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Posts Tagged ‘rotating the seed pattern’

The Snail Shell Labyrinth

The labyrinth next to the Cretan is the Snail Shell labyrinth. These two labyrinths have the same seed pattern. And they are the only ones with this seed pattern. Well, how then do we draw a Snail Shell labyrinth in the Man-in-the-Maze (MiM)-style? Very simple: we just use the Cretan labyrinth in the MiM-style from our last post (see related posts below). This labyrinth rotates clockwise.

The Cretan labyrinth in clockwise rotation

The Cretan labyrinth in clockwise rotation

And now let’s rotate the seed pattern, whilst keeping everything else in place.

The Snail Shell labyrinth in clockwise rotation

The Snail Shell labyrinth in clockwise rotation

Rotating it by one step in anticlockwise direction connects the center to the next intermediate space on the same quadrant of the seed pattern. This generates the Snail Shell labyrinth in clockwise rotation.

The Snail Shell labyrinth in anticlockwise rotation

The Snail Shell labyrinth in anticlockwise rotation

If we rotate the seed pattern one step further, the center is connected to the second next intermediate space. This again generates a Snail Shell labyrinth, however in anticlockwise rotation.

The Cretan labyrinth in anticlockwise rotation

The Cretan labyrinth in anticlockwise rotation

And if we rotate the seed pattern one more step further, we will receive the Cretan labyrinth again, but also in anticlockwise rotation.

The MiM-style thus provides an actual layout of a labyrinth which enables us to do exactly the same as we did here on a more theoretical base, i.e., to rotate the seed pattern (a more detailed description of the whole process is provided here). This theoretical analysis was performed using the seed pattern for the Ariadne’s Thread of the Cretan-type labyrinth. It predicted, that by rotating the seed pattern, only two different figures – the Cretan and the Snail Shell labyrinth – could be generated, each of them in clockwise and anticlockwise rotation. With the MiM-style labyrinths we have now an empirical proof of this result. Of course, it does not matter, whether the seed pattern for the walls or for the Ariadne’s Thread is used. Both lead to the same result although represented either by the walls or by the Ariadne’s Thread. In my theoretical analyses I prefer to use the representation with the Ariadne’s Thread as it is easier to read than the representation with the walls.

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Readers who have been visiting this blog for some time, will know the following figure.

Figure 1: Snail Shell Labyrinth

Figure 1: Snail Shell Labyrinth

Erwin calls it the Snail Shell Labyrinth. He had generated this labyrinth by completing the seed pattern for the Ariadne’s Thread of the Cretan-type labyrinth. But instead of the usual end he completed it around an other one of its ends. So he did, what I refer to as shifting the center. Somebody else also had the same idea, as can be seen in this source on page 14 (scroll down on the link page). Both shifted the center to a neighbouring end in the same quadrant, but did not continue with this process.

What might be unknown to the most is, that the Snail Shell Labyrinth is the only other figure that can be generated by rotating the seed pattern of the Cretan. With the statements from my previous posts about the shifting of the center (see related posts below), we are able to demonstrate this.

Figure 2: Seed Pattern

Figure 2: Seed Pattern

The seed pattern for the Ariadne’s Thread of the Cretan has 16 ends (fig. 2). So there are 16 possibilities to place the center. However, these reduce to four, as the seed pattern is made up of four similar quadrants. After four rotational steps, the seed pattern is self-covering.

Figure 3: Symmetry

Figure 3: Symmetry

In addition, each of these quadrants is symmetric in itself (fig. 3). This reduces the range of figures to two pairs. Each pair is made up of the same figure rotating either clockwise or anti-clockwise.

Figure 4: The ends and the pairs of figures

Figure 4: The ends and the pairs of figures

This is shown in fig. 4. Connecting the first end with the center generates the first figure in clockwise rotation. I therefore label this end as 1 →. If we connect the second end with the center, this generates figure 2, in clockwise rotation (2 →). The third end connected to the center generates also figure 2, although in anticlockwise rotation (← 2). And finally, the fourth end generates figure 1 rotating anticlockwise (← 1). Figure 1 is the Cretan, figure 2 the Snail Shell Labyrinth. We now have completed the first quadrant. The fifth end of the seed pattern is the first of its second quadrant and with this, the whole process begins anew.

