solely of elements with two nested turns (ill. 1, central figure), see also part / 4 of this series. The larger of the two labyrinths with 7 circuits shown in part /3 has two elements with four nested turns (ill. 1, right figure).
Most labyrinths, however, have a mixed – I therefore term it: heterogenous – seed pattern. I will show this here with two examples. The first example is the labyrinth, I am wont to use for purposes of demonstrations, so to speak my demonstration labyrinth. It is this the labyrinth that corresponds with Arnol’d’s figure 5, I have already presented in this post.
The seed pattern of this labyrinth is composed of several different elements. The right half consists of an element with three nested turns (ill. 2, right figure). The left half is composed of two other elements. One of it is a single turn (ill. 2, left figure), the other has two nested turns (ill. 2, central figure).
How many rings now do we need for such a seed pattern in the MiM auxiliary figure? From the previous posts we know that seed patterns with single turns take one circuit, such with two nested turns take 2 circuits and so forth. Therefore it is reasonable to assume we will need three circuits, as many as are required to cover the three nested turns of the right half of the seed pattern.
And indeed this is true. The single turn on top left of the seed pattern covers the innermost circuit of the auxiliary figure (ill. 3, left figure). The two nested turns on bottom left cover two circuits (ill 3, central figure). The three nested turns of the right half of the seed pattern cover three circuits (ill. 3, right figure). Thus, the number of circuits required is determined by the element with the most nested turns.
In addition there is another effect on the apparence of the seed pattern in the MiM-style .
In heterogenous seed patterns, not all the ends lie on the same ring of the auxiliary figure. (ill. 4, left figure). Despite this, of course, three circuits of the auxiliary figure are needed for the seed pattern. Therefore it makes sense to prolong the ends on the inner rings so that they all end on the same ring of the auxiliary figure. Some of the dots are prolonged to lines and also some of the lines are prolonged (ill. 4, central figure). The result can be seen in the right figure of ill. 4.
In order to draw my demonstration labyrinth in the MiM-style, we thus need 5 circuits for the labyrinth, 1 circuit for the center, and 3 circuits for the seed pattern.
The second example offers me the opportunity to draw the attention to a very beautiful historical labyrinth.
The illustration shows the seed pattern and its variation to the MiM-style of the Cakra-vyuh labyrinth. This labyrinth has 11 circuits and is self-dual. However, according to the classification by Tony Phillips, it is an uninteresting labyrinth. Of interest for our purpose is, that the seed pattern of this labyrinth is composed of elements with single (not nested) and with two nested turns. Four of its 24 ends, the four dots, lie on the second innermost ring. The other 20 ends, 16 lines and 4 dots, lie on the third inner ring. The four innermost dots are therefore prolonged to lines so that all ends lie on the third inner ring.
In order to draw the Cakra-vyuh type labyrinth in the MiM-style, an auxiliary figure with 24 spokes and 15 rings (11 circuits for the labyrinth + 1 for the center + 2 for the seed pattern = 14 circuits), i.e. 15 rings is needed.
- How to Draw a Man-in-the-Maze Labyrinth / 4
- How to Draw a Man-in-the-Maze Labyrinth / 3
- Un- / interesting Labyrinths
- Considering Meanders and Labyrinths
P.S.: Fortunately the seed pattern of the Cakra-vyuh type labyrinth in the MiM-style covers the same number of circuits as the seed pattern of the Otfrid type. So for the drawing we can use the walls (black) of the Otfrid type labyrinth in the MiM-sytle (from part / 4) and only have to exchange the seed patterns (blue).