The topic of this blog is the labyrinth. Under nearly all aspects, I would like to arouse your interest on the fascinating lines and the meaning of this old object. Being an old surveyor I put my focus on the geometrical shape.
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Most of the pictures and graphics were created by Andreas Frei and me (Erwin Reißmann), unless stated otherwise, and are provided under license CC BY-NC-SA 4.0.
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.
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 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).
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).
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.
The subject of closely related labyrinths has been treated on this blog already repeatedly. When deriving the complementary labyrinth, I have encountered the group of the four related labyrinths. This occurred when I derived the complementary of the base labyrinth and then derived the duals of the base and the complement (see: related posts 1, below). As a fourth labyrinth there indirectly resulted the dual of the complement what is nothing else than the transpose.
Richard Myers Shelton, however, has published this group of closely related labyrinths already earlier°. In his article, he introduced the concept of the transpose. He derived the transpose of the base labyrinth and subsequently derived the duals of the base and the transpose. Thus, as a fourth labyrinth there indirectly resulted the dual of the transpose what is the same as the complementary labyrinth.
We are thus presented with two different representations of the same situation. In the following, I want to examine this further and in addition will present a third version of the same situation. I will again base my considerations on the labyrinth with one axis and five circuits I had already used when deriving the complementary labyrinth (related posts 2). This is the second of the eight different alternating labyrinths with one axis and five circuits and the simplest labyrinth for which a complete group of four closely related labyrinths exists. This is the “base” labyrinth and has the three relatives, the “dual”, the “transpose”, and the “complement” as shown in fig. 1.
In fig. 2 I show, how we can directly derive the other three labyrinths from the base labyrinth. For this, the pattern of the labyrinth is used. The pattern is obtained from the Ariadne’s Thread in the rectangular form (related posts 3).
In the first line, the derivation of the dual pattern is shown. For this purpose, the base pattern is rotated by 180 degrees. The connections to the outside (triangle) and to the center (bullet point) are interrupted. After rotation, these ends are reconnected, but the connections are exchanged.
The second line shows how the transpose pattern is derived. For this, the base pattern is mirrored horizontally (against the vertical). Again, the connections to the exterior and the center are interrupted. After mirroring, the ends point to the wrong direction. They have to be flipped to be reconnected with the exterior and the center. These two operations mirroring and flipping are combined in icon symbolizing the transpose operation.
In the third line, the derivation of the complementary pattern is shown. This is obtained by vertically mirroring the pattern (mirroring against the horizontal). Similarly, the connections to the exterior and the center are interrupted and after mirroring, the ends point to the wrong direction. They have to be flipped to be reconnected with the exterior and the center. These two operations mirroring and flipping are combined in icon symbolizing the complement operation.
These are the three operations that can be used to derive the dual, the transpose and the complementary labyrinth from the base labyrinth. A direct application of two of these operations is sufficient to indirectly effect the third one. Thus, the group of the four closely related labyrinths can be represented in three different ways. More about this in the next post.
° Shelton, Richard Myers. 2015. „Wayland’s New Labyrinths“ Caerdroia 44, 44-55.
During my Sweden tour in 2007, I also noticed a special traffic sign that I really liked. It indicates points of interest and shows a looped square.
The looped square has been around for a long time, in different forms and in many cultures. It is an ornament, known as St. John’s Arms, in heraldry as a Bowen knot, it is used as a marker and as a keyboard symbol on (Apple) computers.
This picture stone from Stora Havor from around 400 – 600 AD shows a very nice example, which is now in the Fornsalen Museum in Visby (Gotland):
Or this Cox mound gorget from the Mississippi culture, Tennessee (USA), from around 1250 – 1450:
Now another example from heraldry from an English book by Hugh Clark from 1827, where the looped square can be seen in the Bowen knot:
So we have an uninterrupted line in front of us as we find it in the labyrinth, but without a beginning and without an end. Could that be a suggestion for a labyrinth? There are also no branches, even if there are changes of direction. We just have to see or think in three dimensions. And we need a beginning and an end.
Here is the initial figure:
The looped square labyrinth could look like this:
The line arrangement is clear, because the intersections of the lines are not intersections where we could branch off. All we have to do is go forwards and forwards and follow the curves in the process. One could also think of overpasses or underpasses, as with motorways. In the example above, the underpasses and overpasses are easy to see. However, it can also be done without these precise delimitations.
The construction of such a labyrinth would also be a challenge? Who dares to do it? There is also a design drawing for a kind of prototype:
Here the layout drawing as a PDF file to print, save or to look at. To do so, click >> on the top right of the document.
With the 42 sector patterns it is possible to generate a vast number of different sector labyrinths with 7 circuits. If we don’t just want to aribitrarily combine some sector patterns, we should develop an idea for the design and set restrictions that limit the range of the possible combinations. So, for example, our exclusive use of triple barriers in all (side) axes is a strong limitation (see: related posts 1, below).
An other limitation would be to use only one single- and one double barrier in each side-axis. The single barrier occupies two, the double barrier four circuits side by side. Only one circuit remains for the transition from one sector to the next.
Now we could even restrict further and specify that the single barrier must not lie directly side by side with the double barrier, that is to avoid that the axes look as shown in fig. 1.
Then, the passage between the sectors always must lie on one of the circuits 3 or 5. I have identified sector patterns that comply with these requirements on at least one side in fig. 2. Interestingly, this is again the case in 14 different sector patterns. In 12 of these on one side, and in 2 on both their sides.
I have also attributed these sector patterns to quadrants, according to the sector they can be placed in and to the cirucits by which they are connected with other sectors. Figure 3 shows, that we exactly have the same baseline situation – with different sector patterns – as we already had with the triple barriers (related posts 1). The sector patterns that are represented in both quadrants A and D correspond with the pattern of the classical labyrinth with one axis. Those represented in both quadrants C and B correspond with the complementary of it with an S-shaped course of the pathway (related posts 2). Only these two patterns can be placed in any sector and can be used for the „inner course“.
Thus, it can already be concluded at this point, that there are again 16 different patterns for each of the four possible „inner courses“ AB, CD, CB, and AD independent of the number of axes the sector labyrinth will have. Here, I only want to pick out two of them. The first is a self-transpose labyrinth with 6 axes and a course AB (shown with a line with dimensioning marks). The second is a self-dual labyrinth with five axes and a course CB (shown with a line with arrows).
In order to generate the patterns, I simply connect sector patterns such that the base lines of the red triangles fit to each other. I abstain from finalizing the drawings of the patterns, as they are already clearly recognizable and my intention is to highlight / illustrate the method how they are combined.
The labyrinth with six axes and a course AB only consists of the two sector patterns at all that can be placed in any sector. This is the only possibility to design a self-transpose labyrinth for the course AB in which the entrance and the access to the center do not lie either on the outermost or innermost circuit. I consider these latter labyrinths uninteresting. In addition there exists the complementary with a course CD.
Finally, figure 5 shows the self-dual labyrinth with five axes that results if we place the two sector patterns indicated by the line with arrow endings of quadrants C and B in the first and last sector and connect them with three sector patterns for the „inner course“. Also in this course, a self-dual labyrinth can be generated by soley using the two inner sector patterns. In addition, however, another self-dual labyrinth can be designed, in which the path enters on the innermost and reaches the center from the outermost circuit. This labyrinth is shown in fig. 5. Furthermore, two self-dual labyrinths can be generated with the path entering on the outermost and reaching the center from the innermost circuit and therefore in my opinion are uninteresting labyrinths.