Archive for the ‘Design’ Category

Sector Labyrinths

At the end I will also transform a sector labyrinth into the MiM-style. What is special in sector labyrinths is, that the pathway always completes a sector first, before it changes to the next. As a consequence of this, the pathway only traverses each side-arm once. Thus it seems, that sector labyrinths may be easier transformed into the MiM-style than other labyrinths with multiple arms. I will use as an example a smaller labyrinth with four arms and five circuits. There exist several labyrinth examples of this type. I have named it after the earliest known historical example, the polychrome mosaic labyrinth that is part of a larger mosaic from Avenches, canton Vaud in Switzerland.

Figure 1. Sector Labyrinth (Mosaic) of Avenches

Figure 1 shows the original of this labyrinth (source: Kern 2000: fig 120, p 88). It is one of the rarer labyrinths that rotate anti-clockwise. On each side of the side-arms it has two nested turns of the pathway and 3 nested turns on each side of the main axis. The pattern corresponds with four double-spiral-like meanders arranged one after another – Erwin’s type 6 meanders (see related posts 2). When traversing from one to the next sector the pathway comes on the outermost circuit to a side-arm, traverses this on full length from outside to inside and continues on the innermost circuit in the next sector.

In order to bring this labyrinth into the MiM-style, first the origninal was mentally rotated so that the entrance is at bottom and horizontally mirrored. By this it presents itself in the basic form, I always use for reasons of comparability. Fig. 2 shows the MiM-auxiliary figure.

Figure 2. Auxiliary Figure

This has 42 spokes and 11 rings what makes it significantly smaller than the ones for the Chartres, Reims, or Auxerre type labyrinths. The number of spokes is determined by the 12 ends of the seed pattern of the main axis and the 10 ends of each seed pattern of a side-arm.

In fig. 3 the auxiliary figure together with the complete seed pattern including the pieces of the path that traverse the axes is shown and the number of rings needed is explained. For this the same color code as in the previous post (related posts 1) was used.

Figure 3. Auxiliary Figure, Seed Pattern and Number of Rings

As here the angles between the spokes are sufficiently wide, it is possible to use all rings of the auxiliary figure for the design of the labyrinth. We thus need no (green) ring to enlarge the center. Only one (red) ring is needed for the pieces of the path that traverse the axes – more precisely: for the inner wall delimiting them –, four (blue) rings are needed for the three nested turns of the seed pattern of the main axis, one ring (grey) for the center, and five rings (white) for the circuits, adding up to a total of 11 rings.

Fig. 4 finally shows the labyrinth of the Avenches type in the MiM-style.

Figure 4. Labyrinth of the Avenches Type in the MiM-Style

The figure is significantly smaller and easier understandable than the labyrinths with multiple arms previously shown in the MiM-style. Overall it seems well balanced, but also contains a stronger moment of a clockwise rotation that is generated by the three asymmetric pieces of the pathway and of the inner walls delimiting these on the innermost auxiliary circle.

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For Year 2019 we wish you all the best and interesting encounters with the labyrinth.

Labyrinth with 7 Axes, 15 Circuits and 6 Nested Turns at Both Sides of Each Axis, and yet no Sector Labyrinth – Rather the Opposite, Self-dual



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Wishing all visitors of this Blog a Merry Christmas and a Happy New Year!

An 11 circuit Christmas tree labyrinth

An 11 circuit Christmas tree labyrinth

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Type Reims and Type Auxerre

Using the example of the Chartres type labyrinth it could be shown that also labyrinths with multiple arms can be designed in the MiM-style (see related posts 1, below). Generally, all labyrinths with four arms and 11 circuits require in the MiM-style an auxiliary figure with 90 spokes (see related posts 2). This is determined by the number of arms and of circuits.

The number of rings for the Chartres type labyrinth amounts to 22. This number can vary for different types of labyrinths with four arms and 11 circuits. This number depends on the number of circuits, the depth of nestings of the turning points of the path (see related posts 5) and the number of pieces of the pathway traversing the axes. This is explained more in detail in fig. 1.

Figure 1. Auxiliary Figure of Chartres Type – Rings

For the outer walls delimiting the pathway of the 11 circuits (white), 11 rings are needed. One more ring is used for the center (grey). (This space could also have been saved. My first draft of this labyrinth for the New Year did not contain a separate ring for the center yet – see related posts 3 -. However, in such a case no space would be provided for the center. Therefore in the final design I have added a separate ring for the center). There are turns of the pathway with a maximum depth of two nested turns at the main axis. For this, three rings are required (blue). The side-arms all have only single turns of the pathway. So for them, tow rings would be sufficient. For the pieces of the pathway traversing the side-arms, an additional five rings are needed (red). In order to ensure that sufficient space is left for the pathway at the narrowest point between two walls delimiting the path, two rings in the center of the figure are added that are not even used for the lines of the labyrinth. These just serve to enlarge the center (green) such, that the figure can be reasonably drawn at all. Thus, for all this together, 22 rings are needed.

