Feeds:
Posts

## The Pattern and the Dual Labyrinths

Let us consider the duality once more based on the pattern. The two labyrinths that are dual to each other have the same pattern.

Figure 1. The Pattern and the two Dual Labyrinths

The pattern is no labyrinth. It has no closed form and is indifferent with respect to the outside and inside. It can be transformed into a labyrinth in two directions. In the last post (see related posts below: the Dual Labyrinth) we have unrolled the Ariadne’s Thread of the original labyrinth from below and obtained the pattern using method 2 (see below: From the Ariadne’s Thread to the Pattern – Method 2). Then, we have re-curled in the pattern to the other side, i.e. upwards, and thus obtained the Ariadne’s Thread of the dual labyrinth. This, however, lay with the entrance on top. In order to compare the original and the dual labyrinths, we have rotated the dual so that its entrance was from below.

In rotating a labyrinth we rotate the pattern too. By the way, this can be already seen from my earlier post (see below: What’s the Use of the Pattern?). In this post, fig. 5 showed the Ariadne’s Thread of the Chartres type labyrinth in a representation by Niels Mejlhede Jensen with the entrance on the right side. This is one quarter of a circle anticlockwise against our usual orientation with the entrance from below. Consequently in this figure the pattern was also rotated by one quarter of a circle and standing on its left outer side.

In our case here we have the dual labyrinth rotated by half a circle lying on its head. Here I want to show how by rotating the labyrinth, the pattern is rotated too.

Figure 2. Isolating the Dual Labyrinth

In fig. 2 we first isolate the dual labyrinth and also carry-over the pattern lying on it.

Figure 3. The Pattern of the Original and Dual Labyrinths

Then we rotate the isolated labyrinth with the pattern on it (fig. 3) and place it next to the original labyrinth. Both labyrinths now lie with their entrances from below and the pattern placed on top of the figure. The pattern of the dual labyrinth is the same as the pattern of the original labyrinth, however, rotated by half a circle.

Figure 4. From the Original to the Dual Labyrinth

An important consequence arises from this. As shown in fig. 4 it is also possible to proceed as follows in order to transform the original into the dual labyrinth: In a first step we generate the pattern from the original labyrinth. Then we rotate the pattern by half a circle. Finally we can curl it in again downwards and by this generate the dual labyrinth.

Related posts:

### 2 Responses

1. […] The Pattern and the Dual Labyrinths […]

Like

2. […] The Pattern and the Dual Labyrinths […]

Like

This site uses Akismet to reduce spam. Learn how your comment data is processed.