Discussion

It took me some time to answer due to other priorities, but here I am. I know nothing of Naert's book, or Conty, but I am quite happy to leave it that way. After all, my passion lies more with the Cretan and my own variations with that cadence, than with the Chartres labyrinth. For me, you both are far more knowledgeable on Chartres than I am. However, I have, due to my cadence approach, some points to make on Jacques' points of view.

I sketched the cadence of the old and new Sens, and restudied my drawing of the Reims and Chartres cadence. First it is important to note that my graphic representation is meant to make the radial distance to the centre visible. My cadence is about the distance to the centre because I thought (and think) that the centre is in the walkers mind while he walks. The interesting difference between Chartres on the one hand and Sens, Reims, and the Cretan on the other hand, is that you walk Chartres after a short "drink" from the centre, to catch a glimpse of the inspiration, outwards, and that you walk the others from the outside gradually to the centre.

In my graphics, a quarter path is a dot, and a half circle is two dots on the same distance-line. It is the switching of one circuit to another that is depicted by lines. Meaning: the walking on a circuit line is just the quiet part, the turns and jumps are the exciting stuff. That's were you find yourself suddenly closer to the centre, or more far away. Walking the circuit is digesting the effect of that move.

From that perspective, I also find what Jacques calls "spiral course" fascinating, because it is a sequence where you continuously move toward the centre, or away from it. But for that same reason, I may choose to find in the Reims labyrinth two sequences most fascinating, namely where one moves continuously from circuit 1 till 8, and then somewhat later, from 4 to 11. Just imagine the rising expectation, every turn brings you closer. So, for me, Reims would be: small up and down, big stairway, take a deep breath, another big stairway, small up and down, centre! With Jacques focus on spiral or folded courses, he misses the big stairway.

That is not meant to state the approach inferior or something, but just to say: a way of seeing is a way of not seeing.

Second example: Jacques shows that the Sens labyrinth is more Chartres-like than Reims-like, because it has three spiral courses. That is true. But when I look at the three cadences, obviously Sens and Reims are more alike, because both paths move "generally" from outside to inside and the Chartres path moves from inside to outside.

Looking from the cadence point of view, I came to find the Chartres to consist of two 5-circuit labyrinths, connected through the sixth circuit. I still think the Chartres to be unique in its regularity: you cannot divide the other labyrinths into an inner and outer labyrinth. They can be canonical or self-dual, but only Chartres is doubly self-dual.

I go to some length on this, because I want to make a point that I don't believe that the labyrinths were conceived, based on the rhythmical theory. It may be true, but just as well not be true. Looking from another perspective than Jacques' rhythm, one can also come up with plausible patterns of thought that organize different labyrinths into groups. I can see that J's work is an interesting way of looking at labyrinth patterns, and I think you, Jacques did a good job in working it out in its consequences. But for me it is not the final answer to the reason behind the medieval designs.

I remember that I once formulated the hypothesis that if you take a Cretan-like pattern and if you want to use that pattern in a four-quadrant labyrinth (because of the cross, because of the number four, because of the Roman style, I don't know), you need 11 circuits to make it reversible or self-dual. And it is in that context intriguing that the Cretan is also doubly self-dual, just like the Chartres. Frankly, I don't know why the Medieval labyrinths are like they are. But I can think their elegance of design makes them worthwhile to keep and to walk, because connecting with beauty is a good thing (to my opinion).

I hope this is what you, Jacques, meant by a straight comment or reaction on your big effort, because it must have been a big effort to create the theory and book.

Great! I like it like that!

I just read Willem's note.

My book's main objective was to be short.
(well, not really, but...)
What Willem notes in terms of large movements in the labyrinths I have noticed but not described in the book.

Several other canonical labyrinths have longer continuous sequences than Reims.

Five have continuous sequences covering the 11 lanes (all five are clockwise):

No 1 (Chartres historical): centrifugal, from lane 11, right.

No 5 (Reims II): centrifugal, from lane 11 right.

No 8 (inverted Reims II): centripetal from lane 1 right.

