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.

Related Posts:

  1. The Labyrinths with 4 Real Double-barriers, 5 Arms, and 5 Circuits
  2. Classifying the Labyrinths with 3 Double Barriers, 4 Arms, and 5 Circuits

The Labyrinths with 4 Real Double-barriers, 5 Arms and 5 Circuits

Up to now I have examined only sector labyrinths with four arms. The real double-barrier, however, originates from the five-arm labyrinth type Gossembrot 51r. As we know, this is not a sector labyrinth and it has 7 circuits. Now I want to find out, how many sector labyrinths with five arms and exclusively real double-barriers there are. Such labyrinths must have five cirucits. Therefore, we can use the same six sector patterns we have already used for the four-arm sector labyrinths.

In these four-arm labyrinths, only two sector patterns could be placed in every quadrant, i.e. sector patterns no. 3 and no. 8. Four sector patterns could only be placed in the quadrants next to the main axis (related posts 2). Now this is not different either in labyrinths with five arms. However, we then have to fill not only four quadrants, but the five sectors I til V with sector patterns. Sectors I and V lie next to the main axis. In the three sectors II, III, and IV between them, only sector patterns no. 3 or no. 8. can be placed. These can be arranged in only two different sequences, 3 8 3 or 8 3 8.

Figure 1 shows how the first sequence can be completed with patterns for sectors I and V. In sector I the two sector patterns no. 5 or no. 8 can be connected to the sequence 3 8 3. In sector V, sector patterns no. 7 or no. 8 can be attached.

Figure 1. Combinations with the Sequence 3 8 3 in the Sectors II – IV

In fig. 2 we can see, how the second sequence can be completed to a full five-arm labyrinth. Here, in sector I the sector patterns no. 3 or no. 4, in sector V the sector patterns no. 2 or no. 3 can be attached to the sequence 8 3 8 between them.

Figure 2. Combinations with the Sequence 8 3 8 in the Sectors II – IV


Figure 3 shows the four patterns and labyrinths that can be generated with the first sequence (3 8 3 from fig. 1).

Figure 3. Patterns and Labyrinths with the Sequence 3 8 3 in the Sectors II – IV


Figure 4 shows the four patterns and the corresponding labyrinths that can be generated with the second sequence (8 3 8 from fig. 2).

Figure 4. Patterns and Labyrinths with the Sequence 8 3 8 in the Sectors II – IV


Just as in the four-arm labyrinths, there exist also 8 different five-arm labyrinths made-up exclusively of real double-barriers. Even though they have only five circuits, they strongly remind us to the labyrinth type Gossembrot 51 r. We can also name them following the same rule as for the four-arm labyrinths (related posts 1). The name, thus, is composed of an uppercase letter followed by a number of marks. However, since these labyrinths have four side-arms, also four marks have to be attached to each uppercase letter. These marks indicate how the sectors are connected to each other. We have here exclusively real double-barriers with direct connections. Therefore, each name is composed of an uppercase letter followed by four horizontal marks.

Related Posts:

  1. Classifying the Labyrinths with 3 Double-barriers, 4 Arms, and 5 Circuits
  2. The Labyrinths with 3 Double-barriers, 4 Arms, and 5 Circuits

Classifying the Labyrinths with 3 Double-barriers, 4 Arms, and 5 Circuits

There exist 64 patterns of labyrinths with 3 real or pseudo double-barriers, 4 arms, and 5 circuits (see: related posts 1, below). Up to now, only a few of them have been published in any form at all, and among those probably most were published in this blog. The earliest one, however, is a Roman mosaic labyrinth. Now I am tempted to attribute all examples that have been shown last year in this blog to one of the 64 patterns.

Until now, I have used the following names for the patterns:

A – H for the 8 labyrinths with exclusively real double-barriers
A’ – H’ for the 8 labyrinths with exclusively pseudo double-barriers

Therefore, so far, only 16 of the 64 patterns have been given a name. These names have evolved during the elaboration of the previous posts. In order to attribute a name to each of the 64 patterns a more systematic approach is needed. For this, the naming has to be refined. This can be done by drawing upon the tree-diagram (related posts 1). I have given the name D to the uppermost pattern with exlusively real double-barriers, and D’ to the lowermost that uses exclusively pseudo double-barriers. Now we need a differentiation that provides unique names to all eight patterns, i.e also to the six remaining patterns.

Fig. 1 shows (still with the example of the basic labyrinth D) the kind and the sequence of connections between the sectors. Direct connections (real double-barriers) are represented with a horizontal (–), indirect (pseudo double-barriers) with a vertical (|) mark. This order of the combinations is not arbitrary, but systematically ordered. The uppermost combination consists of only direct connections and is represented by the sequence – – –. In the second combination, the last direct connection is replaced by an indirect connection, resulting in the sequence – – |. The third combination replaces the central direct by an indirect connection and results in the combination – | –. The fourth combination uses indirect instead of direct connections in the central and last connection (– | |). And so forth.

