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For every labyrinth exists a second or dual one. And in special cases the dual one looks like the original one. Then this is a self-dual labyrinth.

These connections should be explained here.

Andreas Frei has done this on his website under the topic “Grundlagen” (basics), to this day only in German. I expressly recommend to take a look at it, there are some meaningful drawings also.

Here again we will see it from the practical side. Hence, it is a continuation of the post from the 1st of September, 2013 about the circular 7 circuit labyrinths. Through the dual labyrinths here we will get six more to add to the seven there. So we will have 13 new labyrinths in all.

How we will reach for that, should be shown step by step. Maybe a little bit awkwardly, but I hope, understandably.

We number all labyrinths from the outside inwards in black. “0” stands for the  outside and “8” for the center. The path sequence, that is the order in which we walk through the circuits to arrive at the center, is noticed on the bottom left in black.
Then we number all circuits once again from the inside outwardly in green. “0” is now the center and “8” is now the outside. We write down the circuits in the order in which we walk them while going backwards from the middle. This path sequence is noticed on the bottom right in green.

Typ 032147658Typ 034765218Typ 034567218Typ 036547218Typ 054367218Typ 056723418Typ 056743218Typ 056741238

As already mentioned, there is to every (original) labyrinth a second (dual) one. And this arises when we interchange inside and outside, when we turn inside out. The path sequence which we will get, is normally different from the one of the original labyrinth.

If it is the same, we speak of a self-dual labyrinth. Then an internal symmetry is given. Or differently expressed: The rhythm and the motion sequence is the same when stepping inside or outside.  In our examples this applies to the first (well-known Cretan) labyrinth, and to the last, a new labyrinth.


The remaining six have another path sequence and, hence, are to be taken for new, different labyrinths.
Here the six new types (click to enlarge, print or save):

Typ 076321458Typ 076123458Typ 076143258Typ 076125438Typ 074561238Typ 076541238

These examples shows that always at first the middle is circled around. After that one moves inside the the labyrinth and finally one enters the center from the 3rd or the 5th circuit.

In the case of the types introduced in the last article the entry into the center was always from the outermost, the first circuit. Here we have the circling around the middle immediately after stepping into the labyrinth.

The motion sequences are completely different.
It would be of interest exploring that by a temporary or even a permanent labyrinth. Worldwide there are still no labyrinths of this kind.
The shape must not necessarily be perfectly circular. It is important only to adhere to the path sequence.
For the rest, they can be as simply build in sand like the types introduced in the below mentioned post.

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In former posts (see below) Andreas and I had presented new and up to know unknown 7 circuit labyrinths. We described them from the theoretical point of view. Therefore the practical part shall be of interest in this post.

I had made some “exercises” in wintertime. On snow it is best to construct directly the path of a labyrinth, Ariadne’s thread.

If one wants to experience these labyrinths in sand on the beach for example, it is better to draw the limitation lines.

The circular shape is very good suited for this. I present the labyrinths in some drawings and instructions.
I would be glad if I could inspire as many as possible labyrinth enthusiasts to try out the look and feeling of these labyrinths.


In the drawings the contained seed pattern is set down in colour. In the circular labyrinth with the path sequence 3-2-1-4-7-6-5-8 (the classical or Cretan type) one can recognise that the right half of the pattern is shifted upwards one path width.

The circular classical 7 circuit labyrinth (path sequence 3-2-1-4-7-6-5-8)

The circular classical 7 circuit labyrinth (path sequence 3-2-1-4-7-6-5-8)

The seed pattern is simplified a little bit. The horizontal bars of the central cross and the angles are circular, so that one needs only the vertical lines of the pattern. To complete the labyrinth one can connect, as usual, from the left to the right (or vice versa) all free points and free line ends with each other.

In sand one scratches meaningfully all eight circles entirely on the ground and constructs afterwards the vertical lines. Besides, one should pay attention to the situation of the main axis. The superfluous parts of the circles can be easily “rubed out”. The real boundary lines can be intensified.

In the design drawings the way is 50 cm wide, the middle has 2 m of diameter (= the fourfold of the path width). Thereby one gets a whole diameter of 9 m. Then the line length amounts to about 138 m and the way into the center will be about 120 m.
The whole can be scaled. That means: Should the labyrinth become twice as big, e.g., the diameter should have 18 m, one multiplies all values by two. Then the path would be 1 m wide and one would have to cover 240 m to reach the center.

The different labyrinth types can best made out by the path sequence. This is the order in which the single circuits are worked through.

Here as an arrangement all eight types (Click to enlarge, print or save):

Type 3214 7658Type 3476 5218
Type 3456 7218
Type 3654 7218

Type 5436 7218

Type 5672 3418

Type 5674 3218Type 5674 1238

The last example, the type 5674 1238, is the only self-dual one beside the classical (Cretan) labyrinth (type 3214 7658). This shows its high quality.
Hence, it would be worth to be built as a walkable, permament labyrinth. Worldwide there is not yet one, as far as I know. Who starts?