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Once again I come back to the shifting of the center. This time I want to show the result of rotating the seed pattern of the Näpfchenstein labyrinth.

Illustration 1: The Näpfchenstein labyrinth and seed pattern

Illustration 1: The Näpfchenstein labyrinth and seed pattern

Ill. 1 shows this labyrinth and its seed pattern for the Ariadne’s Thread.

Rotating this seed pattern results in two figures. In fact this is only one figure, the labyrinth itself, either in clockwise or anti-clockwise rotation. This pair of figures is then repeated five times.

Illustration 2: the two figures

Illustration 2: the two figures

A closer look at the seed pattern shows the reasons. The seed pattern is not only composed of 2 similar halves, but made up of six identic sixth parts. Moreover, each of these sixths is symmetric in itself (Ill. 3). This explains why the two figures differ only in their rotational direction (clock- or anticlockwise).

Illustration 3: symmetry

Illustration 3: symmetry

A seed pattern that is composed of multiple similar elements, after a certain number of rotation steps will look exactly the same as in its original position. The Näpfchenstein seed pattern is very well suited to illustrate this.

Illustration 4: rotation steps

Illustration 4: rotation steps

In ill. 4 we first fix the original position of the seed pattern (grey) in fig. a. Then, we place a copy (black) on top of it (fig. b). In the original position, this copy exactly covers the original. Therefore only the black copy can be seen. Third, let’s rotate the (black) copy by one step and connect the next end of it with the center (fig. c). Now the original is uncovered and becomes visible. In fig. d we rotate the copy one step further, and it completely covers the original again.

After only two rotation steps the seed pattern is self-covering. This, of course, generates the same figure as in the original position.

The seed pattern of Rockcliffe Marsh is self-covering after six rotation steps. The seed pattern of my demonstration labyrinth needs a full circle rotation.

The number of rotation steps needed for a seed pattern to be self-covering corresponds with the number of figures that can be generated with it. With the seed pattern of my demonstration labyrinth, 12 figures, with the one of Rockcliffe Marsh 6 figures, and with the Näpfchenstein seed pattern, 2 figures were generated. It has to be kept in mind that the same figure in clockwise and anti-clockwise rotation is counted as two figures. In some seed patterns with an inherent symmetry in each element, the number of different figures may be further reduced.

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In an earlier post I have rotated the seed pattern for the Ariadne’s Thread of my demonstration labyrinth and generated 12 different figures. Six of these figures rotate clockwise, the others anticlockwise.

Each labyrinth with five circuits has a seed pattern with 12 ends. Thus, the framework presented in my earlier post can also be used to rotate the seed pattern of other labyrinths with five circuits. I have done this with the core-labyrinth of Rockcliffe Marsh (Arnol’d’s figure 8).

Illustration 1: Rockcliffe Marsh

Illustration 1: Rockcliffe Marsh

Ill. 1 shows the Rockcliffe Marsh labyrinth on the left with its core-labyrinth marked. On the right, the script version of the core-labyrinth is shown.

Illustration 2: Seed Pattern of Rockcliffe Marsh

Illustration 2: Seed Pattern of Rockcliffe Marsh

Ill. 2 compares the seed pattern of my demonstration labyrinth (left figure) with the one of Rockcliffe Marsh (right figure). The seed pattern of Rockcliffe Marsh is made up of 2 similar halves. This is a characteristic of self-dual labyrinths. In my demonstration labyrinth, the figure that results when connecting the end 7 of the seed-pattern with the center (figure 7) is the dual of figure 1. Self-dual means, that the two duals are identic. Therefore, in Rockcliffe Marsh, figure 7 is identic with figure 1. The same holds for figure 2 and 8 and so forth. It is therefore sufficient to only connect the first six ends of the Rockcliffe Marsh seed pattern with the center, as the ends 7 to 12 will simply reproduce the figures 1 – 6. In the seed pattern of Rockcliffe Marsh the ends 7 – 12 therefore were not numbered.

