Feeds:
Posts
Comments

Posts Tagged ‘variants’

The seed patterns of the heart labyrinth by Mario Höhn were even more difficult to identify for me than those of the Nîmes labyrinth.

Figure 1. Seed Patterns of the Heart Labyrinth

Figure 1. Seed Patterns of the Heart Labyrinth

As fig. 1 shows, what is special about this labyrinth is not so much its heart-shaped form. This alone would not much affect the seed pattern. Very special, however, is that the circuits 7 and 6 are restricted to the first (right) half of the heart. Despite this, the labyrinth looks symmetric. This is because an additional circuit in the form of a turning loop was added in the second (left) half. This fills the space that was left empty by the missing circuits 7 and 6.

Figure 2. Variations to the Seed Pattern

Figure 2. Variations to the Seed Pattern

Therefore, two variations are made to the seed pattern. This is shown in fig. 2:

  • First, the turn from the 7th to the 6th circuit is moved from the 3rd to the 2nd quadrant (A). This turn in the seed pattern for the walls (blue) is marked by an angle or an arc and a dot placed within it. In the seed pattern for the Ariadne’s Thread (red) it is designed as an arc. Therefore, these turns of the pathway are doubled in the second quadrant, the first is embedded between the arms of the central cross, the second lies mirrored somewhere opposite to it.
  • In the right heart chamber the pathway reaches the 7th circuit coming from the 4th circuit. This is as usual in the Cretan type of labyrinth. But then, without leaving the second quadrant, it turns to the 6th and again to the 5th circuit. Only this circuit leads to the left heart chamber. And there, the path does not reach the center from below, but spirals in before it branches to the turning loop, the additional 8th circuit. This is marked in the figure by the red circle, which indicates the situation of the turning loop and the black bullet indicating the center of the labyrinth. This course of the pathway causes an additional spiralling wall to be laid-out in the left heart chamber. These additional modifications in the area of the center have to be added to the seed pattern (B).
Figure 3. How to Complete the Seed Pattern

Figure 3. How to Complete the Seed Pattern

Since these modifications cause strong biases to the seed pattern, I have indicated in fig. 3 which ends have to be connected to each other in order to complete the seed patterns to the whole labyrinth. As usual, completion begins with the two ends that connect to the innermost wall / circuit. I numbered these ends with 1. We then continue following the ascending numbers and thus proceeding from the inside out. Since circuits 7 and 6 are restricted to the first half of the labyrinth and due to the additional features around the center, the two innermost circuits / three innermost walls are not connected with an arc around the center. Only the five outer circuits are connected around the center.

This variant of the Cretan type labyrinth causes variations to the seed pattern such that it misses its original purpose. The purpose of a seed pattern is, that it provides a simple graph, that can be easily kept in mind and completed straightforward to the whole labyrinth.

Related posts:

Read Full Post »

Here I present two more variants of the seed pattern of the Cretan-type labyrinth. I used the same variants as in this post, but present them in a different order from the simplest to the most complex seed pattern.

Figure 1. Abhuyumani Tantra

Figure 1. Abhuyumani Tantra

Fig. 1 shows the seed patterns of the labyrinth-drawing from Rajasthan. The seed pattern for the walls is coherent. The central cross is distorted but well recognizable. The four angles have mutated to four arcs. And they are not embedded between the arms of the cross, but all are aligned horizontally. The inner two turns lie above, the outer aside and slightly below the horizontal bar.

The seed pattern for the Ariadne’s Thread is less coherent. This is because the turn of the pathway from the first to the fourth circuit covers the whole 3rd quadrant. Therefore, the Ariadne’s Thread is directed somewhere through the middle of this quadrant. By this, the segment representing the opposite outer turns falls apart to two pieces.

Figure 2. Half-heart Labyrinth

Figure 2. Half-heart Labyrinth

The seed patterns of the half-heart shaped labyrinth show four peculiarities (fig. 2).

