# How to Generate the Seven Times Seven Labyrinth

For New Year I have presented the Seven Times Seven labyrinth (see: related posts 1, below). Erwin has immediately commented and noticed the similarity with the type Gossembrot 51 r. This is correct. I wanted to develop a self-dual labyrinth based on this type. And I wanted to preserve the typical characteristics of the course of the pathway. Typical for Gossembrot’s labyrinth are not only the double-barriers, but also the manner in which the path is directed through all segments. It is not a sector labyrinth, but rather in about the opposite of that.

In figure 1, I show the pattern of the Gossembrot 51 r type labyrinth. This serves as the starting point (a) and is presented in grey. I have already described earlier, what characterizes the course of the pathway (related posts 2). This happens in segments III to V. Another speciality is the meander in segment II. This meander lies on the circuits 2 – 6. So there is one more circuit each outside and inside of the meander.
First, I isolate the segment that contains the meander (b). The meander itself is self-dual. And, since there are added one more circuit each, at the outside and inside, the whole figure (b) is self-dual too. To this figure are attached on the right side segments III – V. These contain the typical course of the pathway by Gossembrot. From the fact, that segment II is self-dual, it also follows that one of its sides can be connected with a figure that is the dual to the figure connected with its other side. In a second step, therefore, I pick out the figure in segments III – V and place it to the right side of segment II. Figure (c), thus, shows nothing else than segment II not connected with segments III – V of the pattern of Gossembrot 51 r.

Figure 1. Preparation

This figure (c) forms the basis for the generation of the Seven Times Seven labyrinth, or of it’s pattern respectively. The process is shown in fig. 2. Here we begin in the third row with the figures colored in grey (c). In a third step, the figure from segments III – V is now duplicated (d). This duplicate is then rotated by 180 degrees in a fourth step. This produces the dual figure of it (e). Then we shift it downwards and can see: it can be connected to the left side of the figure with the meander from segment II (f). Now we only have to really connect these elements with each other and by this obtain in figure (g) the pattern of the Seven Times Seven labyrinth.
This whole pattern is self-dual. The number of segments has increased from the five segments of the labyrinth type Gossembrot 51r to new seven segments. The dual of Gossembrot’s segments III – V covers the new segments I – III, the meander with its additional circuits inside and outside follows in central segment IV, and Gossembrot’s original segments III – V are here shifted to segments V – VII.

Figure 2. Generation of the Pattern

Figure 3 shows the labyrinth in the basic form without the heptagram in the center and without the heptagon at the periphery. These are add-ons and have to be attributed to the style, rather than to the type of labyrinth.

Figure 3. The Labyrinth in Base Form

A very well balanced labyrinth. The main axis looks the same as in the basic type. Opposite to the main axis, in the central segment IV, lies the meander. In three segments before and after the meander, the typical course of the pathway can be found. The path proceeds in wrapping or wrapped curve through all segments, thereby passing the meander and arrives in a backward movement through sectors VII – V in sector IV, through which it continues as meander, then continues its backward movement through sectors III – I, from where it leads in forward direction through all segments to the center.

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# Sector Labyrinths with Double-barriers – Summary

Today I want to come to an end for the moment. I have begun with 4 arm labyrinths with double-barriers the same as were used by Gossembrot (see: related posts 3). Later I have referred to these as real double-barriers. Subsequent to a comment by Erwin I have also considered labyrinths with pseudo double-barriers. There already exists a historical type of such a labyrinth, the Avenches type. Real and pseudo double-barriers can also appear in combinations in the same labyrinth (related posts 2).

Labyrinths with double-barriers and five circuits must be sector labyrinths. Double-barriers can only appear in side arms. Therefore, a labyrinth with a double-barrier must have at least 2 arms. For each number of 2, 3, 4 and 5 arms there are always 8 different labyrinths using real double-barriers.

This suggests the conclusion that the number of different labyrinths with only real double-barriers is independent of the number of arms. It depends solely from the four sector patterns that can be placed in the first and the four that can be placed in the last sector. Two of the four sector patterns for the first sector are connected on the outermost, two on the innermost circuit with the following sector. Likewise, two of the four sector patterns each for the last sector are connected on the outermost or the innermost circuit respectively with the sector before them. This results in the 8 different labyrinths with tow arms and one double-barrier (related posts 1).

The number of arms can be only increased by inserting additional sectors between those two sectors next to the main axis. In these sectors between the first and the last sector only sector patterns no. 3 or no. 8 can be placed. And, in addition, these must be ordered in alternating order. Therefore, for the sectors between the first and the last sector, in any case, only one interface is available. Two complementary arrangements are possible for any number of arms. One of them begins with sector pattern no. 3, the other with sector pattern no. 8. For each of these arrangements, four different patterns can be generated by connecting them with the appropriate sector patterns of the first and of the last sector.

