Triple Barriers – Patterns for the First and the Last Sector

In the previous post, I have shown the four possibilities how the pathway can take it’s course along all side-arms (see: related posts 1, below). In order to design the main axis, now the sector patterns for the first and the last sector have to be completed. These were only drawn by one half and were used as placeholders. For these, we need the sector patterns that have only one half of a triple barrier at one side.

We know from earlier posts, that there exist 42 different patterns of alternating labyrinths with one axis and seven circuits. These patterns can be used the same way as we did this in the sector labyrinths with double barriers. We therefore have to draw them as sector patterns without the connections to the exterior or to the center. Tony Phillips has compiled the patterns of the interesting including the very interesting labyirnths on his website. These have also already been shown in this blog (related posts 4).

Figure 1 includes only 14 patterns of interesting labyrinths. Interesting are labyrinths that show no trivial circuits. Trivial circuits here are circuits that are attached either inside or outside in the manner of serpentines. This implies, that the pathway in interesting labyrinths must not enter on the first circuit nor reach the center from the last circuit. Labyrinths in which this is the case are uninteresting labyrinths.

However, in the figure information on 22 different patterns is included. Below each pattern the sequence of circuits is indicated. In interesting labyrinths, also the sequence of circuits of the dual labyrinth is indicated in parentheses (a). There are 8 patterns of interesting labyrinths. In such labyrinths, also the dual is an interesting labyrinth too. Thus, together with their duals, there are 16 patterns of interesting labyrinths.

In the very interesting labyrinths, the abbreviation „s.d“ meaning self-dual (b) is indicated. There are 6 very interesting labyrinths.

Among the very interesting labyrinths, two were especially highlighted (c). It is this the basic type and the complement of it, that have just recently been presented here in the Knidos style by Erwin (related posts 2, 3). These are the ony two complementary patterns in this figure. Among the remaining patterns none is complementary to any other. So we now have information on the 16 interesting and 6 very interesting labyrinths. Two of these are complementary. Remain the patterns of 16 interesting and 4 very interesting labyrinths. That is 20 patterns among which none is complementary to any other one. If we look up the complementaries of these, we will obtain 40 patterns. Together with the two complementary very interesting labyrinths this results in 42 patterns. Therefore, based on figure 1 we can reconstruct all 42 patterns (by the way: this is also possible without the information in a), b) and c) using a merely graphical approach, but with this information it can be calculated in advance).

Remains to mention, that Tony Phillips draws the patterns from top right to bottom left (d).

Figure 1. Patterns of Interesting Labyrinths with 7 Circuits by Tony Phillips
Figure 1. Patterns of Interesting Labyrinths with 7 Circuits by Tony Phillips

We need in any case also the patterns of the uninteresting labyrinths. Because they will not be used for labyrinths with one axis, but as sector patterns instead. The connections of sectors always occur on the outermost or on the innermost circuit. A sector that is connected to the previous one on the outermost circuit or connected on the innermost circuit with the subsequent will have a sector pattern, which would in the case of a labyrinth with one axis with connections to the exterior or to the center result in an uninteresting labyrinth.

In the following I will present the patterns in my form of representation (fig. 2). I do not draw them from top right to bottom left as Tony Phillips (grey), but from top left to bottom right (red) instead.

Figure 2. Representation of the Pattern
Figure 2. Representation of the Pattern

In fig. 3, all 42 sector patterns for sector labyrinths with seven circuits are depicted. I have not labelled them further, but only highlighted those patterns with red triangles that can be applied for sector labyrinths using exclusively triple barriers. Among these, 12 patterns have on one side, and the two already known patterns from the last post on both sides one half of a triple barrier. 

Figure 3. The 42 Patterns
Figure 3. The 42 Patterns

In fig. 4, these patterns were assigned to four groups, according to the sector they can be placed in and by which circuit they are connected with other sectors. The two patterns with triple barriers on both sides can be placed in every sector. Furthermore, those patterns that can be placed in the first sector and connect to the next sector on the outermost circuit are assigned to quadrant A. The patterns that can be only placed in the last sector and connect to the previous sector on the outermost circuit are assigned to quadrant B. To quadrant C are assigned the patterns that can be placed in the first sector only and connect to the following sector on the innermost circuit. And finally, quadrant D contains the patterns, that can be placed in the last sector only and are connected on the innermost circuit. 

Figure 4. Patterns by Sectors
Figure 4. Patterns by Sectors

This provides us with all material we need to generate any sector labyrinth with seven circuits using only triple barriers in the side-arms. 

Related Posts:

  1. Sector Labyrinths with Triple Barriers
  2. Tutorial on How to Draw a Classical 7 Circuit Labyrinth in Knidos Style
  3. The Complementary (Transposed) Classical 7 Circuit Labyrinth in Knidos Style
  4. Complementary and Interesting Labyrinths

1 thought on “Triple Barriers – Patterns for the First and the Last Sector

  1. Pingback: Triple Barriers – Combinations of Sector Patterns | blogmymaze

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