For all diligent labyrinth builders: Here are design drawings as PDF files of all types to see, print or copy:

classical_32147658.pdf

classical_34765218.pdf

classical_34567218.pdf

classical_36547218.pdf

classical_54367218.pdf

classical_56723418.pdf

classical_56743218.pdf

classical_56741238.pdf

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Today we will consider the snail shell labyrinth with a bigger center. I like to call it the Knidos labyrinth.
The forms shown here arise if one spreads the ways uniformly – and the boundary lines equally broad as the ways. (This was not the case in the preceding postings of the parts 1 to 3). By this method an empty space (which I have already called fontanel) sometimes is left through the overlapping of the boundary lines. This space looks differently, depending on the path sequence. The whole form of the labyrinth is given when the fontanels shall be very small and the labyrinth very compactly. Besides, the center has the fourfold of the dimension between axes. The whole is, so to speak, a labyrinth on minimal possible space. Also it shows especially well how entwined the labyrinth is. There is no straight line in it any more. However, the labyrinth also does not become absolutely circular.

Knidos Labyrinth Type 1254 3678

Knidos Labyrinth Type 1254 3678

At first the type 1254 3678 after the path sequence without any crossings of the axis.


Now we will play with the crossings of axes. The “original” snail shell labyrinth was developed from the basic pattern, and it had the crossings of the axis at the beginning and at the end.

Knidos Snail Shell Labyrinth with 2 crossings of the axis

Knidos Snail Shell Labyrinth with 2 crossings of the axis

There are two turning points which are embedded in two other circuits. Thereby the changes of direction are made in the middle section. The spiral movement proceeds clockwisely. One can lay the entrance into the labyrinth and the entry into the center on the same vertical line.


Now we will cross the axes in the middle section.

Knidos Snail Shell Labyrinth with 2 crossings of the axis in the middle section

Knidos Snail Shell Labyrinth with 2 crossings of the axis in the middle section

This makes four turning points which lie spatially closely together. The moving inside the labyrinth has changed completely.


Here we will have more motion forwards and only two turning points which lead to changes of direction. All together we cross the axes four times.

Knidos Snail Shell Labyrinth with 4 crossings of the axis

Knidos Snail Shell Labyrinth with 4 crossings of the axis

For me this is still a labyrinth. There are two changes of direction, for the rest only motion forwards. How it is in a snail shell – or in a spiral.

If I have properly counted, there are a total of 12 variations for a labyrinth with the path sequence 1-2-5-4-3-6-7-8. Since one can combine the possible crossings of axes quite differently.

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Today we will consider the snail shell labyrinth in circular form. First we see the labyrinth drawn according to the path sequence without crossing any axis. The following drawing shows the boundary lines in black. We can recognise very well that the middle axis is not crossed.

The Snail Shell Labyrinth without crossing the axis

The Snail Shell Labyrinth without crossing the axis

However, this is not any more the original snail shell labyrinth, even if it shows the same path sequence. Due to the the oscillating movements it is “closer” to a true labyrinth and constitutes an own type. This is why one should call it better type 1254 3678.


Now we will play with the crossings of the axis. Since they constitute the original snail shell labyrinth, although it can be developed from the seed pattern. The crossings of the axis are possible at different places: Here they are at the beginning and at the end.

The Snail Shell Labyrinth with crossings of the axis at the  beginning and at the end

The Snail Shell Labyrinth with crossings of the axis at the beginning and at the end

There are two turning points which are embedded in two other circuits. Thereby there are some changes in the motion sequence in the middle section. As to the rest the spiral movement towards the center in a clockwise direction predominates. One can place the entrance into the labyrinth and the entry into the middle from the last circuit along the middle axis. By fliping the figure vertically one could generate a counterclockwise moving.


Now the crossings of the axis are additionally executed in the middle section.

The Snail Shell Labyrinth with crossings of the axis in the middle section

The Snail Shell Labyrinth with crossings of the axis in the middle section

This looks not so nice in circular form, because one must, so to speak, hook. There are two changes of direction less,  and instead more moving forwards.


What happens if we only bank on motion forwards and don’t care about the changes of direction? And if  we cross all axes possible? The following drawing shows it:

A circular spiral

A circular spiral

This is, by the way, no real spiral in the mathematical sense. The right half has only other radii than the left one.
We have no more a labyrinth. The path sequence which was identical in all previous examples has vanished. So we can see that the changes of direction belong substantially to the labyrinth. A simple motion forwards into the center don’t result in labyrinth.

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In my last post I have pointed to the special layout of this roman mosaic labyrinth (see related posts below).

Figure 1. Mosaic labyrinth of Nîmes

Figure 1. Mosaic labyrinth of Nîmes

In the meantime I had a closer look at it and found six peculiarities.

