# The Dual Labyrinth

Labyrinths have the following well known properties:

• Closed form
• one entrance and a center
• a pathway that leads from the entrance to the center and is the only way back
• the path is free of crossings or junctions and has no dead ends

However, labyrinths have another property too, that is less well known

• they can be turned their inside out

Inverting a labyrinth in such a way results in the dual labyrinth of it. I refer to the baseline labyrinth as the „original“ and to the labyrinth resulting after the inversion as the “dual” labyrinth. The meaning of „original“ here is only in relation with the transformation process to the dual labyrinth. Every labyrinth can be used as baseline and in this respect can be „original“.

Figure 1

We have already shown how the pattern can be obtained from the Ariadne’s Thread (fig. 1 left image). Bending the pattern back downwards (fig. 1, right image) reverses this process and brings us back to the original Ariadne’s Thread.

Figure 2

However, generating the pattern from the original labyrinth is also the first half in the inversion process (fig. 2). So let us continue with it.

Figure 3

For this purpose we bend over the pattern and re-curl it in to the other side from where it was uncurled, that is upwards. This results in another labyrinth, which is the dual and lies with the entrance on top (fig. 3).

Figure 4

In order to compare both labyrinths, we rotate the dual labyrinth and place it next to the original labyrinth (fig. 4). As can be seen, the two dual labyrints are different, but they have a resemblance. The dual labyrinth has the same pattern. This pattern, however, is followed in the opposite direction. The way out of the original labyrinth corresponds with the way into the dual and vice versa.

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## 4 thoughts on “The Dual Labyrinth”

1. Pingback: Self-dual Labyrinths | blogmymaze

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