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Posts Tagged ‘5 circuit labyrinth’

In dealing with the double-barrier technique in recent posts, I found this installation of Mark Wallinger’s Labyrinths on the London Underground:

The labyrinth 233/270 at the station Hyde Park Corner, Photo: credit © Jack Gordon

The labyrinth 233/270 at the station Hyde Park Corner, Photo: credit © Jack Gordon

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

The special feature of this is that two double barriers are located next to each other in the upper part of the central axis. In the routing chosen by him you move at the transition from the 2nd to the 3rd quadrant first away from the center.

I’ve changed that so much that you would “experience” a movement to the center in a walkable labyrinth.

This is what it looks like:

A new labyrinth in concentric style

A new labyrinth in concentric style

I have also moved the side double barriers and this makes the routing in all quadrants also different. So a new type of labyrinth is born.

Here in Knidos style:

A new centered sector labyrinth in Knidos style

A new centered sector labyrinth in Knidos style

Why not as a two-parted labyrinth?

A new two-parted 5 circuit labyrinth

A new two-parted 5 circuit labyrinth

The left part has the path sequence: 3-4-5-2-1 and the right part: 5-4-1-2-3, so there are two 5 circuit labyrinths in it.

And here again in Knidos style:

A new two-parted and centered 5 circuit labyrinth in Knidos style

A new two-parted and centered 5 circuit labyrinth in Knidos style

The remarkable thing about this type is that both the entry into the labyrinth in the 3rd lane takes place, as well as the entry into the center.

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There are, as now everybody knows, 8 different possibilities for a 5 circuit, one-arm labyrinth (where the axis is not crossed).

Tony Phillips has furnished evidence for this on his website. He is a Professor of Mathematics at the State University of New York and he approaches mathematically to the question.

Some of these versions are known and were already built as labyrinths to walk. But some just not. Andreas Frei has already shown these 8 types. Tony Phillips describes the curves of the Russian mathematician Arnol’d which lead exactly to these variations.

Now I would like to introduce once again all 8 variations. In the meantime, I have developed a method to construct a labyrinth according to the path sequence. This path sequence is also a good name and a differentiation sign for a certain labyrinth type.

The different labyrinths are not developed from the well-known pattern. Even not from the pattern for Ariadne’s thread.

The figures of the eight labyrinths, first in round shape and with a bigger middle, then in square form. The contained pattern is emphasised in colour. Indeed, it has been determined after the construction of the labyrinth. The ways and the boundary lines are equally wide. The path sequence should serve as a marking of the type.

0-5-2-3-4-1-6

0-5-2-3-4-1-6

0-5-2-3-4-1-6

0-5-2-3-4-1-6

0-3-4-5-2-1-6

0-3-4-5-2-1-6

0-3-4-5-2-1-6

0-3-4-5-2-1-6

0-1-2-5-4-3-6

0-1-2-5-4-3-6

0-1-2-5-4-3-6

0-1-2-5-4-3-6

0-3-2-1-4-5-6

0-3-2-1-4-5-6

0-3-2-1-4-5-6

0-3-2-1-4-5-6

0-5-4-3-2-1-6

0-5-4-3-2-1-6

0-5-4-3-2-1-6

0-5-4-3-2-1-6

0-5-4-1-2-3-6

0-5-4-1-2-3-6

0-5-4-1-2-3-6

0-5-4-1-2-3-6

0-1-2-3-4-5-6

0-1-2-3-4-5-6

0-1-2-3-4-5-6

0-1-2-3-4-5-6

0-1-4-3-2-5-6

0-1-4-3-2-5-6

0-1-4-3-2-5-6

0-1-4-3-2-5-6

Here some more explanations to the path sequence: The paths inside a labyrinth are numbered from the outside inwards, to the center. The order in which the paths are walked from the entrance up to the middle displays the path sequence. The digit “0” stands for outside (or beginning) and the last number marks the center itself.
The first number after “0” must always be an odd number, so 1, 3, or 5 (and so on). Otherwise it does not work.
The row of numbers must be composed of alternating even and odd digits. Otherwise it will not be a labyrinth. The mathematician expresses this differently, but for us this information should be enough.

Who would like to build a new 5- circuit labyrinth, can get here suggestions.
I personally like the second variation with the path sequence 0-3-4-5-2-1-6 best of all. Who builds it?

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In a series of recent posts Erwin has elaborated on meanders in labyrinths. He has found a meander best suited for labyrinths. This particular meander exists in various forms. Erwin refers to these as “type” followed by an even number, e.g. type 4, type 6 and so forth.

Indeed, these types of meanders can be often found in existing labyrinths. However other figures occur in labyrinths too that can be denoted as meanders. Generally a broad range of figures is referred to as meanders. And it seems not so clear what a meander is at all.

