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## The Level Sequences of Dual Labyrinths

The original and the dual labyrinth have the same pattern, although rotated by 180° (see related posts below). The pattern indicates the course of the pathway. On its course, the pathway covers the circuits in a certain sequence. This level sequence indicates which circuit is covered by the pathway as the first, which circuit follows as the second, third and so forth until the last circuit.

But how do we have to label the circuits? Normally they are numbered in ascending order from the outermost to the innermost circuit. This enumeration corresponds with the direction into the labyrinth. However, each path can be followed in two directions – i.e from the outside into the labyrinth and from the inside out.

Figure 1. Numbering the Circuits

If we follow the direction into the labyrinth (fig. 1, left image), we first meet the outermost and last the innermost circuit. On the way out of the labyrinth (fig. 1, right image), however, we first meet the innermost and last the outermost circuit. The direction in which the path is followed, thus, determines how we have to number the circuits.

Figure 2. Coloring the Circuits

Next, let us determine the level sequence as follows (fig. 2): We color the circuits in the sequence they are covered by the pathway with the colors yellow, green, blue, magenta, and red. Thus, the circuit covered by the path as the first will be colored yellow, the second green, and so forth til the last circuit in red color. In the order of these colors we then can read the numbers of the circuits.

Figure 3. Level Sequences of Original Labyrinth

Fig. 3 shows on the left image the level sequence into and on the right image the level sequence out of the original labyrinth. On its course into the labyrinth, the pathway first covers circuit 3, then circuit 4, circuit 5, circuit 2, and finally circuit 1. The level sequence into the labyrinth therefore is: 3-4-5-2-1. However, the level sequence out of the original labyrinth is: 5-4-1-2-3.

Figure 4. Level Sequences of Dual Labyrinth

Fig. 4 shows the same for the dual labyrinth. The level sequence into the dual labyrinth is: 5-4-1-2-3. The level sequence out of the dual labyrinth is 3-4-5-2-1. As can be seen: the level sequence into the original labyrinth is the same as out of the dual labyrinth: 3-4-5-2-1. Likewise, the level sequence into the dual labyrinth is the same as out of the original labyrinth: 5-4-1-2-3.

Figure 5. Level Sequences in the Pattern

A closer look at the pattern can explain this (fig. 5). The patten can be followed either from top left to bottom right or in the opposite direction. The direction from top left to bottom right corresponds with the pathway into the original and out of the dual labyrinth. The directions from bottom right to top left on the other hand corresponds with the way into the dual and out of the original labyrinth. As we walk into the original labyrinth we walk out of the dual labyrinth and vice versa.

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