Figure 1. Sequence of Connections

If we substitute „–„ with „0“, and „|“ with „1“ , we can see, that the order of the combinations simply corresponds with the binary numbers from 000 to 111. Namely, these are the first eight numbers from Zero to Seven written in the binary system.

Figure 2. Order of the Combinations

With this we can give a unique denomination to each of the 8 patterns that have been derived in fig. 1 (starting from labyrinth D). And not only that. This denomination also provides insight into how the sectors are connected one with another. In this new denomination, I name the first pattern D – – –. It’s previous name was D. The second pattern previously had no name yet and it’s new name is D – – | and so on til the seventh pattern, all without previous names. The lowermost, eighth pattern prevously was named D’ and it’s new name is D | | |. This systematics is independent of the basic labyrinth. We can apply it to all labyrinths A – H. By this, we can give an unique denomination to each of the 64 patterns made-up of a uppercase letter followed by three horizontal or vertical marks.

Now I will attribute some real examples.

Three labyrinth examples can be attributed to one of the patterns from tree diagram D. The earliest one is Roman mosaic labyrinth of the Avenches type (related posts 5). This has the lowermost pattern D | | |.

The second example was introduced by Erwin in his post from August.2019 (related posts 4) and has the uppermost pattern D – – –.

The third example is the one by Mark Wallinger 233/270 shown in the post from October 2019. This has the third pattern D – | – (related posts 2).

Figure 3. Labyrinths of Group D

The labyrinths from fig. 4 cannot be attributed to any variant of the basic labyrinth D. These are all labyrinths with only real double-barriers. That means, they are the basic labyrinths and have the uppermost patterns of other tree diagrams.

The first labyrinth was shown in Erwin’s post from August 2019 and has the pattern G – – – (related posts 4).

The second labyrinth from the post of Erwin from September 2019 has the pattern F – – – (related posts 3).

Third is the labyrinth 10/270 by Mark Wallinger from the same post, and this has the pattern A – – –.

Figure 4. Classification of Other Labyrinths with Real Double-Barriers

The new sector labyrinth in fig. 5 from Erwin’s post from October 2019 has the pattern G – | – (related posts 2). Thus, it is one of the 48 patterns with mixed real and pseudo double-barriers.

Figure 5. Labyrinth from Group G with Real and Pseudo Double Barriers

The labyrinth shown in fig. 6, however, cannot be attributed to any one of the 64 patterns, because it does not have double-barriers in all side-arms. This labyrinth originates also from the post of Erwin from September 2019 (related posts 3). This example shows very well, that it is not possible to generate double-barriers using sector patterns no 1 and no 6.

Figure 6. Not a Labyrinth with Only Double Barriers

However, this was not the intention either. Erwin just wanted to once make use of all eight sector patterns in a four arm sector labyrinth.

Related Posts:

  1. The Labyrinths with Real or Pseudo Double-barriers, 4 Arms, and 5 Circuits
  2. New 5 Circuit Labyrinths with Double Barriers
  3. A new Generation of Sector Labyrinths
  4. A new Type of Sector Labyrinth inspired by Gossembrot
  5. How to Draw a Man-in-the-Maze Labyrinth / 15

The Labyrinths with 3 Pseudo Double-barriers, 4 Arms, and 5 Circuits

In my last post I have shown, that there exist 64 labyrinths with 3 real and / or pseudo double-barriers, 4 arms, and 5 circuits (see: related posts, below). But, how many of these do have exclusively pseudo double-barriers?

This question could actually be answered with the material from the last post. In order to show this, I once again make use of the tree diagram (fig. 1). This shows the combinations that can be obtained based on labyrinth D. There we can see, that the uppermost combination results in a pattern made-up exclusively of real double-barriers (that is, labyrinth D). This is the only one of the eight patterns using only real double-barriers. Similarly, the lowermost combination results in the only pattern made-up exclusively of pseudo double-barriers. I will term this D’. The six combinations in between all result in patterns with mixed combinations of real and pseudo double-barriers.

Figure 1. Combinations with Real, Pseudo, and Mixed Double-barriers

Now, if we proceed the same way as in fig. 1 for all labyrinths A – H, we always will obtain a lowermost combination made-up exclusively of pseudo double-barriers. These patterns and the corresponding labyrinths are shown in fig. 2. I have termed them A’ – H’. Labyrinths with the same uppercase letter belong to the same tree diagram.

Figure 2. The 8 Labyrinths with 3 Pseudo Double-barriers, 4 Arms, and 5 Circuits

Thus, we can conclude, that among the 64 labyrinths there are

  • 8 labyrinths with only real double-barriers
  • 8 labyrinths with only pseudo double-barriers
  • 48 labyrinths with real and pseudo double-barriers

Related Posts: