Crossing Labyrinths with Multiple Axes

In addition to the three labyrinths with one axis from my last post (see: related posts 1, below) there are also 7 historical labyrinths with multiple axes and with their pathway crossing the main axis. Of these, I want to present here four very different examples from Roman times until the 18^th century together with their patterns. I have already shwon on this blog how the pattern can be obtained in crossing labyrinths (related posts 2). 

The oldest crossing labyrinth with multiple axes is the polychrome mosaic labyrinth in the Roman proconsul’s residence, House of Theseus, at Kato Paphos, Cyprus dating from 4 CE (fig. 1). Presented is the Ariadne’s Thread as a guilloche ribbon. The pathway starts from a dead-end on the first circuit. After completion of the full circuit, it crosses the main axis and describes a sector labyrinth with four axes on circuits 2 – 6. Then follows a full 7^th circuit that leads into a closed 8^thcircuit. 

Figure 1. Theseus

Figure 2 shows the labyrinth of Bayeux Cathedral from the 13 CE. This has 4 axes and 10 circuits. The pathway crosses the main axis on the innermost circuit. 

Figure 2. Bayeux

A strange labyrinth is depicted on a plaquette from Italy of the 16^th century. It has 6 axes that are distributed irregularly. There is a flaw between the third and fourth axis, where there is an encapsuled piece of a pathway that is not accessible. This piece circulates on the second and third circuit but has no connection with the pathway that leads from the entrance to the center of the labyrinth. Furthermore, the pathway crosses the main axis three times. This labyrinth can be easily reduced to three axes. 

Figure 3. Plaquette

Also in this design for a hedge labyrinth from year 1704, the pathway crosses the main axis twice and then ends peripherally in a dead-end (fig. 4). 

Figure 4. Liger

All these crossing labyrinths with multiple axes show particularities. Theseus has no entrance and no center, Bayeux is uninteresting, as it has simply a complete circuit added at the inside. The plaquette is drawn faulty and unnecessary complicated. And in Liger, no center can be spotted. 

Related Posts:

1. Crossing Labyrinths
2. The Pattern in Non-alternating Labyrinths

Complementary, Uninteresting Labyrinths with Multiple Arms

Among the one-arm labyrinths we have not found any pairs of uninteresting labyrinths complementary to each other (see related posts, below). In labyrinths with multiple arms, however, such pairs do exist, at least if we consider labyrinths as uninteresting in which the path enters on the outermost circuit or reaches the center from the innermost circuit. This is shown in the following example.

Figure 1. Complementary, Uninteresting Labyrinths

Labyrinth a has 2 arms and 3 circuits. The pathway enters on the outermost circuit. Therefore it is an uninteresting labyrinth. The path also reaches the center from the outermost circuit.

The complementary of it, labyrinth b, is also an uninteresting labyrinth. In this, the path enters the labyrinth on the innermost circuit and also reaches the center from the innermost cirucit.

So far, this is nothing special. But in this labyrinth we can observe another special feature. This can be seen, if we also view the two duals of these labyrinths. This is shown in the already familiar manner in figure 2.

Figure 2. The Dual and the Complementary Labyrinths are the Same

The dual (b) to the original labyrinth (a) ist the same as the complementary (c). The dual (d) to the complementary (c) is the same as the original (a). The two labyrinths that are dual-complementary to each other are the same.

Now this is not valid for all pairs of complementary uninteresting labyrinths. However, other labyrinths exist, in which this is also the case. In figure 3 I show two such examples of labyrinths and their patterns (only originals). In these labyrinths also, the complementary and the duals are the same.

Figure 3. Other Labyrinths with this Property

Related Posts:

• Complementary and Uninteresting Labyrinths

The Seed Pattern in Labyrinths with Multiple Arms (continued)

In my last post I have shown that a seed pattern can also be drawn in labyrinths with multiple arms (see related posts). With the seed pattern for the Ariadne’s Thread it is possible to connect the arms in the shape of a flower (or a propeller). For each arm, a separate petal is needed.

Figure 1. Borderlines

These auxiliary figures can all be drawn in one continous  line.

Figure 2. How to Draw the Borderline

Fig. 2 illustrates this with a three-arm labyrinth. Other flowers with multiple arms can be drawn the same way.

Each of the petals contains a part-seed pattern. I will show this using an example of one of my five-arm labyrinth designs. I choose the design KS 2-3 that has been installed as a temporary labyrinth on the square of Magdeburg cathedral. This labyrinth can be actually seen in the header of this blog. Otherwise there is an image of it here.

Figure 3. KS 2-3

Fig. 3 shows the labyrinth in a drawing by Erwin of the walls and with the Ariadne’s Thread (red) inscribed.