No 11 (inverted Chartres I): centripetal from lane 1 right.

No 19 (inverted Meta-Reims II): centrifugal, from lane 11 top.

Two have 9-lane sequences (both are counterclockwise and centrifugal):

No 3 (Chartres III): from lane 10 lower right.

No 6 (Reims III): like No 3.

An easy way to find these long sequences is to walk each labyrinth starting from lane 6 on axis "C" clockwise to its beginning, then walk it all counterclockwise.

It should be mentioned that this sequence quality, which is indeed remarquable, is also found in some labyrinths with the entry on lane 1: No 19 (11-lane), No 3 and No 6 (9-lane). This should be an argument to keep them in the "canonical" club.

Some other points I will have to think about a little longer.

Indeed, "a way of seeing is a way of not seeing".
That is why I value so much your "straight" comments (both of you).

I think John James' use of canonical and your definition are consistent.

In many geometry studies, there have been identified certain canons, which is to say, "rules," for proportions. A typical and recognizable one would be Egyptian figures. There have been many others.

I heard a presentation by someone who received a doctorate, basing his dissertation on a canon developed by a German monk (whose name escapes me.) We drew figures using the given proportions. I remember that the female was 7/8 as tall as the male. In another instance, there was a distinction between hips and shoulders. The two proportions are reversed for male and female, so that the same measurement applied to a male's hips and a female's shoulders, while a larger measurement applied to the man's shoulders and the female's hips.

Within certain traditions there are certain fixed values that have come to be used to symbolize certain things. Dividing a circle into four quadrants, for example.

So when John James is implying that there was already a canon, a set of rules, for the labyrinth, as it had been around for some 300 years prior to Chartres (minus the lunations and petals, which were a Chartres innovation). Therefore, following the rules and being canonical seem to mean the same thing within this context.

I think "canon" meaning a round of singing, or a certain symmetrical arrangement of a labyrinth pattern, is a different use of the same word.

I guess I don't have it on my website, but I have done a piece on the conversion of the Classical labyrinth to the Chartres or medieval type.

When you make the seed pattern for the classical 7-circuit, in order to make the center circular (rather than the normal semi-circle), you must create more space within the seed pattern by extending the brackets vertically by one additional path width, so that the bracket extends one path width beyond the dot. Also, to make the pattern symmetrical, you must off-set the two sides, so that the arms of the cross are not opposite each other (the left side is lower by one path width). At that point, you can make a circular 7-circuit that is doubly self-dual.

To make the Chartres seed pattern, you simply stack another bracket and dot beyond the first set. This is different than creating a classical 11-circuit, in which added bracket is nested inside the first bracket. If you do this and try to add the internal turns, you will get some islands. By stacking rather than nesting the additional brackets, the pattern changes and no islands result. Stacking the brackets is simply like doubling the seed pattern with twice as many brackets.

One of the distinctive features of the Chartres labyrinth is the two parallel entrance paths. None of the turns reach the lower vertical axis. There is always a path separating axis and turns. But when you enter into the labyrinth at the third path from the outside, as with Sens and Reims, then the pattern is changed so that some of the turns come to the vertical axis. In the case of Reims, most of them come to the axis.

By entering in the center of the pattern, the Chartres design divides the turns so that there are two pairs of turns both above and below the entrance points. That must be why it is so uniquely double double. Some thing that I see in many designs irks me, in that they mix the two types of seed patterns, incorporating both stacked brackets and nested brackets. I think they represent different things and don't find that they go well together.

I feel the urge to say that since I developed my theory independently of yours and before being aware of it, I have an instinctive reflex of always reformulating what you say into my own terms, which may of course be quite disturbing to both of you. I just want to say that I am aware of it.

I also want to insist on my understanding of Willem's short sentence: "a way of seeing is a way of not seeing". I feel it is true, and my interest in our discussion is precisely to bridge the gaps between what each one of us is seeing, particularly in regards to what I don't see or don't integrate into what I see.

