The theme of this blog is the labyrinth in almost all aspects. It has been around since 2008. Since 2012 Andreas Frei from Switzerland is part of it. About once a month a new post should appear.

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The last crossing labyrinths I want to show were all designed by Dom Nicolas de Rely. This clergyman from Benedictine abbey Corbie near Amiens has produced eight drawings with own labyrinth designs, all in pen and ink. Three of them are crossing labyrinths. I have ordered them by the number of axes and labelled them Rely 2, 3, and 4.

Rély 2 has 15 circuits. It is designed on a layout with 8 axes; however by shifting of one (real) single barrier, it can be reduced to 7 axes. The pathway crosses the main axis from the 7th to the 12th circuit. And it reaches the center from the innermost 15th circuit, which is a complete attached trivial circuit. Therefore it is an uninteresting labyrinth (fig. 1).

Because of its pseudo single barriers, Rely 3 has been already shown on this blog (see related posts, below). It has 9 axes and 5 circuits. The pathway crosses the main axis from the 4th to the 1st circuit and reaches the center after a full circle on an attached trivial 5th circuit. Thus, also this labyrinth has to be described as uninteresting (fig. 2).

The third crossing labyrinth, Rély 4, is designed on a layout with 14 axes and 15 circuits (fig. 3). This, however, can be reduced to 10 axes. The pathway crosses the main axis from the 6th to the 13th circuit. The entrance to the labyrinth is from the left side and (erroneously?) closed. The center is not reached at the main axis, but from the third side-axis on the innermost circuit. Therefore there remains a short piece of the pathway leading into a dead-end at the end of the last circuit.

I will have a closer look at the two labyrinths Rély 2 and Rély 4 in a later post.

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.

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.

In addition to the three labyrinths with one axis from my last post (see: related posts 1, below) there are also 7 historical labyrinths with multiple axes and with their pathway crossing the main axis. Of these, I want to present here four very different examples from Roman times until the 18^{th} century together with their patterns. I have already shwon on this blog how the pattern can be obtained in crossing labyrinths (related posts 2).

The oldest crossing labyrinth with multiple axes is the polychrome mosaic labyrinth in the Roman proconsul’s residence, House of Theseus, at Kato Paphos, Cyprus dating from 4 CE (fig. 1). Presented is the Ariadne’s Thread as a guilloche ribbon. The pathway starts from a dead-end on the first circuit. After completion of the full circuit, it crosses the main axis and describes a sector labyrinth with four axes on circuits 2 – 6. Then follows a full 7^{th} circuit that leads into a closed 8^{th}circuit.

Figure 2 shows the labyrinth of Bayeux Cathedral from the 13 CE. This has 4 axes and 10 circuits. The pathway crosses the main axis on the innermost circuit.

A strange labyrinth is depicted on a plaquette from Italy of the 16^{th} century. It has 6 axes that are distributed irregularly. There is a flaw between the third and fourth axis, where there is an encapsuled piece of a pathway that is not accessible. This piece circulates on the second and third circuit but has no connection with the pathway that leads from the entrance to the center of the labyrinth. Furthermore, the pathway crosses the main axis three times. This labyrinth can be easily reduced to three axes.

Also in this design for a hedge labyrinth from year 1704, the pathway crosses the main axis twice and then ends peripherally in a dead-end (fig. 4).

All these crossing labyrinths with multiple axes show particularities. Theseus has no entrance and no center, Bayeux is uninteresting, as it has simply a complete circuit added at the inside. The plaquette is drawn faulty and unnecessary complicated. And in Liger, no center can be spotted.

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:

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.

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:

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:

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.