Thus, there are 8 different labyrinths using exclusively real double-barriers for any number of more than one arms. For each of these labyrinths, a tree diagram can be constructed. We have shown this with the example of labyrinth D (related posts 2). The tree diagram contains patterns for labyrinths with only real, only pseudo and for mixed double-barriers. The tree diagram has shown us, that there are the same number of patterns with exclusively pseudo as with exclusively real double-barriers. The uppermost branch of a tree diagram is made-up of the patterns with only real, the lowermost branch of those with only pseudo double-barriers. This too is true independent of the number of arms.

However, this is not valid for the number of labyrinths with combined double-barriers. This number rapidly increases as the number of arms increases. So, in two arm labyrinths there are only either 8 with a real or 8 with a pseudo double-barrier, but none with combined double-barriers, as these labyrinths can have only one double-barrier. In 3 arm labyrinths, again there are 8 with only real, 8 with only pseudo and, in addition, 16 with combined double-barriers. In 4 arm labyrinths we have seen, that there are 48 labyrinths with combined double-barriers. Finally in 5 arm labyrinths the number of labyrinths with combined double-barriers increases to 112 etc.

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# The Two Arm Labyrinths with a real Double Barrier and 5 Circuits

In my last post, I have shown, how we can obtain three-arm labyrinths from five-arm labyrinths by removing two of the three central sectors (see: related posts 1, below). Using a similar approach, we also can generate two-arm labyrinths from the four-arm labyrinths (related posts 3). The sector patterns next to the main axis then lie next to each other and can be directly connected. Again, there are two groups of four patterns each, that can be generated this way.

Figure 1 shows, for the first group, how the two sector patterns for the first sector can be combined with the two sector patterns for the last. These are connected on the outermost circuit.

Figure 1. The Combinations Passing on the Outermost Circuit

Figure 2 shows the same for the two pairs of sector patterns that can be combined passing through the innermost circuit.

Figure 2. The Combinations Passing on the Innermost Circuit

Figure 3 shows the four patterns and labyrinths that were generated with the combinations from fig. 1.

Figure 3. The Four Patterns and Labyrinths of the First Group Generated from Figure 1

Figure 4 shows the four patterns and labyrinths that were generated with the combinations form fig. 2.

Figure 4. The Four Patterns and Labyrinths of the Second Group Generated from Figure 2

Thus, there exist also 8 different two-arm labyrinths with one real double-barrier. These are named following the well-known rule (related posts 2). The name is composed of an uppercase letter, followed by a horizontal mark.

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# The Labyrinths with 2 Real Double-barriers, 3 Arms, and 5 Circuits

There exist 8 five-arm labyrinths using solely real double-barriers. In four of these, the sequence of the sector patterns in sectors II – IV is 3 8 3. In the other four, this sequence is 8 3 8 (see: related posts 1, below). It takes only a small jump from these five-arm to the respective three-arm labyrinths. As a matter of fact, of the three central sectors, simply two subsequent sectors have to be removed. Then, there remains only one central sector between the two sectors next to the main axis.

Figure 1 shows how for the first four patterns the sequence is reduced from five to three sector patterns. The sector patterns of sectors III and IV are removed (hatched area). Sector pattern no. 3 in sector II is directly connected with sector pattern no. 7 or no. 8 of the former sector V (red line). This becomes now the new pattern of the third sector.

Figure 1. Reduction of the Sequence of Sector Patterns from 3 8 3 to Sector Pattern no. 3

The same approach is also used in fig. 2 in order to reduce the second sequence from 8 3 8 to a central sector with the sector pattern no. 8.

Figure 2. Reduction of the Sequence of Sector Patterns from 8 3 8 to Sector Pattern no. 8

In both cases, this doesn’t change the remaining sequences of the patterns. The patterns of the sectors III and IV are simply skipped.

Figure 3 shows the four patterns and labyrinths, that have been obtained by reducing the first sequence. These all have sector pattern no. 3 in the central sector.

Figure 3. The four Three-arm Labyrinths with Sector Pattern no. 3 in Sector II

Figure 4 shows the four patterns and labyrinths, that have been obtained by reducing the second sequence. These all have sector pattern no. 8 in the central sector.

Figure 4. The four Three-arm Labyrinths with Sector Pattern no. 8 in Sector II

Just as in the five-arm labyrinths, there exist also 8 different labyrinths with exclusively double barriers and three arms. They are named following the rule known from previous posts (related post 2). Thus, the name is made-up of a capital letter followed by two horizontal marks.

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