Variants of the Cakra Vyuh Labyrinth

In the last post I have presented four variants of the seed pattern of the Cakra Vyuh type labyrinth. Perhaps somebody might be interested, how the matching complete labyrinths look like. Here I will show them.

I thus add three other examples to the only example (Original) of this type of labyrinth that has been well known until now. Or, more exactly, only two of them are really new: the examples in the Classical and in the Concentric styles. I had already published the example in the Man-in-the-Maze style previously on this blog. Furthermore it has to be considered, that the original labyrinth rotates anti-clockwise. I have horizontally mirrored the three other examples. It is still the same labyrinth then, although rotating clockwise. I use to show all my labyrinth examples in clockwise rotation so they are more easily comparable.

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Type or Style / 9

A Brief Stylology

In my last two posts I have described six styles.  Of course, these can also be used to order labyrinths. I will show this here in an illustrative manner. I do not take the effort to group all or as many as possible labyrinths into the different styles. I will just use a few examples of each style to illustrate how such a grouping would work.

klassisch

 

Labyrinth Examples in the Classical Style

 

 

konzentrisch

 

Labyrinth Examples in the Concentric Style

 

 

MiM

 

Labyrinth Examples in the Man-in-the-Maze Style

 

chartres

 

Labyrinth Examples in the Chartres Style

 

reims

 

Labyrinth Examples in the Reims or Bastion Style

 

knidos_stil

 

Labyrinth Examples in the Knidos Style

 

Labyrinth Examples in other styles

Of course, with the six styles described above it is not possible to cover the entire spectrum of all labyrinths. Therefore I have added another group to capture other styles and attributed some examples of labyrinths to it. Among the many labyrinths that cannot be attributed to one of the six styles above, it is possible to identify other styles. This particularly applies to labyrinths of which several examples exist in the same style. This, for instance, applies to the last two examples shown in the other styles group (Other 8, Other 9).

We have now ordered the individual labyrinth examples by styles. The result is also a typology or at least an approach to a typology. The only criteriun we have used for the definition of the types is the style. We thus have defined: type = style.

Because style cannot be defined clearly and unambigously, to me it is not well suited as a criterion for the constitution of a typology. Based on style it is not possible to form a complete range of mutually exclusive groups or types of labyrinths. Furthermore, style does not show the essential of a labyrinth.

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Type or Style / 7

What is a Style?

What determines a style cannot be defined as easily and clearly as a type of labyrinth. Style can be described as a trailblazing way of the design of labyrinths. Usually various types of labyrinths can be designed in the same style. And, vice versa, the same type of labyrinth can be realized in various styles.

In the following I will show some styles. Please give attention to the figure and do not care about what is in it. Regarding the style, the numbers of arms, circuits or the level sequence have no importance. These are important for the type. We can also say: The style is not the content but the form. Or more sensually: What’s the wine for the enologist is the type for the labyrinthologist, and similarly the vessel is the style. Therefore I deliberately show the styles using a figure that is no labyrinth in the strict sense. This figure has one entrance and one access to the center, but only one circuit. It can be considered as a predecessor of a labyrinth. This also helps me to show that the style is something truly different, complementary to the type.

The classical style results if we start from a seed pattern and finalize a labyrinth quasi freehand. This results in a particular form of layout. There is no exact definition of this layout. Rather it may vary between almost circular and rectangular according to the drawer or the type of labyrinth shown in this style.

klassisch

Figure 1. The Classical Style

Fig. 1 shows the essential features of this style. The center of the labyrinth is narrow and formed like a dead-end. The circuits on this side and the opposite side of the axis are shifted. Every one-arm labyrinth can be designed in this style; it is even possible to realize labyrinths with multiple arms in the classical Style.

The example in the concentric style shows best which figure I used for the presentation of the various styles.

konzentrisch

Figure 2. The Concentric Style

The essential characteristics of this style are that the middle of the figure and the center of the labyrinth are matching. Also, the center is somewhat enlarged. The axial wall can but does not need to lie on a radius aligned to the center. A point in the center may but does not need to be visible. This style can often be found in labyrinths of manuscripts. In some of them one can see the central point where the compass has been applied.

The Man-in-the-Maze style has already been extensively described on this blog. It is a very good example for what determines a style: This is the original way of a graphical realization – in this case on a strict geometric grid.

MiM

Figure 3. The Man-in-the-Maze Style

Although they lie on a template of concentric circles, the MiM labyrinths are eccentric. In this style, the center of the labyrinth cannot lie in the middle of the figure. The middle of the figure matches with the center of the seed pattern.

Also the extraordinary design of the labyrinth in Chartres cathedral illustrates well what may constitute a style.

chartres

Figure 4. The Chartres Style

What particularly characterizes this style are the lunettes in the center and the lunations at the exterior of this labyrinth. Several labyrinth examples exist that adapted the Chartres style either in the use of the lunettes or the lunations or both elements of this style together. So, Chartres is also a style! We therefore have to deal with a Chartres type and a Chartres Style. We will have to come back to this later.

