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).
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.
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 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.
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“.
- 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.
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 8 now shows the labyrinths corresponding to the patterns of fig. 6.
Finally, figure 9 shows the labyrinths corresponding to 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.