  • They fall apart into two halves. This is due to the layout of this labyrinth, that is not a closed form, but enrolled to a half-open form. The upper half of the seed pattern represents the turns of the pathway that in a closed form would lie on the opposite side next to the turns of this side of the axis.
  • As a consequence of the (half-)heart shape, the inner circuits do not describe half an arc but only about one quarter of an arc of a circle. Therefore, the upper inner turns of the pathway are not aligend in parallel with the upper outer turns, but rotated anticlockwise by ca. 1/4 of a circle.
  • As a result of the artwork of this labyrinth, the situation at the entrance is ambigous. The pathway can enter from above or from below into the third circuit of the labyrinth. For the course of the pathway to be un-ambigous, a part of the wall is missing. I added this as a dashed line on the seed pattern for the walls.
  • Likewise, as a result of the artwork, the major part of the inner space is not accessible, but is separated by the upper boundary of the (half-) heart form. The pathway ends above this boundary, where it reaches the center. By this, all ends of the lower half of the seed pattern are more distant from the center than the ends of the upper half.

Until now the rule for the completion of the seed pattern to the full labyrinth said: “Connect the two ends next to the center with an arc around the center, proceed in the same way with the next two ends and by this add one wall / circuit from the inside out.“ (By the way this also works for both seed patterns combined. First comes the innermost wall, then the innermost circuit, next the second inner wall, then the second inner circuit and so forth.) This rule refers to seed patterns of closed forms.

The rule in the general sense also applies to this variation of a seed pattern of a half-open form. However, the above mentioned specialities must also be considered. As all ends of the upper / opposite half of the seed pattern are closer to the center than the ends of the lower / this-side half, the rule needs to be formulated more precisely as: “connect the nearest end of the upper half to the center with the nearest end of the lower half to the center with a partial arc around the left side of the center.“

As the figure is half-open, the ends of the seed pattern cannot be connected with a full arc, but only with the part of an arc. And, since both halves of the seed pattern are not aligned symmetrically next to each other, it has to be made explicit, that each connection is made between an end of the lower (this-side) and the upper (opposite side) half of the seed pattern. Without being explicitly stated, this is also the case in seed patterns of closed figures. Each arc connects one end of this side with one end of the opposite side of the axis.

Related posts:

Read Full Post »

The same type of labyrinth can occur in different individual variants. This is also valid for the seed pattern of a labyrinth. In order to show this, I went the opposite way than usual. Instead of completing a seed pattern to the labyrinth, I inscribed the seed pattern in different variants of the Cretan-type of labyrinth. And, as there exist two forms of the seed pattern, I marked them both, the seed pattern for the walls (blue) and for the Ariadne’s Thread (red). Then I extracted the seed patterns and separated them. And, for orientation, I also indicated the localization of the center.

Figure 1. Pylos

Figure 1. Pylos

The Pylos labyrinth (fig. 1) has a classical seed pattern for the walls with a central cross, four angles and four dots. This well-known seed pattern thus can be traced back to the earliest securely dated labyrinth at all. Whether the Pylos labyrinth originally was designed from the seed pattern, as many suggest, or not, I leave unanswered here.

Of course, the seed pattern for the Ariadne’s Thread fits into the seed pattern for the walls, since the Ariadne’s Thread follows the pathway between the walls of the labyrinth. The segments of the seed pattern for the Ariadne’s Thread consist of an arc nested into a straight with two rounded angles. These segments are pairwise arranged in vertical and horizontal order.

This arrangement of the segments corresponds with the alignment of the turns of the pathway. In this variant of the Cretan-type labyrinth – on the classic eccentric layout of the Pylos labyrinth – the turns of the inner circuits are aligned along the cross-bar, whereas the outer turns are aligned along the axis. This alignment is characteristic of the classical eccentric layout. (By the way, this can only be observed by viewing the seed pattern for the Ariadne’s Thread. Looking at the seed pattern for the walls, one would rather expect all turns of the pathway to be arranged around the center.)

Figure 2. Silver Coin, Knossos

Figure 2. Silver Coin, Knossos

Fig. 2 shows the seed patterns marked in the labyrinth of the silver coin from Knossos. The central cross of the seed pattern for the walls is easily recognizable. Also, even though not as precise as in the Pylos labyrinth, one can see that the inner turns of the pathway are aligned horizontally, the outer vertically along the axis of the labyrinth. The seed pattern shows, that this labyrinth still preserves essential characteristics of the classical eccentric layout of the Cretan-type labyrinth.

Figure 3. Walahfrid

Figure 3. Walahfrid

This changes if we mark the seed patterns in the labyrinth of Walahfrid. This fully concentric layout of the Cretan-type labyrinth causes two variations to the seed pattern.