In order to transform a labyrinth with four arms and 11 circuits into the MiM-style, a considerable effort is required. To draw the figure with sufficient exactness, compass and ruler or a drawing application will be needed. Now once we have designed the figure for the Chartres type labyrinth, we can easily bring certain other types of labyrinths into the MiM-style. Such types must have four arms, 11 circuits, 5 (and less) pieces of the pathway traversing the axes, as well as two (and less) nested turns of the pathway. This is true, among others, for the two other very interesting historical types of labyrinths, the Reims type and the Auxerre type (related posts 4). In order to transform these into the MiM-style, we can start from Chartres type in the MiM-style. All the lines delimiting the pathway on all spokes and auxiliary circles outside the seed pattern can be left unchanged. Only the sed patterns have to be amended at certain places and the connections to the walls delimiting the pathway have to be adjusted correspondingly.

Figure 2. Labyrinth of the Reims Type in the MiM-Style

Fig. 2 shows the Reims type in the MiM-style. The seed pattern of the main axis has two nested turns only in two places next to the entrance and next to the center of the labyrinth. Otherwise there are only single turns of the pathway at the main axis. There are five pieces of the pathway traversing the first and the third side-arm, and three traversing the second side-arm, just the same as in the Chartres type labyrinth. However, the pieces traversing the first and third side-arme are distributed differently over the side-arms than in the Chartres type labyrinth.

Figure 3. Labyrinth of the Auxerre Type in the MiM-Style

In fig. 3 the Auxerre type in the MiM-style is depicted. The seed pattern of the main axis of this type is somewhat different from that of the Chartres type. The seed patterns of the side arms and thus the pieces of the path traversing the side-arms are the same as in the Chartres type.

Other types of labyrinths can be transformed into the MiM-style in the same way too, e.g. the complementary of Reims. These will all be based on an auxiliary figure with 90 spokes and 22 rings.

However, in other types of labyrinths with 4 arms and 11 circuits, this does not work that easy. So, for instance, the complementaries of Auxerre or Chartres on the main axis have also three nested turns of the pathway. Therefore, for the seed patterns of these types of labyrinths, four (blue) rings are needed. The auxiliary figure for these labyrinths has 23 rings. Thus, the center and all eleven circuits would have to be shifted one ring further outwards. In order to draw these labyrinths, again the seed patterns would have to be amended, and the connections appropriately adapted. In addition each piece of a wall delimiting the pathway outside the center would have to be shifted and modified.

I refrain form drawing these types of labyrinths in the MiM-style. Already from the presently available figures it can be seen, that the style clearly dominates the look of the labyrinth and that a quite careful closer view is needed, if we want to identify the differences between these types in this style.

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In the previous articles on this topic, I have already explained the method of the stamp-folding calculation brought into play by Tony Phillips.

Now it should go on here. Namely, it is possible to generate further variants of labyrinths by simply rotating the polygon used.

I take again the net with the polygon from the last post on this topic (part 2).

The net with the polygon

This diagram can be used to create four different labyrinths. Two directly (line 2 and 3), the other two by a simple calculation.

Other constellations can be gained by rotating the network 12 times by 30 degrees. Or in other words, it’s just like changing the clock for the summer or winter time.
Since only interesting labyrinths are of interest here, I omit all positions where the lines would point to the first and / or last circuit. So from the 12 you should not reach the 1 or the 11. Only the “times” are interesting, which point farther away, that is, run more sharply.
That would be in the above net the 1, 5 and 6. So I turn only to these times. In other words, I bring the 1, 5, and 6 into alignment with the 12. I turn the net by 30, 150, and 180 degrees. To rotate is the net with the polygon, the numbers stay in place.

Here’s the first turn:

rotated by 30 degrees

rotated by 30 degrees

I get four completely different path sequences than in the original above.

The second rotation:

rotated by 150 degrees

rotated by 150 degrees

I get four new variants again.

The last rotation:

rotated by 180 degrees

rotated by 180 degrees

Here I just get a different order of the sequences than in the original polygon. So there are no new variants, just another arrangement. This is because the rotation of 180 degrees corresponds to a symmetrical reflection.

It is not always possible to find new variants. With the help of this net I have generated a total of 12 different path sequences for 12 new labyrinths.

The path sequences can be directly converted into a labyrinth drawing.
Here only one (again in concentric style) is to be shown (the 2nd path sequence from the first polygon above):

A new 11 circuit labyrinth

A new 11 circuit labyrinth

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Fine Tuning

A careful view of the Chartres type labyrinth in the Man-in-the-Maze style (see related posts 1, below) revealed that the design can still be improved. In addition to the not accessible middle, this figure has another 16 smaller segments that are not covered by the pathway. These are highlighted as crossed-through areas in fig. 1. Among others one requirement is that the pathway in a labyrinth should cover as much of the space of the figure as possible and should not let remain any „dead spots“. Such uncovered segments are therefore not liked to be seen in labyrinths.

