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 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:

The Labyrinths With Real or Pseudo Double-barriers, 4 Arms and 5 Circuits

As is known, there exist 8 labyrinths with 3 double-barriers, 4 arms and 5 circuits (see: related posts 1, below). In fig. 1, I show the patterns and labyrinths again and give them subsequent names from „A“ to „H“.

Figure 1. The 8 Labyrinths with real Double-barriers

The most important restriction when deriving these labyrinths was, that the double-barrier must look the same as in Gossembrot’s labyrinth. Erwin has commented and wants also to include courses of the pathway, in which the path may change from the outermost to the innermost circuit or vice versa. I don’t consider these as double-barriers. Furthermore, this principle for the design has already occurred in earlier historical labyrinths. I will now refer to the new double-barrier, consistently used by Gossembrot as „real double-barrier“ and term the older course of the pathway „pseudo double-barrier“. Undoubtedly, the pseudo double-barrier is also an interesting element for the design. In it’s pure form it has been realized in the labyrinth type Avenches. Similarly, the real double-barrier in it’s pure form occurs in the 8 labyrinths shown in fig. 1. It is also possible to mix both principles of design, as has been done by Erwin and Mark Wallinger (see related posts 2).

Here I am interested in the question, how many labyrinths there are if real and / or pseudo double-barriers are used. The most important fundamental concepts for this have been elaborated previously (see: related posts 1). However, what changes is, that now not only options a) or b), but also options c) or d) are allowed for the connection of the sectors (fig. 2).

Figure 2. Admissible Connections

The other constraints, however, remain still valid. Sector patterns no. 1 and no. 6 cannot be used at all. The four one-sided sector patterns no. 2, no. 4, no. 5, and no. 7, can still be placed only in quadrants I and IV. This, because we want to preserve both halves of a double-barrier in all side-arms even if the path changes the circuit when traversing a side-arm. Therefore again, also only patterns no. 3 and no. 8 can be placed in every quadrant.

From this it follows, that we can start with the eight already known labyrinths with real double-barriers. In order to illustrate the following considerations, I pick out the labyrinth D. This has the sequence of patterns 8 3 8 3.

If we now also allow for pseudo double-barriers, this results in not only one but two possibilities at each side arm for the connection of one sector with the next: one on the same circuit, that I will term „direct“ connection, and another one with a change from one extreme circuit to the other, that I will term „indirect“ connection. Since at each side-arm both options are at the disposal, this leads to a much greater number of possible combinations.

Figure 3 shows the possible combinations if we start with labyrinth D and allow also indirect connections. In the first quadrant stands pattern no. 8. At the first side-arm there are two possibilities for a connection between quadrant I and quadrant II. Pattern no. 8 from quadrant I can be directly connected with no. 3 or also indirectly connected with no. 8 in quadrant II. Pattern no. 8 is the complementary of no. 3. An indirect connection requires the pattern complementary to the one for a direct connection. This is a general rule. At the 2nd side-arm for each of the patterns from quadrant II, there are again two possibilities to connect them with quadrant III, and similarly, the same applies to the 3rd side-arm. Whereas previously there were only 1*1*1 = 1 combinations for a direct connection, there are now together 2*2*2 = 8 possible combinations for a direct and / or indirect connection of all quadrants.

Figure 3. Possible Combinations with Direct or Indirect Connections Based on Labyrinth D

Each combination results in a new 4-arm sector labyrinth. I will illustrate this in fig. 4 with the first combination. This results in the already known labyrinth D with exclusively real double-barriers and the sequence of patterns 8 3 8 3.

Figure 4. The First Combination: Exclusively Direct Connections with Real Double-barriers – Labyrinth D

As a second example I show in fig. 5 a pattern that is formed by a combination of real and pseudo double-barriers. Namely, this has pseudo double-barriers with indirect connections at the first and third side-arms, whereas there is a real double-barrier with a direct connection at the second side-arm. This combination results in a sequence of patterns of 8 8 3 3.

Figure 5. The Sixth Combination: Mix of Real and Pseudo Double-barriers

Finally, fig. 6 presents all eight patterns that can be obtained starting with labyrinth D by combining real and pseudo double-barriers.