Figure 2. The Peculiarities

Figure 2. The Peculiarities

  1. Here, there is no diagonal (as opposed to the 3 fine dashed lines from the middle to the other three corners). Along each of these diagonals, the pathway is bent by 90° degrees. Therefore, all circuits only make three bends of 1/4 of a circle.
  2. Accordingly, the turns of the pathway that normally lie beyond the axis (on the side opposite to the entrance) are oriented horizontally, not vertically.
  3. Moreover, they are not arranged in one line as normally, and as are the turns of the pathway on this side of the axis too.
  4. The three inner circuits (circuits 5 – 7) lie entirely on this side of the axis and thus cover only quadrants 1 and 2.
  5. Correspondingly, quadrants 3 and 4 are only covered by the four outer circuits (circuits 1 – 4).
  6. And, as if this were not enough, the center makes one exception. Instead of entering the center axially, the pathway makes one additional turn of 1/4 of a circle before it reaches the center. Therefore it enters the center in parallel with the way into the labyrinth. First I thought, the designer might have wanted to mislead the observer about the true 3/4 – nature of this labyrinth. A closer look at it, however, reveals, that this last turn of the pathway is an inevitable result of the chosen layout for the course of the pathway.

It surprises me again and again how interesting some of the historical labyrinths are.

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Variants of the Cretan Labyrinth

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Today we will see the Snail Shell labyrinth in square form.
We build the labyrinth by strictly using the given path sequence, but we don’t cross the central axis. Thus the path several times changes the direction. This again produces the pendular movement how it is demanded for a “right” labyrinth. There arise four turning points.
A more exact view also shows the hidden type Knossos inside, and that only at the beginning and at the end two circuits with change of course were added. Andreas has already pointed to that.
The following drawing shows the walls in black, the seed pattern is highlighted in colour.

The Snail Shell Labyrinth with 4 turning points

The Snail Shell Labyrinth with 4 turning points

Nevertheless, another new type of labyrinth turned up indirectly via the derivation of the Snail Shell labyrinth from the seed pattern by the construction after the path sequence. Though with the creative defects of the beginning on the first circuit and the entry into the middle from the last circuit. However, at least, it is self-dual.
The seed pattern for this type I developed belatedly. It looks quite different as the meanwhile well-known pattern for the Classical 7 Circuit Labyrinth.


Much different looks the labyrinth if one uses all the four crossings of the axis which are possible, even though keeping up the path sequence.

The Snail Shell Labyrinth with 2 turning points

The Snail Shell Labyrinth with 2 turning points

There remains only two turning points with changes of direction: When changing from the 5th to the 4th circuit, and when changing from the 4th to the 3rd. For the rest one advances forwards in a spiral-shaped movement, just as in a snail shell.
If one arranges the entrance edgeways, one can fill up the whole square. One could still procreate other shapes by twisting and mirroring the figure.

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The “original” Snail Shell labyrinth was created from the seed pattern for Ariadne’s thread. To do that only the first curve to be drawn had to be shifted one “unity” to the right. Then all points were connected with each other. Thus a new type for a 7 circuit labyrinth appeared.

However, this type can also be derived from the well-known seed pattern for the walls. The construction goes as usual, only that everything is shifted to the right.
The following drawing shows the walls in black, the seed pattern is highlighted in color.

The Snail Shell Labyrinth made from the seed pattern

The Snail Shell Labyrinth made from the seed pattern

In the meantime, Andreas has also posted something to this labyrinth. He has explained the pattern in the labyrinth, and has pointed to the fact that the path crosses twice the axis. Thus, in the terminology of Tony Phillips it is a non-alternating uninteresting Labyrinth.

The “pattern” is for Andreas not the seed pattern, but the structure of the labyrinth, as best to be seen in the rectangular form.  Hence, “uninteresting” in the terminology of Tony Phillips means that inside this labyrinth the type Knossos is hidden to which only some circuits are added. And the fact that one enters the labyrinth on the first circuit and reaches the middle from the last one.

For me it is interesting that developing the Snail Shell labyrinth from the seed pattern produces the cruising axes. This is ordinarily not the case when using this method. Nevertheless, a new type of labyrinth appears.


If one constructs a labyrinth by only using the path sequence, and without cruising the axes, one will get another labyrinth again. Thus it looks:

The Snail Shell Labyrinth made from the path sequence

The Snail Shell Labyrinth made from the path sequence

This is quite an other type of labyrinth, although it has the same path sequence. Moreover, it is self-dual, because you may count the circuits from inside outwards and you will get the same path sequence.
This shows once more that only the path sequence is not sufficient to classify the type. Unfortunately, I must say, because this makes the categorization even more difficult and more complicated.

One receives even more variations if one includes the crossing of axes or chooses other forms (circle, square). Of which more later.

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