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## 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).

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.

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