Figure 8. All Eight Combinations Based on Labyrinth D

The same approach as for labyrinth D is also feasible for the seven other labyrinths, i.e. labyrinths A, B, C, E, F, G, and H. In sum, this then results in 8 * 8 = 64 different types of labyrinths with three real or pseudo double-barriers, four arms and five circuits.

Related Posts:

  1. The Labyrinths with 3 Double-barriers, 4 Arms, and 5 Circuits
  2. New 5 Circuit Labyrinths with Double Barriers

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

Sigmund Gossembrot has used the double-barrier as a new element for the design of labyrinths. His five-arm labyrinth on fol. 51 r (see: related posts 5) and the four-arm labyrinth hidden in the design on fol. 53 v (related posts 4) are made up exclusively of double-barriers in all side-arms. They have 7 circuits and are no sector labyrinths.

In a series of posts, Erwin has introduced new sector labyrinths with four arms, five circuits and double-barriers (related posts 1, 2, 3). He has based these designs on the 8 possible courses the pathway can take in a one-arm labyrinth with five circuits. Sector labyrinths can be obtained by stringing together such courses of the pathway. Theoretically there exist 4096 variations for an arbitrary stringing together of four out of 8 courses of the pathway. Erwin has shown some of them. However, not all did consistently employ the principle of the double-barriers.

Here I will address the question how many sector labyrinths there are with four arms, five circuits and consistently employing double-barriers. I also start with the 8 possible courses of the pathway. These are based on Arnol’ds’ meanders in fig. 1 (related posts 6).

Figure 1. The Meanders by Arnol’d

In fig. 2 I show the patterns that correspond with the meanders. The patterns bear the same numbers as the meanders they were derived from. The left side of the figure shows the patterns of all alternating one-arm labyrinths with five circuits. Each of hese patterns also contains a connection from the outside into the labyrinth (from top left) and a connection to the center (to bottom right). These connections are represented in grey. In order to be used as segments (sectors) in sector labyrinths, these patterns at first have to be considered without the grey connection lines. The question here is the course of the path inside the sector. In a sector labyrinth, multiple such patterns are stringed together. Only the first pattern contains a connection to the outside and only the last a connection to the center. The patterns representing the 8 possible courses in a sector are displayed in the box on the right side.

Figure 2. The Corresponding Patterns – Left Half: Patterns of the One-arm Labyrinths; Right Half: Patterns of the Sectors

Next we intend to string together such patterns of sectors and to generate four-arm labyrinths using exclusively double-barriers. Let us first have a look at such a double-barrier in the labyrinth type Gossembrot 51 r. Figure 3 shows the labyrinth with the Ariadne’s Thread drawn in (red). Except for the one-arm labyrinths, an axis always lies between two segments, is formed by two different segments. Let us choose the double-barrier at the third side-arm. This connects segments III and IV and lies on the outermost four circuits. In the scaled-up section the seed pattern for the walls delimiting the path is drawn-in in blue color. It can be seen, that two nested turns of the Ariadne’s Thread are mirrored symmetrically against the central piece of the wall delimiting the pathway. Four circuits are needed for the double-barrier. In a labyrinth with five circuits, only one circuit remains free for the passage from one sector to the next. From this it becomes clear, that labyrinths with five circuits and using exclusively double-barriers must be sector labyrinths. There is only one circuit for the path to traverse the arms. This implies that the pathway must have completed the previous sector entirely before changing to the next.

Figure 3. The Double-barrier as by Gossembrot

Figure 4 shows the admissible connections between the sectors. (Pro memoria: the lines represent the pattern, i.e. the Ariadne’s Thread in rectangular form). The double-barriers occupy four circuits side by side. So they can lie in two places on circuits 2 – 5 or on circuits 1 – 4. Only connections on the same circuits are allowed, that is, the two options on the outermost (a) or on the innermost (b) circuit. If we would consider to change to another circuit when connecting the segments, as shown in options c or d, this would result in the insertion of an additional axial piece of the path between the halves of the double-barrier and these halves would be shifted by one circuit one against the other. But this is not a double-barrier any more.