So, what is a meander?

On his website, Tony Phillips cites Arnol’d, a Russian mathematician. Independent of the labyrinth, Arnol’d wanted to investigate how many meanders there are. He defines a meander as follows:

  • Connected oriented curve,
  • that does not intersect itself
  • and intersects a fixed, oriented line in several points.

He illustrates this with the example of a curve that intersects the fixed line five times. He found that there exist eight different such curves. I have copied the illustration from Tony’s website, enumerated the 8 curves and display it below.

8 curves

8 curves

It can easily be recognized that these curves are very closely related with the labyrinth. One only has to rotate them by a quarter of a circle in clockwise direction and straighten them out somewhat. This results in the patterns of 8 one-arm labyrinths with 5 circuits each. Therefore, let us have a closer look at these figures.

figure 1

figure 1

This figure represents a serpentine that leads from the entrance to the center of the labyrinth. It is this the pattern of the historical Näpfchenstein labyrinth. This labyrinth is self-dual.

figure 2

figure 2

This curve depicts the pattern of the labyrinth type Löwenstein 5b. This is the dual of the labyrinth shown in figure 4. Dual labyrinths have the same pattern, although the pattern is rotated by half a circle, and the entrance and center are exchanged. The “Rockery Labyrinth“, designed by Erwin is also of this type.

figure 3

figure 3

This curve includes a pattern of the Knossos type (3 circuits) to which are attached an additional circuit on both, the outer and the inner side. I am not aware of any existing labyrinth of this type. This labyrinth is self-dual.

figure 4

figure 4

This is the pattern of the labyrinth type Löwenstein 5a. It is dual to the labyrinth shown in figure 2. The “Pilgrim Hospices” labyrinth, designed by The Labyrinth Builders is of this type.

figure 5

figure 5

This figure contains the pattern of the labyrinth I use for my investigations and presentations, so to speak my demonstration labyrinth. It has the following properties that are important for this purpose: the path does not enter on the first circuit, it does not reach the center from the last circuit and the labyrinth is not self-dual. The dual to this labyrinth is shown in figure 7.

figure 6

figure 6

This pattern corresponds with a serpentine from the inside out. The path enters along the axis and first encounters the innermost circuit. From there it winds itself out circuit by circuit until it reaches the first (outermost) circuit. Then it is directed axially to the center. Erwin has discovered this type of labyrinth (Chartres 5 classical) by omitting the side-arms of the Compiègne-type labyrinth. This labyrinth is self-dual.

figure 7

figure 7

This is the dual of my demonstration labyrinth shown in figure 5.

figure 8

figure 8

This curve represents Erwin’s meander best suited for labyrinths. It is a type 6 meander. This is the pattern of the core-labyrinth of the historical Rockcliffe Marsh labyrinth. Rockcliffe Marsh is a very special labyrinth. First it has an unusual layout. The figure is opened along the axis and unrolled to a segment of a circle. Second, it is made up of a core-labyrinth (the inner 5 circuits) that is enclosed by a spiral outside.

Conclusion

Arnol’d’s definition of a meander is closely related with the labyrinth. His curves correspond with the patterns of all one-arm labyrinths in which the pathway does not cross the axis. The number of intersections between the curve and the fixed line corresponds with the number of circuits in the labyrinth. This was demonstrated in detail for labyrinths with 5 circuits.

  • Thus there exist 8 different patterns for a labyrinth with one axis and five circuits with the pathway not crossing the axis.
  • With an increasing number of circuits, the number of different pattern increases dramatically. E.G. there are 14 possible patterns for a labyrinth with 6 and 42 for a labyrinth with 7 circuits.
  • According to Arnol’d’s definition, all 8 figures are meanders. If we follow Erwin’s definition, only figure 8 is a meander suited for labyrinths.
  • If we adopt the definition of Erwin, we will capture the most common and essential labyrinths. However, we will also miss a broad range of existing and potential patterns of labyrinths.
  • If we adopt the definition of Arnol’d, every pattern of a one-arm labyrinth, in which the way does not cross the axis is referred to as a meander. This definition seems too broad and can be further differentiated.

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The thing with the faultily drawn silver coin in the previous article has bothered me and I have more exactly looked at them once again:

Square labyrinth with five circuits and drawing mistakes 350-300 BC

Square labyrinth with five circuits and drawing mistakes 350-300 BC / Source: Hermann Kern, Labyrinthe, 1982, pict. 54 (rotated), German edition

It is remarkable that the four usual turning points for the 7 circuit classical labyrinth are to be seen. Also the central cross of the wellknown seed pattern is to be recognised. However, which lines are the walls, and which the way?
A little bit was also cheated with the line widths, because the both lower turning points do not lie on the same height.