The same applies to the labyrinth of Reims too. We could also speak of a Reims style or a Bastion style.

reims

Figure 5. The Reims Style

The labyrinth that has been laid out in the Reims cathedral also has a pioneering design. I do not primarily mean the lawful proportions and composition of the octagonal forms. This per se also deserves attention. However, what constitutes the Style is the bastions. Such bastions, also with varied, rounded shapes, have been adapted in many other labyrinths.

Many other labyrinth examples have special graphical features, e.g. Nîmes, Ravenna, Al Qazwini, Cakra-vyuh. Some of these are singular examples. What constitutes a style cannot be conclusively resolved. Which element may characterize a style, and whether only one or multiple elements be required to characterize a style will be controversial. However, what seems a central requirement to me is that a style can be found in various examples of labyrinths. So that it has influenced other labyrinths.

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How to Draw a Man-in-the-Maze Labyrinth / 7

The Formula

Many Man-in-the-Maze (MiM) illustrations show the Cretan-type labyrinth. Many others show figures that look alike but are no true labyrinths. With the seed pattern and the MiM auxiliary figure we are now able to draw every type of a one-arm labyrinth in the MiM-style. This can be done by varying the seed pattern, placing it into the center of the auxiliary figure and completing it to the whole labyrinth. For this, however, we have to know exactly, how many spokes and rings the auxiliary figure for a particular type of labyrinth must have. Both, the number of spokes and rings are determined by the seed pattern of the labyrinth to be drawn in the MiM-style. These numbers can be found by trying out. But they can also be calculated with a formula. Figure 1 shows this with the example of my demonstration labyrinth (see also post / 5 of this series, related posts below).

MiM_KS_Formel

Figure 1. The Formula

Where

S is the number of spokes of the auxiliary figure
E is the number of ends of the seed pattern
R is the number of rings of the auxiliary figure
V is the number of nestings of the element with the most nested levels

Fig. 1 shows the seed pattern for the walls (blue) with all turns of the pathway (red) drawn in. (By the way: these red lines are nothing else than the seed pattern for the Ariadne’s Thread.)

To determine the number of spokes we use the entire seed pattern for the walls. Just count the ends. The number of spokes is the same as the number of ends of the seed pattern: S = E.

The calculation of the number of rings is somewhat more complicated. For this, the number of spokes is needed and, in addition, the element of the seed pattern for the Ariadne’s Thread with the most nested levels. In figure 1, this is the element of the right half with three nested turns. The number of rings is equal to the number of spokes divided by two plus the number of nestings plus 1, thus: R = S/2 + V + 1.

The auxiliary figure for my demonstration labyrinth in the MiM-style therefore needs 12 spokes and 12/2 + 3 + 1 = 10 rings. Let us test this formula with some of the other labyrinths of earlier posts of this series. The auxiliary figure for

  • the Cretan requires 16 spokes and 16/2 + 2 + 1 = 11 rings (see post / 1, related posts below)
  • the Löwenstein 3 – type labyrinth requires 8 spokes and 8/2 + 1 + 1 = 6 rings (see post / 4, related posts below)
  • the Otfrid-type labyirnth requires 24 spokes and 24/2 + 2 + 1 =15 rings (see post / 4, related posts below)

The number of spokes depends only from the number of circuits of the labyrinth. All auxiliary figures of labyrinths with the same number of circuits have the same number of spokes.

The number of rings depends from the number of circuits too, but in addition also from the depth of the nestings of the seed pattern. Auxiliary figures of labyrinths with the same number of circuits can have different numbers of rings (see post / 3, related posts below).

At this point a precision with respect to the number of nestings has to be made. The element with the most nestings of my demonstration labyrinth has three nested turns, i.e. three nestings. However, there exist also seed patterns in which several turns on the same level are nested by a turn on the next level.

MiM_KS_Verschachtelung

Figure 2. Nested Levels

Two such examples are shown in fig. 2. The left figure has 4 elements, but only 3 nestings, as the two turns aligned vertically to the right lie on the same level. The right figure has 5 elements but also only 3 nestings. So, what counts is the number of nested levels, not the number of turns.

Let’s show this with an example that has not yet been shown in the MiM-style. For this, the alternating labyrinth with five circuits that corresponds with Arnol’d’s figure 6 and has already been repeatedly presented on this blog, is well suited.

KS_A6

Figure 3. Chartres 5 classical

The seed pattern of this labyrinth has two elements with 2 nestings. Each element includes 3 turns, i.e. 2 turns on the first level nested by one turn on the second level. According to the formula, the auxiliary figure for this type of a labyrinth has 12 spokes, as is the same with all labyrinths with five circuits, and 12/2 + 2 + 1 = 9 rings. This is 1 ring less than in the auxiliary figure for my demonstration labyrinth although this also has five circuits.

With this formula we now have everything we need to be able to transform any type of a one-arm labyrinth into the MiM-style. This can be done with the following steps:

  • Choice of the type of labyrinth
  • Extraction of the seed pattern
  • Calculation of the number of spokes and rings of the auxiliary figure
  • Variation of the seed pattern into the MiM-style
  • Placing the seed pattern in the center of the axiliary figure
  • Determining the situation of the center of the labyrinth
  • Completion of the seed pattern around the center from inside out to the whole labyrinth.

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