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## The Cretan at the Crossing Point

The Cretan labyrinth is related to other historical labyrinths in two manners. The easiest way to show this is to compare the seed patterns for the Ariadne’s Thread (see related posts below) of these labyrinths. A first line leads from the Knossos- to the Otfrid-labyrinth. For reasons of space, I arrange this in horizontal order and therefore refer to it as the horizontal line. The other (vertical) line leads from the Löwenstein 3- to the Tibble-labyrinth. The first labyrinth in either line is one of the only two existing alternating one-arm labyrinths with 3 circuits.

`  1 / 1                             Löwenstein 3                            1 / 3`
`  Knossos                              Cretan                              Otfrid`
`  3 / 1                              Hesselager                            3 / 3`
`  4 / 1                               Tibble                                4 / 3`

The labyrinths of the horizontal line contain exclusively the single double-spiral like meander (Erwin’s type 4 meander, see related posts below). However, they are made up of a varying number, i.e. 1, 2 or 3 of such meanders. Their seed patterns are composed of a varying number of similar segments. A segment consists of two nested arcs.

• The Knossos-type labyrinth contains one meander. The seed pattern of this labyrinth is made up of two segments. This pair of horizontally aligned segments complete to the meander in the labyrinth.

• The Cretan consists of two meanders that are connected by a circuit between them. The seed pattern is made up of two pairs of segments aligned vertically.

• Finally the Otfrid-type labyrinth is made up of three meanders that are connected by circuits between them. The seed pattern consists of three vertically ordered pairs of segments.

All labyrinths of the vertical line consist of two similar figures that are connected with a circuit between them. They all have a seed pattern made up of four similar quadrants. But the seed patterns differ with respect to the shapes of the quadrants.

• The Löwenstein 3-type labyrinth consists of 2 serpentines. This is reflected in the seed pattern by the four single arcs.

• The Cretan is composed of 2 single double-spiral like meanders (type 4 meander). The quadrants of the seed pattern of this labyrinth consist of two nested arcs.

• The Hesselager type labyrinth is made up of 2 two-fold (type 6) meanders. The quadrants in its seed pattern are made up of three nested arcs.

• Finally, the Tibble-type labyrinth consists of 2 three-fold (type 8) meanders, the quadrants of its seed pattern are made-up of four nested arcs.

The images above are arranged in the form of a table or matrix with 4 rows, 3 columns and 12 fields (frames). Six of these frames contain seed patterns directly related to the Cretan, the others are still void. The relationships of the horizontal and vertical line can also be formulated as follows:

• Progressing (horizontally) one column to the right will increase the number of meanders by one.
• Progressing (vertically) one row downwards will increase the depth of the meander by two. The depth of a meander corresponds exactly with it’s type number – a type 4 meander has depth 4, a type 6 meander depth 6 a.s.f.

With this information we are able to add the missing seed patterns and the corresponding labyrinths. By doing so we will encounter two other historical labyrinths and one figure that is no labyrinth. Of course it is also possible to add more rows or columns to the table and to fill the new frames with the corresponding seed patterns. All figures generated this way are self-dual. The figures of the first row are, in the terminology of Tony Phillips, uninteresting, all other figures are very interesting labyrinths (see related posts below).

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## The Pattern of the Snail Shell Labyrinth

If we dissect a labyrinth along its axis and uncurl it symmetrically on both sides, we can transform it into a rectangular form. The Ariadne’s Thread in the rectangular form is what I refer to as the pattern of a labyrinth. In this pattern, the entrance is at top left and the way into the center at bottom right.

Figure 1 shows this process in abbreviated form for the Snail Shell labyrinth. The labyrinth, represented by its Ariadne’s Thread is dissected along the axis (2 vertical black lines). Both halves of the axis are flipped upwards by half the arc of a circle around the center. By this, the Ariadne’s Thread is transformed from a circular closed form into a rectangular form.

The path of the Snail Shell labyrinth traverses the axis twice. This is indicated with the black circles. When transforming the Ariadnes Thread into the rectangular form, the segments of the pathway that lie on the axis are dissected too and come to lie on both sides of the rectangular form. These segments are drawn as dashed lines in the pattern and also indicated by circles.

Figure 2. Labyrinths Contained in the Snail Shell Labyrinth

In the Snail Shell labyrinth, a Knossos-type labyrinth (single double-spiral like meander) is included. To this are attached at the inside and outside one circuit with the pathway changing direction. Therefore, in the Snail Shell labyrinth, the labyrinth that corresponds with Arnol’d's figure 3 is also included. To this are attached at the in- and outside one circuit without the pathway changing direction. This is where the path traverses the axis. Thus, the Snail Shell labyrinth, in the terminology of Tony Phillips, is a non-alternating uninteresting labyrinth.

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## Seed Pattern – Shifting the Center / 4

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

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

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

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

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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## How to Make Seven New (up to now unknown) 7 Circuit Labyrinths on Snow

A way is made by walking it. This is applicable all the more for the labyrinth. And on snow this is especially nice – and simply possible. Thus I tried to put into practise the theoretical / mathematical considerations of the last blog entries.

For that to happen, I memorized the path sequence of the respective thread of Ariadne, and repeated it over and over again like a mantra while trampling the path into the  snow. And I counted in which circuit I just was and which was the next to come. For one have to pay attention where and what circuits will be made later,  and leave enough place for them. Having a look at the providently printed drawing of the prototype before tracking the path will help.
After arriving the center I traced back one more time the whole long way to the beginning what was sometimes quite strenuous. One should not change the lane, this is a point of honor. And if one makes the distances between the single circuits greater than a hop, this is likewise not possible.

I have tried to implement all 7 new types. I have made some more often. My “favorite type” is at present 5674 1238. The path sequence as an eight digits figure can be noticed quite well in two groups of four. Then the well-known classical labyrinth would be e.g. the type 3214 7658.

Who wants, can investigate more exactly the different types in the below quoted post. And if someone liked to experience the path on one’s own, he may copy and print the drawings.

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