Illustration 3: The three pairs of figures

Illustration 3: The three pairs of figures

Ill. 3 shows the result. The numbers of the figures indicate which end of the seed pattern was connected with the center to generate the figure.

  • First: the number of different figures reduces to six. Three of them rotate clockwise, three anti-clockwise.
  • Second: A closer look reveals that there are only three different figures, each in clockwise and anti-clockwise rotation. These pairs of figures have been arranged on the same line in the illustration (figure 1 and 6, fig. 2 and 5, and fig. 3 and 4).

The reason for this is that the seed pattern of Rockcliffe Marsh not only is made up of 2 similar halves. In addition, each of these halves is symmetric around the dashed line (illustration 4).

Illustration 4: Symmetry

Illustration 4: Symmetry

Self-duality reduces the number of different figures from 12 to 6, the symmetry of the seeds in both halves reduces it further to only three different figures.

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In the first post of this series I have stated, that it is of crucial importance to indicate on the seed pattern where the center of the labyrinth is situated. Generally this is not stated and the seed pattern is oriented in such a way that the entrance is from below and the center lies somewhere on top.

The seed pattern for the Ariadne’s Thread of my demonstration labyrinth is also oriented this way. But what happens, if the center is shifted to an other end?

Figure 1: Shifting the Center

Figure 1: Shifting the Center

In our seed pattern there are 12 possibilities to place the center.

Figure 2: The 12 Ends of the Seed Pattern

Figure 2: The 12 Ends of the Seed Pattern

We could complete a new figure around each of these 12 ends as shown in the first post. This will always result in a figure with an entrance, a center and five circuits. This remains the same and thus can be drawn as one general figure. The only thing that changes is how the circuits are connected to each other. And this is affected by the seed pattern.

Figure 3: The General Figure with Entrance, 5 Circuits and Center

Figure 3: The General Figure with Entrance, 5 Circuits and Center

The general figure has a free circular space in place of the axis. The connections for the five circuits as well as the entrance and the centre are distributed in equal distances on the circle. The circle in fact is only an auxiliary line. Into this circular space we then place the seed pattern for the Ariadne’s Thread.

Figure 4: Rotating the Seed Pattern

Figure 4: Rotating the Seed Pattern

This enables us to rotate the seed pattern and subsequently connect each of its 12 ends with the center. The next figure illustrates this process for the connection of the first three ends of the seed pattern with the center.

Figure 5: Connecting the Ends 1 - 3 with the Center

Figure 5: Connecting the Ends 1 – 3 with the Center

We first connect the end 1 with the center. This is the correct center I usually mark with a bullet point. The resulting figure is our demonstration labyrinth. Another figure results, if we place the center at the end 2 of the seed pattern. This figure is made-up of a core-labyrinth of the Knossos-type. To this are added two consecutive circuits outside, without the pathway changing direction from clockwise to anticlockwise. The path thus traverses the axis twice. Connecting the end 3 with the center results in a figure composed of a core-labyrinth of the Tholos type with two additional ciruits outside and one additional circuit inside. Except for the core-labyrinth the path does not change direction and thus traverses the axis three times.

All 12 resulting figures can be viewed here.

With the same seed pattern 12 different figures could be generated. These were labelled as figure 1, 2, etc. till figure 12. The number indicates which end of the seed pattern was connected with the center to generate the figure.

Figures 1 (and its dual, figure 7) and 5 (11) can be considered as true labyrinths, figures 2 (8) and 3 (9) are composed of a core-labyrinth with added circuits on which the path does not change direction. However, these are not spirals in the strict sense as the path does not continually wind itself in but changes stepwise from one circuit to the next whilst traversing the axis. Figures 4 (10) and 6 (12) are combinations of isolated closed forms on certain circuits and a spiral-like or single pathway that only covers the remaining circuits.

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