Figure 4. Seed Pattern for the Ariadne’s Thread

In fig. 4 the seed pattern for the Ariadne’s Thread is shown per se and completed to the whole Ariadne’s Thread. In order to complete this seed pattern, for each circuit 10 ends have to be connected with five segments of a circuit. Evidently, with an increasing number of arms, the seed pattern comes to look closer to the entire labyrinth. The part-circuits are shortened in relation to the part-seed patterns.

The seed pattern has first and most often been published for the walls of the Cretan type labyrinth. Also for several other one-arm labyrinths, seed patterns have been published. Therefore, the seed pattern does not constitute a characteristic attribute of the Cretan type labyrinth. It is not even a characteristic specific for one-arm labyrinths alone.

However, the use of seed patterns in labyrinths with multiple arms is of minor practical importance. The original purpose and meaning of the seed pattern was, that it enables us to capture the essential of a labyrinth with a simple memorable system of lines that allows us to generate the labyrinth straight away. And this applies best to the seed patterns of the Cretan type labyrinth and its relatives of the vertical line.

Figure 5. Seed Patterns of the Cretan Labyrinth and it’s Relatives of the Vertical Line

Fig. 5 shows the seed patterns for the walls in the first column and for the Ariadne’s Thread in the second column. The corresponding types of labyrinths are on

• row 1: Löwenstein 3
• row 2: The Cretan
• row 3: Hesselager
• row 4: Tibble

 

Related Posts

• The Seed Pattern in Labyrinths with Multiple Arms
• The Cretan at the Crossing Point
• A Temporary Labyrinth on the Magdeburg Domplatz

 

The Seed Pattern in Labyrinths with Multiple Arms

The seed pattern is an extract of the axis of the labyrinth without the circuits. A seed pattern can also be drawn for labyrinths with multiple arms.

Figure 1. The Arms

Fig. 1 shows this with a one-arm and a two-arm labyrinth compared. The one-arm labyrinth is of the Cretan type, the two-arm labyrinth is one of my own designs. For reasons of simplicity I chose the representation with the Ariadne’s Thread.

In labyrinths with multiple arms a separate seed pattern has to be extracted for each arm. Of course, these two parts belong together. This should become directly evident.

The seed pattern for the Ariadne’s Thread is drawn with an auxiliary line that delimits the layout of the seed pattern (see related posts below). This auxiliary line can be used to graphically connect the two part-seed patterns.

Figure 2. Connecting the Part-Seed Patterns

Figure 2 shows how we can prodeed for this. In one-arm labyrinths the center lies beyond the seed pattern. And, strictly speaking, it always has to be indicated, where the center of the labyrinth is situated. In labyrinths with two arms the seed pattern of the side-arm is situated beyond the center opposite to the seed pattern of the main axis of the labyrinth. In a seed pattern of a labyrinth with multiple arms, the center is fixed by the situation of the arms relative to each other. And thus it comes to lie within the seed pattern. With the auxiliary line the sub seed patterns for the Ariadne’s Thread can easily be connected in the shape of an “8”. This can be drawn freehand in one line. By doing so, we have performed a variation of the original circular or elliptic form to a petal-shaped form. This, however, is a minor variation and does not affect the seed pattern itself.

Figure 3. Completing the Seed Pattern for the Ariadne’s Thread

As fig. 3 shows, the seed pattern of a multi-arm labyrinth is completed exactly the same way as a one-arm labyrinth seed pattern (see related posts). The ends of the seeds nearest to the centre are connected first. By this, the innermost circuit is generated. Next, the ends nearest to the first circuit are connected the same way, and so forth. And so, one circuit after another is added from the inside out. The only difference to a one-arm labyrinth is, that in a multiple-arm labyrinth multiple segments have to be generated for each circuit. In a two-arm seed pattern, for each circuit, four ends have to be connected with two sections of circuits between the two arms.

There are two notable differences in the shape of the seed pattern of the main axis and of the side arms.

• The seed pattern of the main axis has two ends more than the seed patterns for the side arms. This is due to the fact that the entrance of the labyrinth and the access to the center lie on the main axis.
• Usually we consider alternating labyrinths where the path does not traverse the main axis (although there are some notable exceptions). In these labyrinths it is indispensable that the path traverses the side arms. Otherwise it would not be possible to reach the area opposite the side-arm and it thus would be impossible to generate the side arm at all.

Figure 4. Completing the Seed Pattern for the Walls

Of course, what is valid for the Ariadne’s Thread works with the walls of the labyrinth too. The seed pattern for the walls, however, looks more complicated and less elegant. The part-seed patterns of the two arms are not graphically connected, as the seed pattern for the walls traditionally is drawn without an auxiliary line. The Ariadne’s Thread is the simpler graphical representation of both, the labyrinth and the seed pattern of it.

Relatecd posts:

• Some Thoughts about the Seed Pattern for the Ariadne’s Thread