  • The cross-bar of the central cross is split in two halves. The bar on the side where the entrance to the labyrinth is situated, is shifted outside by one unit. This is a result of the centralization of the layout. Not only this bar, but all walls on this side are shifted downwards by one unit.

  • All turns of the pathway are now aligned in parallel with the axis.

One could even abstract this seed pattern further and omit all horizontal branches. This would result in a system of vertical lines and the dots of the seed pattern for the walls.

Related posts:

Read Full Post »

The Cretan is the most frequently encountered type of labyrinth, and so for this type we can find a broad range of individual variants. Here I want to show some examples that are of particular interest for various reasons. Unless stated otherwise, all figures are sourced from the book Through the Labyrinth by Hermann Kern. The details can be found here.

Figure 1. Pylos

Figure 1. Pylos

This graffito on a clay tablet from Pylos dates from 1200 BCE at the latest and is the oldest securely dated labyrinth. It shows the Cretan-type on a rectangular layout.

Figure 2. Silver  coin, Knossos

Figure 2. Silver coin, Knossos

This figure shows the labyrinth with a concentric layout on a silver coin from Knossos, ca. 190 100 BCE. The center of the labyrinth covers with the middle of its circuits. The axis, however, is somewhat eccentric, as the pathway reaching the center is aligned centrally.

Figure 3. Walahfrid

Figure 3. Walahfrid

On this drawing from a parchment manuscript by Walahfrid Strabo (808-849), the labyrinth is shown in full concentric form. The axial wall that connects the innermost with the outermost wall of the labyrinth is aligned centrally with the center.

The following examples show, that variants of the layout are not limited to standard forms, such as circles or rectangles.

Figure 4. Heart labyrinth

Figure 4. Heart labyrinth

This heart-labyrith by Mario Höhn is of the Cretan-type, although with an additional closed circuit at the inside. Not all circuits are in parallel course (as with a supposed 7-lane roundabout). Circuits 7 and 6 are limited to the right heart chamber. Circuit 5 leads to the left chamber, where it is connected with the closed 8th circuit.

Figure 5. Double labyrinth

Figure 5. Double labyrinth

An other method to generate a heart labyrinth was used by Marty Kermeen and Jeff Saward. They apply a double labyrinth (DL). This is made up of two identic labyrinths (L) that are mirrored horizonally and connected to each other. So the actual labyrinth is one of these two part-labyrinths. This is a Cretan-type projected on a half-hearted layout.

Figure 6. Abhuyumani Tantra

Figure 6. Abhuyumani Tantra

This tantric drawing from Rajasthan, India, 19th century, shows the labyrinth arranged on three quarters of a circle – most of it actually is unrolled to a semi circle. Only the turn from the first to the fourth circuit covers the whole third quadrant. The fourth quadrant is not covered by the figure.

Figure 7. Nîmes

Figure 7. Nîmes

This roman mosaic labyrinth from Nîmes, France, 1st century, has an inconspicious rectangular outline. But, like no other, it shows that the layout of a labyrinth is not only limited to its outline (circle, rectangle, heart, etc.). It is also important to consider how the course of the pathway is organized within this outline form. And this is really tricky. Just try to identify the seed pattern of this labyrinth. The course of the pathway is special in at least three points.

  • All circuits do not rotate by a full (360°) but only a 3/4 (270°) circle. This is the same as with the Indian labyrinth described above. It is a sort of a 3/4 labyrinth. However, the layout covers all four quadrants.
  • The inner circuits are completely embedded in quadrants 1 and 2. Normally all circuits cover all quadrants.
  • Only the outer 4 circuits cover all quadrants.

These shiftings and transformations vary the layout of the labyrinth so that it is barely recognizable.

But what makes me classify all these different examples as Cretan-type labyrinths? What do all these have in common? What defines a Cretan-type labyrinth has been repeatedly described on this blog and elsewhere:

  • One-arm labyrinth
  • alternating, i.e., the pathway does not traverse the axis
  • 7 circuits
  • level sequence: 3-2-1-4-7-6-5.

It is important to keep in mind that we are dealing with alternating labyrinths. There exist also non-alternating labyrinths. Only among the alternating labyrinths there is exactly one type of labyrinth for each level sequence. The other way round, this allows us to unequivocally describe each type of an alternating labyrinth by its level sequence.

Related posts

Read Full Post »

%d bloggers like this: