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Posts Tagged ‘auxiliary figure’

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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  1. How to Draw a Man-in-the-Maze Labyrinth / 13
  2. How to Draw a Man-in-the-Maze Labyrinth / 9
  3. Our Best Wishes for 2018
  4. The Complementaries of the Three Very Interesting Historical Labyrinths with 4 Arms and 11 Circuits
  5. How to Draw a Man-in-the-Maze Labyrinth / 5

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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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The Rings of the Auxiliary Figure

In the two previous posts I have presented the „Cretan“ and the Snail-Shell labyrinth in the Man-in-the-Maze (MiM-) style (see related posts below). Both have the same seed pattern and the same auxiliary figure. If we now try to draw other labyrinths with 7 circuits in the MiM-style, we’ll soon notice, that not all can be matched with the auxiliary figure with 11 rings. Although all have 16 spokes, some of them need less, others more than 11 rings. What’s the reason for that? I will show this with the two extreme examples.

The Smallest Seed Pattern of any Labyrinth in the MiM-style

The Smallest Seed Pattern of any Labyrinth in the MiM-style

The first labyrinth is comparable with the historical Löwenstein 3 (3 circuits), Näpfchenstein (5 circuits), or Casale Monferrato (6 circuits) labyrinths. I don’t know any historical example of such a labyrinth with 7 circuits. It is one of the 20 uninteresting labyrinths (see related posts below) with 7 circuits, although self-dual. The pattern of this labyrinth is made of a serpentine from the outside in. The seed pattern consists of 8 similar eigth parts. In the auxiliary figure this is the smallest possible seed pattern. This takes one circuit .

MiM Auxiliary Figure with 10 Rings

MiM Auxiliary Figure with 10 Rings

A labyrinth of this type, drawn in the MiM-style, requires an auxiliary figure with only 10 rings.

The biggest Seed Pattern in Labyrinths with 7 Circuits

The biggest Seed Pattern in Labyrinths with 7 Circuits

The second labyrinth is comparable with the historical Knossos (3 circuits), and (the core-labyrinth of ) Rockcliffe Marsh (5 circuits) labyrinths. I am not aware of a historical example of such a labyrinth with 7 circuits either. It is one of the six very interesting labyrinths (see related posts) with 7 circuits. The pattern is made-up of a double-spiral like meander. Whereas the Knossos-type labyrinth consists of a single (Erwin’s type 4) meander and Rockcliffe of a two-fold (type 6) meander, this labyrinth is based on the three-fold (type 8) meander. The seed pattern consists of two identical halves. Each is composed of four nested turns. This is the largest possible seed pattern for a labyrinth with 7 circuits in the auxiliary figure. For this, four circuits are needed.

MiM Auxiliary Figure with 13 Rings

MiM Auxiliary Figure with 13 Rings

A labyrinth of this type, drawn in the MiM-style, requires an auxiliary figure with 13 rings.

Both seed patterns have 16 ends and the auxiliary figures have the same number of spokes. This is the same for all labyrinths with 7 circuits (colored in blue in the figure below).

Single and Nested Turns

Single and Nested Turns

The number of rings needed for the auxiliary figure depends on three factors:

  1. the number of circuits of the labyrinth
  2. plus one additional circuit for the center
  3. the number of circuits needed for the seed pattern

The number of circuits for the seed pattern depends on how deeply nested it is. In a labyrinth with 7 circuits, this can vary from a series of single, not nested, turns up to four nested turns. These two extremes are colored in red in the above figures. The lower extreme requires 1, the upper 4 circuits.

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Recently Erwin informed me, that a new comment had come in to one of his older posts: „Is there an easy way to draw the ‘man in the maze’ labyrinth?“ I was about to answer in the comment box, but it turned out that it was not that easy. However, there is a sure way, and this is not really difficult either. Two elements are needed for this purpose: An auxiliary figure and the seed pattern. Let us have a closer look at what this means, using the labyrinth presented by Erwin.

The Native American Labyrinth

The Native American Labyrinth

As Erwin correctly notes, this is a labyrinth of the Cretan type. All the walls are arranged in a strict geometric order. They all lie on a special grid. This is made-up of a wheel with 11 concentric circles / rings and 16 spokes.

The Auxiliary Figure

The Auxiliary Figure

This grid is the auxiliary figure we need. In his drawing, Erwin has numbered the circuits of the labyrinth from 1 through 10 with the (not accessible) middle being number 11. This same enumeration can also be applied to the walls. These lie on the rings of the grid. Then, the outermost ring has the number 1, the innermost ring number 11.

The second element we need, is the seed pattern. As we want to draw the labyrinth represented by the walls, we also need the seed pattern for the walls of the labyrinth. And, since it is a labyrinth of the Cretan type, we need the seed pattern of the Cretan-type labyrinth.

The Seed Pattern

The Seed Pattern

The seed pattern is first varied such that it fits to the auxiliary figure. All the lines and dots must lie on the grid. Compared with some earlier variations, this is only a slight variation. It is immediately recognizable as a seed pattern of the Cretan type. This is now placed in the middle of the auxiliary figure. The 16 ends of the seed pattern are formed by the intersection points of the 16 spokes with the 9th ring.

The First Step

The First Step

Now it remains to complete the seed pattern to the whole labyrinth. For this, first, the situation of the center of the labyrinth has to be determined. Then, we proceed exactly as described here. First, the two ends next to the center are connected. However, this is not done with an arc around the center, but using segments of lines that lie on the grid provided by the auxiliary figure. By this we draw the innermost wall of the labyrinth.

The Second Step

The Second Step

We then continue with the ends next to first connection and create the second inner wall.

The Third Step

The Third Step

Then follows the third connection in a similar way, and likewise all others follow. And by this, from the inside out, one wall is added after each other.

The Complete Labyrinth

The Complete Labyrinth

Finally, one opening remains. This is the entrance to the labyrinth. If we then remove the auxiliary figure, we can easily view the final result.

Labyrinth without Auxiliary Figure

Labyrinth without Auxiliary Figure

This guidance shows the importance of distinguishing between the type of a labyrinth and the layout. We have here drawn a Cretan-type labyrinth on a Man-in-the-Maze layout, or, let’s say: in the Man-in-the-Maze style (MiM-style). Every type of a one-arm labyrinth can be drawn in the MiM-style.

The type of a labyrinth can be entirely represented by the seed pattern. This, actually, is the meaning and purpose of the seed pattern, that it contains the essence of the whole labyrinth.

In my previous posts I have followed variations to the Cretan type labyrinth and examined their effects on the shape of the seed pattern. Some variants were found, that varied the seed pattern in such a way, that it was hardly recognizable and practically useless.

Here I come back again to the original purpose of the seed pattern. In this case, where we want to draw a labyrinth in the MiM-style, the seed pattern is of essential practical use. Without it, I see no easy way. But using the seed pattern on the auxiliary figure, with some skills, it is even possible to draw a labyrinth freehand in the MiM-style.

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