I may nevertheless seem to be on the defensive and trying to vindicate my point; this is I think some kind of friendly fight, combat or joust, in the manner of very sophisticated martial arts, where our arms are words, sentences and ideas, and no harm is done, but great satisfaction and admiration comes from well-defended positions and well-directed attacks against these positions. But then again, we always have some attachment to our personal positions.

I may surprise you by saying that I have more pleasure in the game itself and in the building of theories and models than in the finding of an elusive truth. Of course, the models and theories have to be operational, which may be the same as being true, but I think not. A good model is one that works like a fine clock. The fact that the clock keeps the time may be some "given" bonus. A clock may also be seen as a working sculpture.

Robert: your "piece" about deriving the Chartres labyrinth from the Cretan is in your Chartres Book, p. 12-13. I think that what you call "seed pattern" is quite exactly what I call "key".

Also your idea of having to "rebuild" the Reims labyrinth from the Cellier drawing into a more regular geometry is in your book, p. 32.

Referring to John James' use of the word canonical, the Chartres labyrinth (or its design) cannot be considered canonical just because it was repeated everywhere. I consider (and I think you also do) Reims and Sens as also canonical but they were not repeated anywhere (that we know of).

Robert and several other authors use the word "classical" to designate the Cretan labyrinth. My reason not to do that is double:

From an historical point of vue, the term classical often refers to classical antiquities, which should also include the Roman period, and obviously the Roman labyrinth. But this meaning would exclude the Medieval labyrinth, which is not correct.

From a graphic-structural point of vue, the classical labyrinth is the one with a single path, built according to some symmetrical geometric scheme. I know of only 4: the Cretan, the Roman, the Medieval, and the Pima-Maricopa-O'Odham (Arizona Indians, which is a very sophisticated geometrical variation of the Cretan labyrinth but I think may be independent). In practice the classical labyrinth is any labyrinth which is not a maze.

Last week Willem mentioned the long continuous sequences in the Reims labyrinth as being an important aspect of its interest. Please tell me if I understood correctly your idea when referring to longer sequences in other canonical labyrinths.

I have not yet "digested" completely the idea of self-dual and doubly self-dual. It seems to make sense. We will have to apply the cadence diagram method to other canonical labyrinths with long continuous sequences to see the real meaning of that idea of self- and doubly-self.

Robert, I think the word canon as designating a musical form (multiple voices singing the same melody and entering successively, sometimes each on a different degree of the scale) can not be applied directly to a certain symmetrical arangement of the parts of a labyrinth. I think it has to be kept on a quite abstract level, as being a rule or a set of rules to which a thing does or does not conform itself. Nevertheless, the disposition of successive spiral-courses on successively different lanes strongly suggests the idea of a musical canon.

I think as an example of graphic canons for drawing the human figure - male and female - you were referring to Dürer, but I cannot verify that easily (yes, of course, Google...).

When I associate Sens with Chartres rather than with Reims, it is from a structural point of view (same "template"). The practical point of view, the one which refers to the actual effect (kinesthetic-psychological) of walking the path, is effectively related with these sequences that Willem mentions and the centrifugal-centripetal and rhythmical effects, which are precisely the objective of constructing a given labyrinth design.

What I call the "key" of the labyrinth is the means to "interpret" a given "template" into a particular labyrinth with specific sequences and rhythmical effects.

Within the next year, you are going to be blown away by the work of a man named Ben Nicholson. An architect by training, he spent a year studying floor patterns in Florence, Italy.

He has done extensive work on meander patterns. And he has designed a whole system of understanding labyrinths according to the key (seed) pattern.

A few of his drawings are currently on display at the Whitney Museum of American Art in New York City as part of an exposition called "Architecture by the numbers." He calls some of his work with Pythagorean principles "archeogeometry." He has a book that should be out in the fall. He will also be putting a lot of his material on a new website.

I'm sure he would have a lot to say about these concepts of canonical, etc. Perhaps I could send him a note and have him look at what you have posted on your website.

Sounds ominous!
Hope I will not be blown away so violently that nothing is left on my table!

I'm glad this discussion goes on well, although it can be remarkably time-consuming for me. Again I had to study to understand some points. The Cretan world seems much simpler, even if I make it more complex with my multiple forms.

