In my last posts I had shown the method of transforming the Medieval labyrinth by leaving out the barriers.
The first possibility to generate a labyrinth is of course the use of the seed pattern. Thus most of the Scandinavian Troy Towns with 7, 11, or 15 circuits were created.
Some years ago I wrote about the meander technique. Thereby many new, up to now unknown labyrinths have already originated.
Andreas still has demonstrated another possibility in his posts to the dual and complementary labyrinths. New versions of already known types therein can be generated by rotating and mirroring.
Now I want to use this technology to introduce some new variations.
I refer to simple, alternating transit mazes (labyrinths). Tony Phillips as a Mathematician uses this designation to explore the labyrinth. He also states the number of the theoretically possible variations of 11 circuit interesting labyrinths: 1014 examples.
The theoretically possible interesting variations of the 3 up to 7 circuit labyrinths once already appeared in this blog.
I construct the examples shown here in the concentric style. One can relatively simply effect this on the basis of the path sequence (= circuit sequence or level sequence). There is no pattern necessary. The path sequence is also the distinguishing mark of the different variations.
I begin with the well known 11circuit classical labyrinth which can be generated from the seed pattern:
The 11 circuit labyrinth from the seed pattern
To create the dual version of it, I number the different circuits from the inside to the outside, then I walk from the inside to the outside and write down the number of the circuits in the order in which I walk one after the other. This is the new path sequence. The result is: 5-2-3-4-1-6-11-8-9-10-7- (12).
In this case it is identical to the original, so there no new labyrinth arises. Therefore, this labyrinth is self-dual. This in turn testifies to a special quality of this type.
Now I generate the complementary version. For that to happen I complement the single digits of the path sequence to the digit of the centre, here “12”.
5-2-3-4-1-6-11-8-9-10-7
7-10-9-8-11-6-1-4-3-2-5
If I add the single values of the row on top to the values of the row below, I will get “12” for every addition.
Or, I read the path sequence in reverse order. This amounts to the same new path sequence. But this is only possible with self-dual labyrinths.
I now draw a labyrinth to this path sequence 7-10-9-8-11-6-1-4-3-2-5-12.
Thus it looks:
The complementary 11 circuit labyrinth from the seed pattern
This new labyrinth is hardly known up to now.
Now I take another labyrinth already shown in the blog which was generated with meander technique, however, a not self-dual one.
The original 11 circuit labyrinth from meander technique
First, I determine the path sequence for the dual labyrinth by going inside out. And will get: 7-2-5-4-3-6-1-8-11-10-9- (12).
Then I construct the dual labyrinth after this path sequence.
This is how it looks like:
The dual 11 circuit labyrinth
Now I can generate the complementary specimens for each of the two aforementioned labyrinths.
Upper row the original. Bottom row the complementary one.
3-2-1-4-11-6-9-8-7-10-5
9-10-11-8-1-6-3-4-5-2-7
The bottom row is created by adding the upper row to “12”.
The complementary labyrinth looks like this:
The complementary labyrinth of the original
Now the path sequence of the dual in the upper row. The complementary in the lower one.
7-2-5-4-3-6-1-8-11-10-9
5-10-7-8-9-6-11-4-1-2-3
Again calculated by addition to “12”.
This looks thus:
The complementary labyrinth of the dual
I have gained three new labyrinths to the already known one. For a self-dual labyrinth I will only receive one new.
Now I can continue playing the game. For the newly created complementary labyrinths I could generate dual labyrinths by numbering from the inside to the outside.
The dual of the complementary to the original results in the complementary of the dual labyrinth. And the dual of the complementary to the dual one results in the complementary one of the original.
The path sequences written side by side makes it clear. In the upper row the original is on the left, the dual on the right.
In the row below are the complementary path sequences. On the left the complementary to the original. And on the right the complementary to the dual one.
3-2-1-4-11-6-9-8-7-10-5 * 7-2-5-4-3-6-1-8-11-10-9
9-10-11-8-1-6-3-4-5-2-7 * 5-10-7-8-9-6-11-4-1-2-3
The upper and lower individual digits added together, gives “12”.
It can also be seen that the sequences of paths read crosswise are backwards to each other.
I can also use these properties if I want to create new labyrinths. By interpreting the path sequences of the original and the dual backwards, I create for the original the complementary of the dual, and for the dual the complementary of the original. And vice versa.
If I have a single path sequence, I can calculate the remaining three others purely mathematically.
Sounds confusing, it is too, because we are talking about labyrinths.
For a better understanding you should try it yourself or study carefully the post from Andreas on this topic (Sequences … see below).
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