Figure 4. Admissible Connections between Sectors

This circumstance strongly limits the range of the possibilities for stringing together the patterns. Figure 5 shows, how the different patterns can be used. The red numbers, characters at the free ends of each pattern indicate with which patterns it can be connected there (number of pattern, E for entrance, Z for center). A four-arm labyrinth has four segments. These are therefore also referred to as „quadrants“.

Figure 5. Possibilities for the Use of the Patterns

  • Two patterns, no. 1 and no. 6 cannot be used at all. With these it is not possible to generate a double-barrier.
  • Four „one-sided“ patterns, that is no. 2, no. 4, no. 5, and no. 7 have only on one side a half of a double-barrier (circled in red). On this side they can be connected with other patterns to double-barriers. It is true that it is also still possible to connect pattern no. 2 with no 5, and pattern no. 4 with no. 7 (not indicated). However, such a connection results in a two-arm labyirinth with one double-barrier only. On the second side of these one-sided patterns the free end lies on the third circuit. There, no double-barrier can be generated. Therefore, on this side only a connection to the entrance or the center is possible. Thus, these one-sided patterns can only be placed next to the main axis. Pattern no. 2 and no. 7 can only be placed in quadrant IV, where they are connected with the center. Pattern no. 2. can further only be connected with no. 8, and pattern no. 7 can be connected with no. 3.
  • Only two patterns, no. 3 and no. 8 can be completed to double-barriers at both sides. And only these can be placed in quadrants II or III. Furthermore, they may also be placed in quadrants I or IV, and thus be connected with the entrance or with the center (not indicated). Patterns no. 3 and no. 8 can be stringed together alternately or they can be connected with other one-sided patterns (pattern no. 3 with no. 5 and no. 7; pattern no. 8 with no. 4 and no. 2).

This provides us with the basis for the generation of the patterns for the sector labyrinths with the double-barriers. We start with the patterns for the quadrants II and III. For these, there are only two arrangements. Pattern no. 8 can be attached to no. 3 (above) or pattern no. 3 can be attached to no. 8 (below). The upper combination can be supplemented with patterns no. 5 or no. 8 towards quadrant I, and with patterns no. 2 or no. 3 towards quadrant IV. The lower combination can be supplemented with patterns no. 3 or no. 4 towards quadrant I, and with patterns no. 7 or no. 8 towards quadrant IV.

With the upper combination of the patterns no. 3 and no. 8 in quadrants II and III, thus, four patterns of labyrinths with four arms, five circuits and a consistent use of double-barriers can be generated. These patterns are shown in fig. 6.

Figure 6. The Patterns with the Combination no. 3 in Quadrant II – no. 8 in Quadrant III

Also with the lower combination of the patterns no. 8 and no. 3 in quadrants II and III, four patterns of labyrinths with four arms, five circuits and a consistent use of double-barriers can be generated. These patterns are shown in fig. 7.

Figure 7. The Patterns with the Combination no. 8 in Quadrant II – no. 3 in Quadrant III

Figure 8 now shows the labyrinths corresponding to the patterns of fig. 6.

Figure 8. The Labyrinths Corresponding with the Patterns of Fig. 6

Finally, figure 9 shows the labyrinths corresponding to the patterns of fig. 7.

Figure 9. The Labyrinths Corresponding with the Patterns of Fig. 7

The question concerning the number of possible labyrinths can be clearly answered:

  • There exist 8 labyrinths with 3 double-barriers, 4 arms and 5 circuits.

Moreover, in addition to this question, we obtain the following findings:

  • Labyrinths with 5 circuits and fully employing double-barriers must be sector labyrinths.
  • Such labyrinths cannot have double-barriers at the main axis. Double-barriers are only possible at the side-arms.

Related Posts:

  1. New 5 Circuit Labyrinths with Double Barriers
  2. A new Generation of Sector Labyrinths
  3. A new type of Sector Labyrinth inspired by Gossembrot
  4. Sigmund Gossembrot / 3
  5. Sigmund Gossembrot / 2
  6. Considering Meanders and Labyrinths