Figure 1. Segments not Covered by the Pathway

These segments have evolved as a result of the construction of the seed pattern (see related posts 2). However, upon a closer view it turned out that they can be resolved. For this, as shown in fig. 2, the outer delimiting line has to be removed and the wall delimiting the pathway in between must be prolonged further to the inside.

Figure 2. Corrections to the Seed Pattern

The necessary corrections are shown for the segment on low left in orange color. Such corrections must be carried out for all 16 segments in a similar way.

The result can be seen in fig. 3. After these corrections, the labyrinth looks even somewhat more dynamic. Finally, only the one non accessible middle is left over. This is just the same as in the alternating one-arm labyrinths in the MiM-style.

Figure 3. The Final Labyrinth

However, now it cannot anymore be seen easily where the pathway traverses the side-arms. Therefore, the gain in dynamics and unity is associated with a loss in transparency.

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Again we deal with the simple, alternating, transit mazes, defined by the New York Professor of Mathematics Tony Phillips. In his calculations he ascertains a number of 1014 theoretically possible variants of interesting 11 circuit labyrinths (12-level mazes).

He also describes a simplified method for calculating these variants, which John E. Koehler developed in 1968 to solve a related problem of stamp-folding.

The following pictures should explain this method. To this I first use the already known path sequence for the 11 circuit labyrinth which can be generated from the basic pattern, namely: 5-2-3-4-1-6-11-8-9-10-7-12.
The path sequence must begin with an odd number and then the row must be composed of even and odd numbers alternately. The center is named with “12”, as it is the outside.

I draw a circle and divide it into 12 parts, as for a dial. Now I have to connect all points with lines, but same-colored lines must not cross.



I start with blue in 12 and go to 5, 2, 3, 4 (Fig. 1). Then from 4 to 1, thereby I change the colour (Fig. 2). I continue with 6, 11, 8, 9, 10 (Fig. 3). I again change the colour and complete the lines from 10 to  7 and 12 (Fig. 4).

But you can do it differently. For example, draw all the lines first in one color and then the intersecting ones in the other. Here again, the same-colored  lines should not cross each other. But more than once, as long as they are different (see 4 – 7).

The web

The web

But since we are looking for new labyrinths, we now go the opposite way: We draw a network of 12 lines, which connects all 12 points according to the above specifications and derive from this the path sequence.

Here is an example:

The web with the polygon

The web with the polygon

I write the first path sequence in line 2 (here in blue), starting at 12 and reading the lower digit, here 5. This is the beginning of the path. Then I follow the polygon until I land at 12 again and get: 5-2-3-4-1-6-11-10-9-8-7-12. That’s the original.
Now I go backwards and write the path sequence in line 3. So from 12 to 7, etc. That gives: 7-8-9-10-11-6-1-4-3-2-5-12. This is the complementary to the original.

I receive the lines 1 and 4 by arithmetic. I add the corresponding numbers of each row to “12”. In line 4, I get the dual to the original. In line 1, I get the complementary to the dual.

I verify this by comparing the numerical columns thus obtained with the others in “reverse”. This applies to the lines 1 and 4, as well as 2 and 3.
This is reminiscent of what has been described before when dealing with the dual and complementary labyrinths (see Related Posts below).

But there are alternatives. I turn the dial around, write the numbers for the 12 dots to the left, counterclockwise.
This is how it looks like:

The web with the two dials

The web with the two dials

The left side shows the dial as before. I start at 5, count to 12 and get the original. Then I start at 7 and count again to 12 and get the complementary to the original.
Now the right dial. I also start at 5 and count to 12 and so get the dual to the original. Then again from 7 to 12 and I get the complementary to the dual.

What should the blue written path sequences mean? They point out that the entry into the labyrinth can be placed on the same axis as the entry into the center. Here on circuit 5 and 7. Walter Pullen calls this that a labyrinth layout is mergeable. This allows you to construct a small recessed spot in the labyrinth, which some name the heart space. Especially in the concentric style, this can be implemented well.

From these two newly obtained path sequences, I now construct two new 11 circuit labyrinths in concentric style:

They have a different pattern of movement than the upt to now known labyrinths. In addition, we see 6 turning points for the circuits.

This is the dual to the previous labyrinth. Again, there is another “feeling”.

Who makes the beginning and builds such a labyrinth?

The other two paths sequences also result in new labyrinths, which I don’t show here. These belong to the remaining 1000 variants that are theoretically possible for 11 circuit labyrinths.

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