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Archive for the ‘Typology’ Category

Quite simply: By leaving off the barriers in the minor axes. I have already tried this with the Chartres labyrinth (see related posts below). But is that also possible with every other Medieval labyrinth?

As an example I have chosen the type Auxerre that Andreas showed here recently. This labyrinth is self dual as are Chartres and Reims, therefore of special quality. And they all have a complementary version.

The Auxerre labyrinth

The Auxerre labyrinth

Here the original with all the lines and the path in the labyrinth, Ariadne’s thread. The barriers in the minor axes are identical with those of the Chartres type. There is only another arrangement of the turning points (the lanes 4, 5, 7, 8) in the middle of the main axis.

The original Auxerre labyrinth without the barriers

The original Auxerre labyrinth without the barriers

The barriers are omitted. When drawing Ariadne’s thread, I found that four tracks could not be inserted. Hence, I have anew numbered the circuits and there remain now 7 circuits instead of the original 11. However, this also means that by changing this Medieval labyrinth into a concentric Classical labyrinth through this method no 11 circuit labyrinth is generated, but a 7 circuit.

The 7 circuit circular Cretan labyrinth

The 7 circuit circular Cretan labyrinth

If one looks more exactly at it, one recognises the well-known path sequence: 3-2-1-4-7-6-5-8. We got a Cretan labyrinth in concentric style.


Now we turn to the complementary labyrinth:

The complementary Auxerre labyrinth

The complementary Auxerre labyrinth

The complementary labyrinth is generated by mirroring the original one. The upper barriers remain, right and left they run differently and in the main axis, the turning points shift. The entrance into the labyrinth changes to the middle (lane 9) and the entrance into the center is from further out (lane 3).

The complementary Auxerre labyrinth without the barriers

The complementary Auxerre labyrinth without the barriers

As with the original, four lanes can not be inserted (4, 5, 7, 8). Therefore, the result is again a 7 circuit labyrinth. I renumbered the lanes and have redrawn the labyrinth.

This is how it now looks like:

The complementary 7 circuit circular Cretan labyrinth

The complementary 7 circuit circular Cretan labyrinth

The labyrinth is entered on the 5th lane, the center is reached from the 3rd lane. The path sequence is: 5-6-7-4-1-2-3-8. This labyrinth is not one of the historically known labyrinths. But it showed up in this blog several times (see related posts below). Because it belongs to the interesting labyrinths among the mathematically possible 7 circuit labyrinths.

The surprising fact is that no 11 circuit Classical labyrinth could be generated through the transformation. But for that  the 7 circuit Cretan labyrinth. Therefore we can say that the heart of the Medieval Auxerre labyrinth is the Cretan (Minoan) labyrinth as it is in the Chartres labyrinth.

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By halving a 7 circuit labyrinth in labyrinthine logic, as it was successful for the 5 circuit Chartres labyrinth.

The 7 circuit Chartres labyrinth

The 7 circuit Chartres labyrinth

The 4th circuit cuts the labyrinth in two parts. Then I receive an external (circuits 1 – 3) and an internal labyrinth (circuits 5 – 7). Both are identical in its path sequences. Even if the “barriers” are at different places.

Two 3 circuit Chartres labyrinths

Two 3 circuit Chartres labyrinths

The path sequence defines the type: 3-2-1-2-3-2-1-4. It is identical for both versions. This 3-2-1-4 reminds very much of the smallest possible labyrinth: the Knossos labyrinth (and of the meander).
If I leave out the barriers, I receive this labyrinth. This once again shows the quality of the Chartres labyrinth.

To make the layout more appealing, I can arrange the barriers in steady distances, in a way make a labyrinth with three arms.

The 3 circuit Chartres labyrinth (Petit Chartres)

The 3 circuit Chartres labyrinth (Petit Chartres)

This is the smallest possible version of a Chartres labyrinth. And there are just two barriers possible for it. Otherwise it does not work. Also three are not possible, but with four barriers it works aganin.

How should one name this type now? I suggest Petit Chartres because it is a sort of a basic element of the Chartres labyrinth. Still other names are conceivable.

It am speaking here about the type and not about the style. The petals in the middle and the lunations around the perimeter belong to the style.

To create this labyrinth is possible in a variety of ways, not necessarily in the manner described. For more read also the related posts below.

There is even a copyrighted labyrinth of this kind: The Story Path©. Warren Lynn and John Ridder of Paxworks have developed it and call the style “3-circuit-triune”. I do not know how they have found it.

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The Chartres labyrinth occurs in many variations. Here I speak of the 11 circuit Chartres labyrinth as a type. Some elements of the original labyrinth in the Cathedral at Chartres, such as the six petals in the middle and the lunations  around the outermost perimeter, belong to the style Chartres.