Thank you, Robert for your reported explanation of the stacked and nested brackets and Jacques for noting where I could read about it.

When I look at a labyrinth from the viewpoint of someone who is concentrating on getting to the centre, stacked brackets mean a gentle way of moving closer or more far to the centre, because you always move to an adjacent path. Nested brackets always mean a more jumpy path. The remarkable pattern of Chartres ensures that you only make jumps at the very beginning and the very end.

When I try to express the feeling of walking a labyrinth where there are many jumps in the distance to the centre, it is like riding a train on a very uneven track. The train moves forwards, of course, but simultaneously you are shaken left and right in an unorderly fashion. Or compare it to a rollercoaster ride, or a cakewalk. People get in these things for fun, but it is difficult to meditate there.

When I perceive labyrinths as a kind of focussing lens, channelling energies between heaven and earth, you can have either a smooth, even encounter with that stream, or an irregular, maybe jolted one, due to the irregularities of the lens. Like distorting mirrors on a fair. May be fun, or it can come as a shock.

Having noted that, I remembered having drawn in 2001 a Chartres labyrinth with entrance and exit exchanged, thereby making the cadence move gently from outside to inside, just like Reims. I have enclosed both pictures in Acrobat pdf format.

There are two funny things about it: I called it at the time "an inverted Chartres", but it is obviously not the same as your inverted Chartres.
And secondly: I could not find the proper reason in Jacques' terminology why this is not a canonical labyrinth, although there is something funny with the AE axis. I can see, by the way, that this is a combination of a nested and stacked seedpattern. But it still looks almost Chartres.

All my thanks to both of you for accepting to continue this discussion.

Willem:
Very interesting, very surprising!

Indeed, your inverted Chartres was also one of mine, except that I did not keep it because some of the crossings along the main axis (the key, or seed pattern) are more than 2 level deep (see my book p. 6, top).

The "template" (the structure of the "B" "C" and "D" axes, that is: the arrangement of the crossings along these axes) is canonical. I also called that template "inverted Chartres", as you did and for the same reason, as also the family of 3 labyrinths that are derived from it through different keys.

Of course we could accept 3-level-deep keys. But the known historical labyrinths that have 3-level keys are not otherwise canonical. And the aesthetic appearance of the drawings is inferior, because somewhat crowded. Also the resulting labyrinths have more long radial crossings along the main axis, which I think you rightly identify as the cause of the "rough ride" effect of some labyrinths.

This last aspect may be relevant in the discussion of stacked and/or nested "brackets" (or crossings) mentioned by Robert.

That being said, I had come across that specific design of your inverted Chartres and several others, both through my deductive method (as explained in the book) and also by reversing, as you did, designs that I already had. I had called that method "sequential inversion". Your method, if I understand correctly, was to reorient the entrance radial paths and then to make a lateral inversion into a mirror image to retrieve the entrance on the left. Mine was to reverse the sequence of the lanes inside out, keeping the entrance path where it was but extending (or shortening) it to the appropriate lane, and relocating all the crossings on the appropriate new lane. The final result is the same (including the inversion of the template, axes "B" and "D").

Two labyrinths that I kept as canonical also happen to be inversions of others, like yours with Chartres (my Nos 7 and 8 are sequential inversions of Nos 4 and 5 respectively). I hesitated to keep them when I noticed that, but since the visit sequence is so different, I decided to keep them.

I was from the beginning fascinated with the Reims labyrinth's A-E axis (main axis) design:

The current discussion about nested/stacked crossings or brackets and long crossings along the main axis as a source of "rough riding" just reminded me that the Reims labyrinth is the one with the simplest key or seed. The only nested part has also the function of bringing the entry to lane 3, instead of lane 1 which would have been an imperfection. This is why the ride is particularly smooth. And the graphical appearance so elegant.

In fact, the single nested crossing along the main axis is necessary for the labyrinth to be operational. A plain key (with no nested crossings) does not yield an operational labyrinth (there is an island on lanes 2-3, lower left quadrant, and also on lanes 9-10, lower right quadrant). None of the 6 possible canonical templates is operational with a plain key. But the "plainest" key is that of the Reims labyrinth.

