The supposed uniqueness of the Chartres labyrinth

June 14th 2004

Résumé en français

La supposée unicité du labyrinthe de Chartres.

Le présent article est une réponse à celui de Pierre Rosenstiehl « Comment le "Chemin de Jérusalem" de Chartres sépare les oiseaux des poissons », publié dans « M. C. Escher : Art and Science » (New York, 1986-1988).

On réfère encore à cet article, que certains semblent considérer comme l'ouvrage définitif sur la mathématique du labyrinthe médiéval. Mon intention est de montrer que ses présupposés sont faux, et que par conséquent ses conclusions sur l'unicité du labyrinthe de Chartres sont aussi fausses. J'ai en effet découvert qu'il existe vingt labyrinthes médiévaux canoniques (ou parfaits), parmi lesquels au moins trois ont existé historiquement : ceux de Chartres, de Reims et de Sens.

Après plusieurs communications avec lui, je dois conclure que, au 13 novembre 2005, Monsieur Rosenstiehl ne semble pas intéressé à poursuivre la discussion.

Object of that paper

The present paper is a response to Pierre Rosenstiehl's paper "How the Path of Jerusalem in Chartres separates Birds from Fishes" published in "M. C. Escher: Art and Science" (Coxeter, Emmer, Penrose, Teuber, ed.; North Holland, Amsterdam, New York, Oxford, Tokyo, 1986, 1987, 1988).

Rosenstiehl's paper is still being referred to (Tessa Morrison: The Typology of the meandering Symbol and The Geometry of History; Vanessa Compton: Thesis: Rosenstiehl at Chartres: The Mathematics of the Cathedral Labyrinth). It seems to be considered as some kind of definitive statement on the mathematics of the Medieval labyrinth. I intend to show that it is built on false assumptions and that its conclusion about the "uniqueness" of the Chartres labyrinth is of course also false. I have indeed discovered that there are 20 possible "canonical" Medieval labyrinths, of which at least 3 have existed historically: Chartres, Reims and Sens.

As of Nov. 13th 2005, after several communications with him, I must conclude that Mr Rosenstiehl is not interested in reopening the discussion. Maybe my readers could help me convince him.

Preliminary considerations

In January of 2004 I have published a short book on the rhythmical structure of the Medieval labyrinth. As a consequence of that rhythmical approach, I had been able to develop the notion of canonical (or perfect) Medieval labyrinth and to derive 19 different such canonical labyrinths, including the Chartres and Reims labyrinths, and the recently discovered design of the Sens labyrinth (which by then I did not know).

Reading Rosenstiehl's article, I realized the importance of the fact that part of my method was empirical (which was indeed acknowledged in my book), and I decided to formalize it into a systematic exploration. I then discovered (to my shame) that my previous method had left out one possible Canonical labyrinth, which brings the total to 20.

But this revision of my method also brought me confirmation of its validity and of Rosenstiehl's errors.

Rosenstiehl's errors

First error: incomplete observation of known models

Rosenstiehl describes the medieval labyrinths as sharing "the same unicursal structure: a single string starting from the outer edge finishing in the center (...), never crossing itself, makes 32 turns with a certain regularity along two orthogonal axes, forming twelve layers including the tip of the string" (...) "even a small alteration of a single detail would disrupt the magical regularity of the maze". (I assume "string" to be a synonym of path, with a slight reference to Ariadne's thread; layer means level or lane).

From this, it seems that the author considers not only the general design of the labyrinth, but also the possible peculiarities of its path. Indeed, he mentions that "even a small alteration of a single detail would disrupt the magical regularity of the maze". However, he does not recognize explicitly such differences as can be found between the paths of the Chartres and Reims labyrinths. Indeed, he goes on: "As there is only one way to trace it, we have called it the dodecamaze". Therefore either he considers the Chartres and Reims labyrinths as similar or equivalent, or else he rejects the Reims labyrinth as not being a "dodecamaze". In both cases, he is in error: the rest of his demonstration can only be tautological and indeed there can be only one Medieval labyrinth.

Second error: the so-called "law of alternation"

In more technical terms, the author says that "turnabouts systematically alternate with straight runs of the path on three semi-axes" and calls that "the law of alternation".

