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Posts Tagged ‘man in the maze’

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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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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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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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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Non-alternating Labyrinths

In all previous posts of this series with the exception of the second part (see related posts below) I have shown alternating labyrinths. In alternating labyrinths the pathway does not traverse the axis. However, there exist also labyrinths in which the path traverses the axis (in multiple-arm labyrinths: the main axis). These are termed non-alternating. A beautiful example of such a labyrinth is depicted in a manuscript from the 10./11. century of the Stiftsbibliothek St. Gallen. Erwin has already presented it on this blog, and I have published on it in Caerdroia 38 (2008).

Illustration 1. St. Gallen Labyrinth

Illustration 1. St. Gallen Labyrinth

From part / 2 of this series, we know that in principle also non-alternating labyrinths can be drawn in the MiM-style, as the Snail Shell labyrinth is non-alternating. The pathway of this labyrinth traverses the axis twice. Once when it skips from the first to the second circuit and second when skipping from the second inner to the innermost circuit.

Illustration 2. The Ariadne's Thread

Illustration 2. The Ariadne’s Thread

The pathway of the St.Gallen labyrinth (ill. 2), however, comes in clockwise from the outer circuit, turns to the right and moves axially to the innermost circuit, where it turns to the left and continues without changing direction (clockwise). How does this affect the seed pattern and its variation into the MiM-style of this labyrinth?

Illustration 3. Seed Patterns Compared

Illustration 3. Seed Patterns Compared

Ill. 3 shows the seed pattern of my demonstration labyrinth from part / 5 of this series (figures a and b) and compares it with the seed pattern of the St. Gallen labyrinth (figures c and d). The seed pattern of the demonstration labyrinth has one central vertical line. This represents the central axial wall to which are aligned the turns of the pathway (fig. a). This is the case with all alternating labyrinths. Variation of seed patterns of alternating labyrinths into the MiM-style leaves the central line and the innermost ring untouched (fig. b). The auxiliary figures of alternating labyrinths all have two vertical spokes and an intact innermost ring.

This is different with the labyrinth of St. Gallen. The seed pattern of this labyrinth has two equivalent vertical lines (fig. c). Between these lines the pathway continues along the central axis. If we vary this seed pattern into the MiM-style, we find no central wall and the innermost ring interrupted (fig. d). The auxiliary figure of the St. Gallen labyrinth therefore has no vertical spoke.

Illustration 4. Labyrinth of St. Gallen in the MiM-style

Illustration 4. Labyrinth of St. Gallen in the MiM-style

Non-alternating labyrinths can be drawn in the MiM-style in the same way as alternating labyrinths. The seed pattern of the St. Gallen labyrinth has two elements with single and two elements with two nested turns, and in addition the segment of the path that traverses the axis. In the MiM-auxiliary figure this seed pattern covers two circuits. This corresponds with the elements that are made-up of two nested turns. The pathway segment traversing the axis needs no additional circuit, as for this the innermost ring is interrupted to let the path continue through the middle of the seed pattern.

Illustration 5. My Logo in the MiM-style

Illustration 5. My Logo in the MiM-style

And, finally, here is my logo in the MiM-style (ill. 5).

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Heterogeneous Seed Patterns

In all my previous posts on the Man-in-the-Maze labyrinth I have shown labyrinths with homogeneous seed patterns (see: related posts, below). I term seed patterns as homogenous, that are composed of a series of similar elements.

Illustration 1. Single, 2 nested, and 4 nested Turns

Illustration 1. Single, 2 nested, and 4 nested Turns

I have shown seed patterns with single (not nested) turns as well as seed patterns with elements of solely two nested turns or four nested turns. The labyrinths of the Löwenstein 3, Näpfchenstein, and Casale Monferrato types and the smaller of the two labyrinths with 7 circuits shown in part / 3 of this series all have single turns (Ill. 1, left figure). The patterns of these labyrinths are made-up of a serpentine from the outside in. The seed patterns of the Knossos, Cretan and Otfrid type labyrinths are composed solely of elements with two nested turns (ill. 1, central figure), see also part / 4 of this series. The larger of the two labyrinths with 7 circuits shown in part /3 has two elements with four nested turns (ill. 1, right figure).

Most labyrinths, however, have a mixed – I therefore term it: heterogenous – seed pattern. I will show this here with two examples. The first example is the labyrinth, I am wont to use for purposes of demonstrations, so to speak my demonstration labyrinth. It is this the labyrinth that corresponds with Arnol’d’s figure 5, I have already presented in this post.

Illustration 2. Seed Pattern with Single, 2 nested, and 3 nested Turns

Illustration 2. Seed Pattern with Single, 2 nested, and 3 nested Turns

The seed pattern of this labyrinth is composed of several different elements. The right half consists of an element with three nested turns (ill. 2, right figure). The left half is composed of two other elements. One of it is a single turn (ill. 2, left figure), the other has two nested turns (ill. 2, central figure).

How many rings now do we need for such a seed pattern in the MiM auxiliary figure? From the previous posts we know that seed patterns with single turns take one circuit, such with two nested turns take 2 circuits and so forth. Therefore it is reasonable to assume we will need three circuits, as many as are required to cover the three nested turns of the right half of the seed pattern.

