Discussion

It (the labyreims.com site) is very interesting, and very nicely executed!.

You should have a look at the article of P. Rosenstiehl cited in my bibliography.
He also had a feeling for the rhythm of the Chartres maze.

I did not get involved in the study of medieval mazes, but I did notice that the Chartres maze has the same "throat" as the s.a.t. maze 0.5.4.3.2.1.6.11.10.9.8.7.12
[In fact in the 860 Auxerre ms. the two configurations are explicitly related!] .

0.5.4.3.2.1.6.11.10.9.8.7.12 is the "double" of 0.5.4.3.2.1.6 just as the Cretan maze is the double of 0.3.2.1.4. I do not think, however, that either of these mazes were known in antiquity.

In contrast the Rheims maze is not related in this way to a more primitive labyrinth.

I should mention also that

1. not all the Roman mazes are exactly as you describe: there is a series of pompeian-style mazes where the fourth quadrant is altered so as to give a more symmetric overall appearance. See my article in Leonardo - also cited in my web bibliography. Daszewski also noticed this.

2. I do not think it is correct to say that the Cretan maze never had a central space; but I am not sure about this point, and I don't have Kern at hand.

3. The Cretan maze does not "spiral," at least not for me. Rather the path oscillates back and forth.

I took some time on your message

I went back to your labyrinth pages to refresh my memory

Now back to your message, which would deserve a long response, which maybe I won't be able to do now, but I will try.

I don't have access now to Rosenstiehl's article.

I like the word "throat", which is new for me in that context.

Indeed, the Chartres labyrinth has the same throat, or key, referring to my terminology, as the 11-lane s.a.t. labyrinth you mention.

Your reference to the Auxerre ms seems to refer to Heiric of Auxerre's labyrinth.

Are the two configurations you mention the same as those I mention at that link?

I am not yet very familiar with the notions of doubles and of a given 11-lane labyrinth being the double of a 5-lane labyrinth. I understand the words but I don't visualize the reality.

Now to your numbered paragraphs:

1.- I have found in Kern (chapter XII) several labyrinths related to the Roman type, but with quadrants symmetrical to each other, all post-Gothic. I cannot find an example with only one quadrant inverted for symmetry.

I would be inclined to consider these as variations away from the typical Roman design with four (or sometimes three or six) identical quadrants.

2.- I also think that the original Cretan design is a very specific notion: it is built from a cross, by extending successively the four arms of the cross in such a way that a labyrinth path results. I think there is only one way to do that (of course, it can be done with 3, 11, 15 lanes). The method of building it from the seed pattern yelds also only one solution if you start with a cross.

3.- The path of the Cretan labyrinth is not spiral, of course, but the design is built over a spiral geometric subjacent template.

This is why the transformation from Cretan to Medieval forces the displacement of the two horizontal arms of the cross.

See the Pontremoli labyrinth (Kern No 278): the exterior and interior circles are concentric but the median "circles" are still aligned and facing each other as the original arms of the cross and therefore they form a spiral (just follow them with your finger).

See also Kern 180 (Wandalbert of Prüm) and 194 (Walahfrid Strabo) which show the obligation of choosing between spiral and concentric, in relation with the central medallion.

The possibility of having a central medallion is related to the concentric subjacent geometry: the spiral geometry does not allow a round medallion.

The spiral construction is quite obvious in the seed pattern method of constructing the Cretan labyrinth.

I see the link and yes, your interpretation agrees exactly with mine.

I'm copying here the level-12 section of the tex-file of a work on SAT mazes that I wrote in 1988 and circulated at the time. Part of it was published as "Topologia dei Labirinti" in "L'Occhio di Horus : itinreari nell'imaginario matematico" edited by Emmer around then.

[K:xxx] refers to figure xxx in Kern (German ed.).
Figure 12 is the maze-type illustrated on my first web page (in an altered form).
Figure 6 is the Chartres pattern.

(Tony's original formulae edited by JH who could not translate them into HTML code).

0~3~2~1~4~7~6~5~8~11~10~9~12 is the product of three half-Cretan mazes; see Figure 12. It occurs in many medieval manuscripts, beginning with the Codex of Otfrid of Weissenburg, written in the period 863-871 [K:172]. Here and in [K:186], the maze has a Christian interpretation; in other examples [K:185, 191, 194], it is related to the story of Theseus and the Minotaur.

0~5~4~3~2~1~6~11~10~9~8~7~12 is a diploid of 0~5~4~3~2~1~6, equipped with gates to make it into the non-simple cathedral maze of Figure 6. It is drawn on a page of a Historical-geographical encyclopedia written around 860 in Auxerre [K:182].

0~5~2~3~4~1~6~11~8~9~10~7~12 is a diploid Cretan maze: its nucleus has twice the number of L's. This maze occurs on tiles from the abbey of Toussaints in Châlons-sur-Marne (abbey destroyed in 1544) [K:254]. It is also common in Scandinavian Troy-towns [K:565, 566, 568-569], and frescoed into church walls in those countries [K:588, 594, 595, 596].

I thought I was the first to think of the small scribbles on the Auxerrre labyrinth as doors or gates to permit the transformation from an S.A.T. into Chartres (and vice-versa, of course).
This is very interesting.

I am working on the methods of drawing the Cretan labyrinth which brought back to me your doubt as to that labyrinth having a spiral structure.

I just posted on my site an old drawing that I had made to show the spiral structure of the Cretan Labyrinth