However, the construction is not completely pointless, it shows even some  astonishing things. Here is a graphic illustration:

Square labyrinth structure

Square labyrinth structure

The walls are the black lines. The paths are the empty area in between.
The four turning points, the starting point and the final point are connected with each other, without crossing as it should be in a classical labyrinth. Hence, here we have three coherent lines instead of two. However, the paths are crossing, this can not be. Also there is no real center, but a bifurcation. However, it is possible, to walk through the labyrinth structure by using all paths.

Try it yourself:
Start in A, go to the right and from the crossroad point X on turn left, also while going out. Then one lands again in the starting point A and has crossed all ways.
Or one goes in A upwards and from X on always to the right. Then one reaches  again A after pacing all ways.

It is a “passageway labyrinth” with bifurcation. This is not allowed for a labyrinth in the strict sense, but in the Baltic wheel or in the Wunderkreis of Kaufbeuren we will find this centuries later. Or of course in the true mazes of the later centuries.

Either a joker or a trainee who has not paid attention so properly in the drawing lessons has stamped this coin?
Or here something is already to be found what appears in the labyrinth history  centuries later again.

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I have found a meander suitable for it in the antique collection of the Martin von Wagner museum of the university of Würzburg on this Aeolian plate  from the time about 575 BC.

Aeolian plate from 575 BC

Aeolian plate from 575 BC

The meander in schematic form looks like this:

Meander with the line sequence 0-5-2-3-4-1-6

Meander with the line sequence 0-5-2-3-4-1-6

We read from the left to the right: 0 is outside, 6 stands for the middle, 1 to 5 for the circuits. We read the path sequence (line sequence, level sequence): 0-5-2-3-4-1-6. This is the order in which the paths are followed.
Notes to the path sequence:
Odd and even integers must alternate.
The first integer after 0 is always an odd number.

We use the path sequence directly to construct a circular labyrinth with a bigger middle:

Ariadne's thread with path sequence 0-5-2-3-4-1-6

Ariadne’s thread with path sequence 0-5-2-3-4-1-6

The labyrinth has 5 circuits. The first step leads me directly quite near to the center, into the 5th circle. Then I turn outwardly to the 2nd circle, I approach the center by turning to the 3rd and 4th circle, from where I turn quite outwardly into the 1st circle, and from there, I finally enter the center.

All formative principles which Hermann Kern (Labyrinthe, 1982, p. 14, German edition) demands for a labyrinth are fulfilled.

Is there a historical labyrinth with this alignment?
So much I could investigate, this does not seem to be the case. (Objections are welcome).

Among the silver coins of Knossos  from the time about 500 BC till 100 BC, mentioned in one of the previous posts, there is a coin with the depiction of a 5 circuit labyrinth which is faulty, unfortunately.

Square labyrinth with five circuits and drawing mistakes 350-300 BC

Square labyrinth with five circuits and drawing mistakes 350-300 BC / Source: Hermann Kern, Labyrinthe, 1982, pict. 54 (rotated), German edition

In the following drawing you see a square classical 5 circuit labyrinth with the path sequence  0-5-2-3-4-1-6 . The walls are black, the path is the white empty place between them.

Square labyrinth with five circuits

Square labyrinth with five circuits

Who wants, can compare the layouts and find out what the old Greeks have made wrong on their coin.
To be fair, I must say that there are still 7 more different versions for a 5 circuit labyrinth.

However, the middle can also become a little bigger. In the following drawing the seed pattern contained in the black walls, is marked in colour.

Square 5 circuit classical labyrinth

Square 5 circuit classical labyrinth

The seed pattern can be simplified very much to 2 dots and 5 lines.

The 5 circuit classical labyrinth with small center

The 5 circuit classical labyrinth with small center

To draw the labyrinth I join the free end of the innermost line in an arc with the free end of the line to the right. Then I go to the left line and join it with the free end of the right line parallel to the first arc. And so on with each line and dot.

Who rather wants the “accustomed” sight, here it is:

The 5 circuit classical labyrinth

The 5 circuit classical labyrinth

The seed pattern looks familiar. If one copies it, turns it around 180 degrees,  and attach it, one receives the seed pattern for the 11 circuit classical labyrinth. Or in other words: Two meander of this type put together result in a 11 circuit classical labyrinth.

An other variation of this meander labyrinth arises if I want a bigger middle, but not the perfectly circular form:

The 5 circuit meander labyrinth

The 5 circuit meander labyrinth

The two turning points on the right and left side form a triangle together with the very center of the center. So the relationship with the Baltic wheel and the Indian labyrinth appears. However, in those labyrinths one doesn’t reach the center directly from the outermost ring outside as some circuits are added around the center. Besides, the Baltic wheel has a second short entrance/exit, disqualifying it as a labyrinth in the strict sense.

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