Illustration 3. Seed Pattern from Illustration 2 in the MiM-style

Illustration 3. Seed Pattern from Illustration 2 in the MiM-style

And indeed this is true. The single turn on top left of the seed pattern covers the innermost circuit of the auxiliary figure (ill. 3, left figure). The two nested turns on bottom left cover two circuits (ill 3, central figure). The three nested turns of the right half of the seed pattern cover three circuits (ill. 3, right figure). Thus, the number of circuits required is determined by the element with the most nested turns.

In addition there is another effect on the apparence of the seed pattern in the MiM-style .

Illustration 4. Seed Pattern from Ill. 3 with Prolonged Ends

Illustration 4. Seed Pattern from Ill. 3 with Prolonged Ends

In heterogenous seed patterns, not all the ends lie on the same ring of the auxiliary figure. (ill. 4, left figure). Despite this, of course, three circuits of the auxiliary figure are needed for the seed pattern. Therefore it makes sense to prolong the ends on the inner rings so that they all end on the same ring of the auxiliary figure. Some of the dots are prolonged to lines and also some of the lines are prolonged (ill. 4, central figure). The result can be seen in the right figure of ill. 4.

In order to draw my demonstration labyrinth in the MiM-style, we thus need 5 circuits for the labyrinth, 1 circuit for the center, and 3 circuits for the seed pattern.

Illustration 5. Demonstration Labyrinth in the MiM-style

Illustration 5. Demonstration Labyrinth in the MiM-style

The second example offers me the opportunity to draw the attention to a very beautiful historical labyrinth.

Illustration 6. Seed Pattern of the Cakra-vyuh Type Labyrinth

Illustration 6. Seed Pattern of the Cakra-vyuh Type Labyrinth

The illustration shows the seed pattern and its variation to the MiM-style of the Cakra-vyuh labyrinth. This labyrinth has 11 circuits and is self-dual. However, according to the classification by Tony Phillips, it is an uninteresting labyrinth. Of interest for our purpose is, that the seed pattern of this labyrinth is composed of elements with single (not nested) and with two nested turns. Four of its 24 ends, the four dots, lie on the second innermost ring. The other 20 ends, 16 lines and 4 dots, lie on the third inner ring. The four innermost dots are therefore prolonged to lines so that all ends lie on the third inner ring.

Illustration 7. Labyrinth of the Cakra-vyuh Type in the MiM-style

Illustration 7. Labyrinth of the Cakra-vyuh Type in the MiM-style

In order to draw the Cakra-vyuh type labyrinth in the MiM-style, an auxiliary figure with 24 spokes and 15 rings (11 circuits for the labyrinth + 1 for the center + 2 for the seed pattern = 14 circuits), i.e. 15 rings is needed.

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P.S.: Fortunately the seed pattern of the Cakra-vyuh type labyrinth in the MiM-style covers the same number of circuits as the seed pattern of the Otfrid type. So for the drawing we can use the walls (black) of the Otfrid type labyrinth in the MiM-sytle (from part / 4) and only have to exchange the seed patterns (blue).

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The Spokes of the Auxiliary Figure

In all earlier posts on this subject I have considered labyrinths with 7 circuits. The auxiliary figures of these labyrinths all have 16 spokes. The number of spokes of the auxiliary figure is determined by the number of the ends of the seed pattern. I will show this here for some selected labyrinths with less or more than 7 circuits. The first two examples are the only alternating labyrinths with three circuits. The third is a labyrinth with 11 circuits.

Type Löwenstein 3

Type Löwenstein 3

The simpler labyrinth with three circuits is of the Löwenstein 3-type. The seed pattern of this labyrinth has 8 ends. The pattern is made-up of a serpentine from the outside in. This labyrinth again contains the smallest possible seed pattern that covers only one circuit in the MiM auxiliary figure. Therefore the auxiliary figure has 8 spokes and consists of three circuits for the labyrinth, one for the center and one more for the seed pattern. For the boundaries of the five circuits, six rings are needed.

Type Knossos

Type Knossos

The other is the well-known Knossos-type labyrinth. The auxiliary figure for this type of labyinth has also 8 spokes. The pattern of this labyrinth, however, is made-up of a single double-spiral like meander (Erwin’s type 4 meander). This has two nested turns on each half of the seed pattern. It is this the largest possible seed pattern for a labyrinth with three circuits in the MiM-style. The seed pattern covers two circuits. Therefore, the auxiliary figure for this labyrinth needs 6 circuits / 7 rings, which is one more than the Löwenstein-type labyrinth.

As a third example I show the Otfrid-type labyrinth in the MiM-style.

Type Otfrid

Type Otfrid

Ths seed pattern of this type of labyrinth has 24 ends as is the case with all other labyrinths with 11 circuits. Thus, the auxiliary figure has 24 spokes. In addition the seed pattern consitsts of six similar sixth parts, each of which is made-up of two nested turns. It therefore covers two circuits. The auxiliary figure thus has 11 circuits for the labyrinth plus one for the center and two for the seed pattern, in all 14 circuits and 15 rings.

Seed Patterns with Single (Erwin's Type 4) Meanders

Seed Patterns with Single (Erwin’s Type 4) Meanders

The seed patterns of the Knossos-, Cretan- and Otfrid-type labyrinths all need two circuits in the MiM auxiliary figure. Remember that the Knossos type labyrinth is made-up of one, the Cretan of two and the Otfrid-type labyrinth of three single double-spiral-like (Erwin’s type 4) meanders. These are the three labyrinths of the horizontal line of the labyrinths directly related with the Cretan labyrinth.

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