Reflections on the Wunderkreis, 2

As we have seen (in part 1), the most different variants of the Wunderkreis can be created. Depending on which part is emphasized more or less, they then look like.
When creating a new labyrinth, of course, it also depends on the size of the available space and the purpose the labyrinth is to serve.

Type 5 a-c
Type 5 a-c

The path sequence, if we go first to the left: 0-3-2-1-4-a1-b2-c1-c2-b1-a2-5-0. To the right we have: 0-5-a2-b1-c2-c1-b2-a1-4-1-2-3-0.
With the digits we have the sequence with odd and even numbers, as we know it from a classical labyrinth.
With the letters, which designate the elements of the double spiral, we can also see a certain systematic: The letters come alternately one after the other. If two identical letters follow each other, we have reached the center of the spiral and the basic change of direction. The additions “1” designate the lower part and the addition “2” the upper part of a transition.
If we take a closer look at the circuit sequences, we can see that the second one (to the right) is opposite to the first one.
So we can say that here two different but related labyrinths of a group are united in one. Depending on which path we choose first.

How many circuits does this Wunderkreis actually have?
That is a little difficult to count. To do this, we divide the figure into three parts, the lower left quarter, the upper half, and the lower right quarter. Let’s start at the bottom left: There are the 3 “labyrinthine” circuits and 3 of the double spiral. At the top we have 4 “labyrinthine” circuits and the 3 of the double spiral. Bottom right: 5 “labyrinthine” circuits and the 3 of the double spiral. So, depending on the angle of view, we have 6, 7 or 8 circuits.
The type designation is the maximum number of “labyrinthine” turns plus the letter sequence for the turns of the double spiral. Adding both gives the number of total circuits. In this example “5 a-c” so 8 in total.
In the file name for the drawings I have tried to express this as well, additionally provided with the indication of the entrance and the exit of the labyrinth.

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Reflections on the Wunderkreis, 1

The Wunderkreis has often been the subject of this blog. Today I would like to bring some basic remarks to it.

As is known, the Wunderkreis consists of labyrinthine windings and a double spiral in the center. Thus, there is no center to reach as usually in the labyrinth and, in addition, an extra exit, but it can also be formed together with the entrance in a branching.

This makes it more difficult to represent all this in a pattern. Also the usual path sequence (or circuit sequence) with the alternating odd and even numbers does not work properly anymore.

Therefore, I suggest to designate the spiral-shaped circuits with letters. This also gives the possibility to better describe the different types.

Here is the smallest Wunderkreis in my opinion:

Wunderkreis Type 3 a
Wunderkreis Type 3 a

A 3 circuit (normal) labyrinth with a double spiral. The path sequence, starting to the left, would be: 0-1-2-a1-a2-3-0. If I move to the right first, the result is: 0-3-a2-a1-2-1-0.

General note on “0”. This always means the area outside the labyrinth. Even if “0” does not appear on the drawings.

Now I can either increase the outer circuits or only the double spiral or both.

Type 3 a-b
Type 3 a-b

This is one more course for the double spiral. The path sequence to the left: 0-1-2-a1-b2-b1-a2-3-0. To the right: 0-3-a2-b1-b2-a1-2-1-0.

And now:

Type 5 a
Type 5 a

The double spiral as in the first example, the outer circuits increased by two. This creates a path sequence with (to the left): 0-3-2-1-4-a1-a2-5-0. Or to the right: 0-5-a2-a1-4-1-2-3-0.

Now follows:

Type 5 a-b
Type 5 a-b

In addition to the previous example, the double spiral is also enlarged. This results in: 0-3-2-1-4-a1-b2-b1-a2-5-0. And: 0-5-a2-b1-b2-a1-4-1-2-3-0.

In the circuit sequences I recognize the regularities as they occur also in the already known classical corresponding labyrinths. And if I omit the double spiral, I also end up with these labyrinths.

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How to Make the Related Labyrinths

I use a different method to generate the related labyrinths than Andreas. But I’ll get the same result. This is how we complement each other.

Essentially, I am using the level or path (= circuit) sequence to get the version of a particular labyrinth I want. Also, I am taking the path sequence to construct the labyrinth, not the seed pattern.

The basic labyrinth
Fig.1: The basic labyrinth

I usually number from the outside in (the left digits in blue), additionally here from the inside to the outside (the right digits in green).
The level sequence for the basic labyrinth is here: 0-1-2-5-4-3-6. “0” stands for outside, “6” stands for the center. We have a 5 circuit labyrinth in front of us. “1” to “5” are the numbers of the circuits (paths), hence the path sequence 1-2-5-4-3 (fig. 1).


To create the dual labyrinth, I just use the green numbers on the right side of the basic labyrinth. I determine the level sequence by going outwards from the center. I get 6-3-2-1-4-5-0. Now I draw a labyrinth using this row of digits, going from the outside to the center. But first I replace “6” with “0” and “0” with “6”, I swap inside and outside as it were. The new level sequence is then: 0-3-2-1-4-5-6 (fig. 2).