For me the type Chartres exists above all in the layout of the paths.  One goes in quickly (on the 5th circuit) and one quickly approaches the middle (6th and 11th circuit). Then follows the wandering through all quadrants. The access of the centre happens from completely outside (1st circuit) quickly about the 6th and 7th circuit into the centre.

Theoretically there are lot of possibilities to build similar types to the Chartres labyrinth. They can be found worldwide. However, the original Chartres labyrinth owns many special qualities which make it an extraordinary example among the Medieval labyrinths. Among others, it is self-dual and symmetrical.

Layout of the 11 circuit Chartres labyrinth

Layout of the 11 circuit Chartres labyrinth

Hence, the original can be divided in labyrinthine mathematics (11:2=5) in two equal labyrinths. I cut it into two parts, by omitting the 6th circuit. Thereby I get two new, yet identical 5 circuit labyrinths in a Chartres-like layout: I quickly reach the middle and finally enter the centre directly from the outermost circuit. The way in between shows the labyrinthine pendular movement, that Hermann Kern describes as characteristic for a labyrinth.

Layout of the 5 circuit Chartres labyrinth (Demi-Chartres)

Layout of the 5 circuit Chartres labyrinth (Demi-Chartres)

How should we now name this type of labyrinth? To me the name 5 circuit Chartres labyrinth seems properly to differentiate it from other 5 circuit Medieval labyrinths with another layout for the paths.
I would like to call it Demi-Chartres.

Just now you may see a nice example for the practical realisation in Vienna on the Schwarzenbergplatz in the temporary plant labyrinth to the European Year of Cultural Heritage 2018:

The temporary plant labyrinth on the Schwarzenbergplatz at Vienna © Lisa Rastl

The temporary plant labyrinth on the Schwarzenbergplatz at Vienna © Lisa Rastl

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By rotating or mirroring one will get dual and complementary labyrinths of existing labyrinths. Or differently expressed: Other, new labyrinths can be thereby be generated.
So I have three more new labyrinths as I can make a complementary one from a new dual labyrinth and I can make a dual one from a new complementary, which are identical. (For more see the Related Posts below).

Seen from this angle I have examined the still introduced 21 Babylonian Visceral Labyrinths in Knidos style and present here the variations most interesting for me. Since not each of the possible dual or complementary examples seems noteworthy.

Many, above all complementary ones, would begin on the first circuit and lead to the center on the last, which is yet undesirable.

Leaving out trivial circuits also will generate new labyrinths. This applies to the last two ones. If you compare the first and the last example you see two remarkable labyrinths: The first with 12 circuits and the last with 8 circuits, but using the same pattern.

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To say it more exactly, here I relate to the 21 row-shaped visceral labyrinths, still known from some of the previous articles (see Related Posts below).

The appearance is defined by the circuit or path sequence. With that one can construct the different and new labyrinth types (here 21). To this I use the once before presented method to draw a labyrinth (see below).

The path and the limitation lines are equally wide. The center is bigger. The last piece of the path leads vertically into the center. All elements are connected next to each other without sharp bends and geometrically correct. There are only straight lines and curves. This all on the smallest place possible. All together makes up the Knidos style.

Look at a single picture in a bigger version by clicking on it:

I think that by this style the movement pattern of every labyrinth becomes especially well recognizable. With that they can be compared more easy with the already known labyrinths.

Remarkably for me it is that only one specimen (E 3384 v_6) begins with the first circuit. And the fact that many directly circle around the middle and, finally, from the first circuit the center directly is reached. Noticeably are also the many vertical straight and parallel pieces in the middle section.

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Here it is about the decoding of the circuit sequences of the row-shaped 21 visceral labyrinths shown in the last article on this subject (see related posts below).

The question is: Can I generate one-arm alternating labyrinths with one center in the middle from them? That means no walk-through labyrinths where the also unequivocal path passes through, but is ending at an aim in the middle.
Maybe one could call them “walk-in labyrinths” contrary  to the “walk-through labyrinths”?

The short answer: Yes, it is possible. And the result are 21 new, up to now unknown labyrinths.

The circuit sequence for the walk-through labyrinth can be converted into one for a walk- in labyrinth by leaving out the last “0” which stands for “outside”. The highest number stands for the center. If it is not at the last place in the circuit sequence, one must add one more number.
This “trick” is necessary only for two labyrinths and then leads to labyrinths with even circuits (VAT 984_6 and VAN 9447_7).

The gallery shows all the 21 labyrinths in concentric style with a greater center.

Look at the single picture in a bigger version by clicking on it:

 

All labyrinths are different. Not one has appeared up to now somewhere. They have between 9 and 16 circuits, the most 11 circuits. They show between 3 and 6 turning points.

In these constellations there are purely mathematically seen 134871 variations of interesting labyrinths, as proves Tony Phillips, professor of mathematics.

There are still a lot of possibilities to find new labyrinths or to invent them.

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Further Link
The website of Tony Phillips

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We hope you had a good start in new year. This year again promises intersting insights into the world of labyrinths.

 

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