The only functional labyrinth with a straignt plain key (no nested crossings) that I know of is the primitive form with no quadrant division (Kern, No 11), similar to the so-called "Otfrid" labyrinth but simpler, that I call the "boustrophedon" because each lane is traveled successively in reversed direction, like the ox used to do when ploughing the field, in the times of non-lateralized ploughs.

I am currently working on Willem's sequences/cadences, which indeed are becoming very interesting.

1 -I have been working on the cadence graphic method. I have been drawing free-hand and very rough. Nevertheless, here is something:

As I said before, five canonical labyrinths have long 11-lane continuous sequences. I tested these labyrinths with the cadence diagram method.

2 - There are 2 different lenghts for these 11-lane sequences:
14 units for labyrinths 1, 5, 7, 11,
16 units for No 19

These units are time-units or angular measure-units corresponding to a quadrant or a quarter-circle.

Is that difference significative?

3 - According to the cadence diagrams, all these labyrinths are self-dual, symmetrical, reversible, retrogradeable and palindromic. (I think that, in our context, these five words are synonymous).

Again according to these diagrams, I think none is doubly self-dual (if I understand correctly the meaning of this expression): - self-dual means having a center of symmetry, and therefore being reversible etc. - doubly self-dual is each half having its own center of symmetry, and therefore being reversible upon itself.

This is what I understand from your (Willem's) Caerdroia article, about the Cretan labyrinth cadence diagram: like a two level hierarchical symmetry.

I have not found that in the Medieval labyrinth.

On the Chartres cadence diagram, there is an important partial section which is symmetrical, starting from the fifth node (not counting the node on line 0), that is: starting with the double node on line 10, to the double node on line 8 just before (2 nodes before) the general center of symmetry.

This section is symmetrical unto itself (same thing for the corresponding section in the second half of the diagram).

But this section is not a complete half of the diagram (or of the labyrinth), neither is it centered on the half.

I have not seen that kind of important partial symmetries in the other diagrams. I will go on studying the other canonicals.

4 - I think that the idea of double self-dual labyrinth has been developed from the kind of diagram on lower page 37 of Robert's book, that I think Craig Wright also used in his book (I don't have the book: I had borrowed it from a library).

I have to say I never was able to understand exactly the purpose of this diagram, except to give an idea of the general symmetry of the path, without any strict logic or intention behind that. I think the idea of double-self-duality cannot be derived from that diagram. I think both of you must have had a different way of deriving that notion.

I found the word self-dual in the article by Tony Phillips. Visually I was triggered by the double meander pattern of the classical labyrinth. In fact, when I wrote the article, I had not walked a Chartres-like labyrinth, other than the Saffron Walden and the Hilton labyrinth. Due to the strange feeling of that Hilton one, I needed to find myself a graphical formalism to depict the pattern, and so I came upon the cadence way. From there on, after understanding the classical pattern, I finally became more familiar with the Chartres one. Before that I had never quite understood how it worked, and had kept myself away from it.

Your statement about the partial sections in the Chartres which do not connect straight on, but via the double node on line 6: Thatıs correct. It seems a kind of transition path is needed. That is also the case in the classical labyrinth, where the 4th circuit serves that purpose and connects two 3-circuit patterns. In the Saffron Walden case, you have it at path 6 and 12. So the Chartres pattern is two 5-circuits, connected by an intermediary circuit. That circuit is also used for the beginning and the end. Just like that, at Saffron Walden, the start of the threefold pattern at circuit 15 is reached by two "stairs" at levels 6 and 12, and ended the other way around.

It does not surprise me that the other canonical labyrinths are not doubly self dual. Maybe a better mathematician (someone like Tony Phillips) would be able to prove that Chartres is the only possible solution.

I will meditate on that some time.

I am not inclined to see these problems as formally mathematical, but simply a matter of clear visualization.

I have sent my www address to Tony Phillips. We will see what he can do with that (if ever he is still interested and available enough).