The dual labyrinth
Fig. 2: The dual labyrinth to the basic labyrinth

The left numbers now indicate the level sequence: 0-3-2-1-4-5-6. If I now read the green numbers on the right side, I of course get the basic labyrinth again.


Now I use another technique to get the transposed labyrinth. I take the path sequence of the dual labyrinth, here: 3-2-1-4-5 and complement all numbers to “6”.
3-2-1-4-5 dual
3-4-5-2-1 transpose
————
6-6-6-6-6
The second line, completed by “0” for the outside and “6” for the center, gives the level sequence for the transposed labyrinth: 0-3-4-5-2-1-6 (fig. 3).

But there is still a different technique to get there: I can read the path sequence from the basic labyrinth backwards and again complete with “0” and “6”.
1-2-5-4-3 base
3-4-5-2-1 transpose
The second line, completed by “0” for the outside and “6” for the center, also gives the level sequence for the transposed labyrinth: 0-3-4-5-2-1-6 (fig.3).

The transpose labyrinth
Fig.3: The transposed labyrinth to the base labyrinth

If I now take the green numbers of the right side, I’ll get the dual of this transposed labyrinth, which is the next, the complementary labyrinth with the level sequence: 0-5-4-1-2-3-6 (fig. 4).


But again there is also the above described technique to get the complementary labyrinth. I take the basic labyrinth and complement the numbers of its path sequence to “6”.
1-2-5-4-3 base
5-4-1-2-3 complement
————
6-6-6-6-6
The second line, completed by “0” for the outside and “6” for the center, gives the level sequence for the complementary labyrinth: 0-5-4-1-2-3-6 (fig. 4).

I can also take the dual labyrinth and read the path sequence backwards, and again add “0” and “6”.
3-2-1-4-5 dual
5-4-1-2-3 transpose
The second line, added with outside and center gives the level sequence for the complementary labyrinth: 0-5-4-1-2-3-6 (fig. 4).

The complement labyrinth
Fig.4: The complementary labyrinth to the base labyrinth

If I now take the green numbers on the right side, I’ll get the dual labyrinth to this complementary labyrinth, namely the transposed labyrinth with the level sequence: 0-3-4-5-2-1-6 (fig. 3).


So we have seen three different ways to transform one labyrinth into another by using the path or level sequence.

However, it only takes two methods to create the appropriate labyrinths. I personally prefer the “transposing” technique and the “complementing” technique.

First we have the basic labyrinth (fig.1). Through transposing the path sequence of the basic labyrinth 1-2-5-4-3 into 3-4-5-2-1 I’ll then get the transposed labyrinth (fig.3).
This transposed labyrinth with the path sequence 3-4-5-2-1 I transform to the dual labyrinth by complementing the path sequence to 3-2-1-4-5 (fig.2).
This dual labyrinth I then transform to the complementary labyrinth by transposing its path sequence 3-2-1-4-5 into 5-4-1-2-3 for the complementary labyrinth (fig.4).
For control purposes, I can transform the basic labyrinth into the complementary by complementing the path sequence 1-2-5-4-3 of the basic into 5-4-1-2-3 (fig. 4) for the complementary.

All of these transformation methods have the same effect as the rotating and mirroring techniques by Andreas.

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The Complementary (transposed) Classical 7 Circuit Labyrinth in Knidos Style

I described this type in the concentric style in my last post (see related posts below). Today it is about the representation of the transposed labyrinth in Knidos style.

The path sequence is: 5-6-7-4-1-2-3-8. The special thing about it is that one enters the labyrinth on the 5th circuit, and the center on the 3rd. circuit.

The walls and Ariadne's thread

The walls and Ariadne’s thread

And yet this type can be aligned to the central axis. This is only possible by editing in the Knidos style.

I come back to the original labyrinth using the same method that I used to get to the complementary type: I add the difference to the last digit (the goal) to the row of numbers in the path sequence. So:
5-6-7-4-1-2-3-8
3-2-1-4-7-6-5-8
8-8-8-8-8-8-8-8
This is then the original, well-known classical (Cretan) labyrinth.

What does the Knidos style actually mean?
By this I mean, above all, that the labyrinth has a larger center than just the width of a path, that it is as compact as possible and, above all, that it is developed from the path sequence and not from the basic pattern for the boundary lines (the walls). So it is Ariadne’s thread, the path in the labyrinth, that determines the construction. And this must be geometrically correct with constant path widths, elements that are as round as possible and as few “spaces” as possible.

Here in another graphic:

The transposed labyrinth in Knidos style

The transposed labyrinth in Knidos style

Here are the drawing instructions for a kind of prototype to be scaled for the axis dimension of 1 m.

The design drawing

The design drawing

Here you might view, print or download it as a PDF file.

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