Let us first examine the horizontal semi-axes. It is easy to see that this observation is erroneous. On the Chartres labyrinth, on the left semi-axis, there is a pair of "straight runs" located on lanes 10 and 11; on the right semi-axis, the pair is located on lanes 1 and 2. On the Reims labyrinth, those pairs are located respectively on lanes 7-8, and 4-5. Therefore there is no "law of alternation". Furthermore, other locations of these pairs are possible and will produce different labyrinths otherwise perfect.

If we now look at the vertical semi-axis, the "law of alternation" seems to hold true, but again there is a possibility of a different disposition of the "turnabouts" and "straight runs" that does not conform to the "law of alternation" and that will also produce different labyrinths otherwise perfect (the other arrangement has "straight runs" located on lanes 1, 6, 11).

Since the author does not acknowledge these possibilities, and considers only his supposed "law of alternation", it follows that his whole argumentation about the uniqueness of Chartres is not valid.

Third error: incorrect formalization and argumentation

The author describes the "throat" (the main semi-axis) of the Chartres labyrinth as "a system of nested pairs of height two" (which is correct); he suggests the possibility of different arrangements but his description of them is inadequate and seems limitative. He then goes on to conclude that there is only one possible labyrinth within this system of geometrical rules (which include those that I have described as erroneous). Of 675 theorical possibilities "tediously" tested, 674 produce islands or closed sections of the path; the remaining one is, of course, the Chartres labyrinth. Given the arbitrary initial limitations in the definition of the "dodecamaze", the contrary would indeed have been very surprising.

My response

After my first, partly empirical, exploration of the possible canonical labyrinths, I had no way of responding to that argumentation, except by saying that I had indeed found 19 of these canonical labyrinths. In my more systematic research, I had to formalize the possibilities of distributing the crossings along the main axis. The default structure of the elementary labyrinth is entirely composed of first-level crossings along the main axis. The building-up of the Medieval labyrinth calls for adding some second-level crossings, which are necessarily nested inside a double-lenght first level crossing. What has to be specified is the location of these second-level crossings.

Because of the necessary symmetry of the path, both sides of the main axis will be similar but inverted. Therefore we have only to consider one side of the axis. The second-level crossings can be located starting on any or all of the odd-numbered lanes (because of their direction, which, along this axis, is always from a counterclockwise lane to a clockwise lane), and since they cover two lanes, only five locations are possible. This is easily represented on a five-column table, the columns being numbered 1, 3, 5, 7, 9, and the lines representing each of the 32 possible "throat" structures or arrangements (that is, 2 exponent 5). Each of these possible structures then has to be tested against each of the 6 general "templates" resulting from the different possible arrangements of the three other semi-axes (mentioned under "second error" and described on my site and in my book). The total number of theoretical possibilities is 192, of which only 20 produce functional labyrinths, which are indeed the 20 canonical labyrinths.

Even if we narrow the research to the Chartres labyrinth arangement of the three secondary axes, there are three different possible "throat" arrangements, resulting in three canonical labyrinths: the historical Chartres, the newly discovered Sens, and a new one with entry on lane one. The author should at least have discovered these.

My intention in using this method was limited to canonical labyrinths, who incorporate important symmetry restrictions. It could of course be generalized to non-canonical labyrinths, which should yield labyrinth designs in very large numbers. And since the author's limitations on symmetry are not explicit, his argumentation should not have excluded these non-canonical labyrinths, which makes his conclusion all the more surprising.

What about Birds and Fishes?

Rosenstiehl's title may look somewhat strange in relation to the content of the paper. One has to remember that the paper was published in a collective book under the general title "M. C. Escher: Art and Science". In the middle of his argumentation, the author applies some topological transformation (which he calls "Escherization-anamorphosis") to the drawing of the Chartres labyrinth. Out of that comes an Escher-like tessellation composed of birds and fishes. This transformation brings into evidence a little known property of the Medieval labyrinth: the fact that splitting the wall along the main axis produces two separate systems of walling (see my "St-Michael's" labyrinth), each of which can be touched with the same hand the whole length of the path (therefore allowing the removal of one of the systems without destroying the path structure of the labyrinth, which is indeed remarquable). But that was not the author's point. I think that this was simply a game to justify the presence of that paper in the Escher collection. What puzzles me most is that the author seems to consider that transformation as part of his argumentation.

Conclusion

Except for its possible interest as a practical joke, and for a couple of interesting remarks made "en passant", this paper about the Chartres labyrinth being the only Medieval labyrinth possible seems to me completely irrelevant.