We take a 7-circuit classical labyrinth and number the single circuits from the outside inwards. “0” stands for the outside, “8” denotes the center. I take this two numbers into the circuit sequence, although they are no circuits. As start and end point they help to better understand the structure of the labyrinth.
The circuit sequence is: 0-3-2-1-4-7-6-5-8
Everybody which already has “trampled” Ariadne’s thread (the path) in the snow knows this: Suddenly there is no more place in the middle, and one simply goes out. And already one has created a walk-through labyrinth. This is possible in nearly all labyrinths.
Then maybe it looks like this:
If one wants a more compact labyrinth, one must change the shape. The internal circuits become, in the end, a double spiral. We can make either two separate ways or join them. So we will get a bifurcation.
We will get the following circuit sequence if we take the left way or the fork to the left:
Now we take first the right way or the fork to the right, then the circuit sequence will be:
Because the two rows are written among each other, they simply can be add up together (without the first and the last digit):
This means: If I go to the left, I am in the original labyrinth, if I go to the right, I cross the complementary one.
It has the circuit sequence 0-5-6-7-4-1-2-3-8.
Or said in other terms: The walk-through labyrinth contains two different labyrinths, the original one and the complementary one.
The 7-circuit labyrinth is self-dual. Therefore I only get two different labyrinths through rotation and mirroring as Andreas has described in detail in his preceding posts.
How does the walk-through labyrinth look if I choose a non self-dual labyrinth?
I take this 9-circuit labyrinth as an example:
Here the boundary lines are shown.
On the top left we see the original labyrinth, on the right side is the dual to it.
On the bottom left we see the complementary to the original (on top), on the right side is the dual to it.
However, this dual one is also the complementary to the dual on top.
The first walk-through labyrinth shows the same way as in the original labyrinth if I go to the left. If I go to the right, surprisingly the way is the same as in the complementary labyrinth of the dual one.
And the second one?
The left way corresponds to the dual labyrinth of the original. The right way, however, to the complementary labyrinth of the original.
Now we look again at a self-dual labyrinth, an 11-circuit labyrinth which was developed from the enlarged seed pattern.
The left one is the original labyrinth with the circuit sequence:
The right one shows the complementary one with the circuit sequence:
The test by addition (without the first and the last digit):
Once more we construct the matching walk-through labyrinth:
Again we see the original and the complementary labyrinth combined in one figure. If we read the sequences of circuits forwards and backwards we also see that both labyrinths are mirror-symmetric. This also applies to the previous walk-through labyrinths.
Now this are of all labyrinth-theoretical considerations. However, has there been such a labyrinth already as a historical labyrinth? By now I never met a 7- or 9-circuit labyrinth, but already an 11-circuit walk-through labyrinth when I explored the Babylons on the Solovetsky Islands (see related posts below). Besides, I have also considered how these labyrinths have probably originated. Certainly not from the precalled theoretical considerations, but rather from a “mutation” of the 11-circuit Troy Towns in the Scandinavian countrys. And connected through that with another view of the labyrinth in this culture.
There is an especially beautiful specimen of a 15-circuit Troy Town under a lighthouse on the Swedish island Rödkallen in the Gulf of Bothnia.
It has an open middle and the bifurcation for the choice of the way. This article by Göran Wallin on the website Swedish Lapland.com reports more on Swedish labyrinths.
For me quite a special quality appears in these labyrinths, even if there is joined a change of paradigm.