Self-dual Labyrinths

Inverting a labyrinth brings us to the dual labyrinth of it. If we now invert the Cretan-type labyrinth, we will obtain another Cretan-type labyrinth, although with the entrance on top.

O-D Kretisch

Figure 1. Inverting the Cretan-type Labyrinth

Fig. 1 shows the process and result of this inversion.

In order to compare the original and dual labyrinths, as previously shown, we isolate the dual labyrinth and carry-over the pattern on it. Then we rotate the dual labyrinth with the pattern lying on it, so that it is directed with the entrance from below, and place it next to the original labyrinth.

SD Kret

Figure 2. The Original and Dual Labyrinths are the Same: Self-dual

As shown in Fig. 2, the original and dual labyrinths are the same. The two labyrinths that are dual to each other have the same pattern, although the pattern is rotated by 180°. This is the case here too. So the right image really shows the pattern rotated by 180°. However, this pattern, after it has been rotated, is self-covering. This is not the case in “normal” dual labyrinths.

Now let us also have a look at the level sequences. As the Cretan-type labyrinth has 7 circuits, we need to use 7 colors.

UF 7 Farben

Figure 3. The Colors of the Circuits

Fig. 3 shows the sequence of the colors. In addition to the first five colors from our last post, we use the color Bordeaux for the circuit that is covered as the 6th and orange for the circuit that is covered as the last by the pathway.

UF Muster Kret

Figure 4. Level Sequences in the Pattern

Fig. 4 shows the level sequence directly on the pattern. As usual, the left image indicates the level sequence into the original and out of the dual labyrinth, whereas the right image indicates the level sequence into the dual and out of the original labyrinth. Both level sequences are identical.

Labyrinths in which the original and dual labyrinths are the same, are referred to as self-dual. These are particular labyrinths and they have a higher inner order than “normal” dual labyrinths.

Muster d sd

Figure 5. Pattern of a Dual (left) and Self-dual (right) Labyrinth

This can also be seen in comparing the patterns of dual and self-dual labyrinths (fig. 5). In dual labyrinths (left image), the courses of the first (grey) and the second (black) halves of the pathways are different, whereas they are congruent in self-dual labyrinths (right image).

Some of the most excellent labyrinths are self-dual, such as the Otfrid, Chartres, Reims, Auxerre, Saffron Walden and some others.

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From the Classical 7 Circuit Labyrinth to the Roman Labyrinth

The oldest known labyrinth figure is the Classical 7 Circuit labyrinth (sometimes also called: the Cretan labyrinth). Its origin is about 1200 B.C. The further development falls in the time of the Roman empire from 165 B.C. till 400 A.D. The general name is Roman labyrinth and there are different types again. They have in common that different sectors (mostly four) are run one after the other.

The Classical 7 circuit labyrinth in square form

The Classical 7 circuit labyrinth in square form

In his book “Labyrinths and Mazes of the World” (published in 2003 by Gaia Books, London) Jeff Saward has described how the development of the Roman labyrinth from the Classical labyrinth is possible. Her I only want to put this across in a few steps.

We begin with the Classical labyrinth in square form.
In the drawings the boundary lines are shown in black. The seed pattern contained therein is emphasized in blue. The ways are put in orange, in the same width as the boundary lines.

The whole figure is reduced to a quarter through a rotation. The vertical parts of half the seed pattern move to a horizontal line.

The quartered Classical labyrinth

The quartered Classical labyrinth

To generate an entire Roman labyrinth from the quartered labyrinth, another two circuits must be inserted in every sector: One around the middle, and one at the outside. In the outer rings one walks to the the next sector, the last path leads to the center.
If one examines exactly the paths, one can recognize that the way is the same as the way back in a Classical labyrinth. Or differently expressed: In a Roman labyrinth one wanders four times the way back of a Classical labyrinth.

The Roman labyrinth

The Roman labyrinth

The path sequence can be understood with the help of the figures.  So one well can see the Classical labyrinth inside the Roman labyrinth.

Even better one recognizes the relationship with the Classical labyrinth in the diagram illustration.

The diagram of the Roman labyrinth

The diagram of the Roman labyrinth

The Roman labyrinth is self-dual like it is the Classical labyrinth. One sees this well in the following graphics. Howsoever the diagram is rotated or mirrored, the path sequence is always the same. Also it plays no role whether one walks in direction to the center or reversed, or whether one fancies the entrance below or on top.

The diagram of the Classical labyrinth in four variants

The diagram of the Classical labyrinth in four variants

There are different historical Roman labyrinth of this kind. The oldest one comes from the second century A.D. and is to be seen on a mosaic in Pont Chevron (France). This is why Andreas Frei calls it type Pont Chevron (see link below).

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How to Make Six New (up to now unknown) Circular 7 Circuit Labyrinths in Sand

For every labyrinth exists a second or dual one. And in special cases the dual one looks like the original one. Then this is a self-dual labyrinth.

These connections should be explained here.

Andreas Frei has done this on his website under the topic “Grundlagen” (basics), to this day only in German. I expressly recommend to take a look at it, there are some meaningful drawings also.

Here again we will see it from the practical side. Hence, it is a continuation of the post from the 1st of September, 2013 about the circular 7 circuit labyrinths. Through the dual labyrinths here we will get six more to add to the seven there. So we will have 13 new labyrinths in all.

How we will reach for that, should be shown step by step. Maybe a little bit awkwardly, but I hope, understandably.

We number all labyrinths from the outside inwards in black. “0” stands for the  outside and “8” for the center. The path sequence, that is the order in which we walk through the circuits to arrive at the center, is noticed on the bottom left in black.
Then we number all circuits once again from the inside outwardly in green. “0” is now the center and “8” is now the outside. We write down the circuits in the order in which we walk them while going backwards from the middle. This path sequence is noticed on the bottom right in green.

Typ 032147658Typ 034765218Typ 034567218Typ 036547218Typ 054367218Typ 056723418Typ 056743218Typ 056741238

As already mentioned, there is to every (original) labyrinth a second (dual) one. And this arises when we interchange inside and outside, when we turn inside out. The path sequence which we will get, is normally different from the one of the original labyrinth.

If it is the same, we speak of a self-dual labyrinth. Then an internal symmetry is given. Or differently expressed: The rhythm and the motion sequence is the same when stepping inside or outside.  In our examples this applies to the first (well-known Cretan) labyrinth, and to the last, a new labyrinth.

The remaining six have another path sequence and, hence, are to be taken for new, different labyrinths.
Here the six new types (click to enlarge, print or save):

Typ 076321458Typ 076123458Typ 076143258Typ 076125438Typ 074561238Typ 076541238

These examples shows that always at first the middle is circled around. After that one moves inside the the labyrinth and finally one enters the center from the 3rd or the 5th circuit.

In the case of the types introduced in the last article the entry into the center was always from the outermost, the first circuit. Here we have the circling around the middle immediately after stepping into the labyrinth.

The motion sequences are completely different.
It would be of interest exploring that by a temporary or even a permanent labyrinth. Worldwide there are still no labyrinths of this kind.
The shape must not necessarily be perfectly circular. It is important only to adhere to the path sequence.
For the rest, they can be as simply build in sand like the types introduced in the below mentioned post.

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