An addition to my book

The rhythmical structure of the Medieval labyrinth

A revision

That short book is about the rhythmical structure of the Medieval labyrinth. As a consequence of that rhythmical approach, I have 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 authentic design of the Sens labyrinth (which by then I did not know). I have already acknowledged that part of my method to arrive at that series of labyrinths was empirical: that is, it was not systematical and did not guarantee completeness.

Recently, I read Pierre Rosenstiehl's 1986 article on the mathematics of the Medieval labyrinth. His conclusion is that the Chartres labyrinth design is the only possible form of the Medieval labyrinth. Of course, I did not agree with this conclusion, since I knew 19 such labyrinths, of which 3 have even existed historically. But in order to write a response ("The supposed uniqueness of the Chartres labyrinth") to that article, I had to revise my own methodology and make 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.

In my first research, the exploration of the general templates and families had been systematic. I now had to formalize the possible distributions of the crossings along the main axis, or "throat", of the labyrinths. 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 first-level crossings. What has to be specified is the location of these second-level crossings. Because of the symmetrical structure of the canonical labyrinth, both sides of the main axis are identical but inverted. Therefore only one side needs to be described.

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 (2 at the fifth power). 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 axes. The total number of theoretical possibilities is 192, of which only 20 produce functional labyrinths, which are indeed the 20 canonical labyrinths.

Since the addition of a new labyrinth to the original series made necessary a modification of the numbering system, I decided to revise it completely, based on the "natural" order suggested by the five-column table read as a five-position binary numeration system. The order of the 6 families stays the same; the order of the labyrinths inside some of the families had to be modified, and of course, the addition of the new labyrinth pushes down all those who follow it.

You will find here the previously missing labyrinth, in its right and left versions. You will also find a revised descriptive-analytical table of all the 20 canonical labyrinths, in their new order and with their new numbering, and also with their old numbering to facilitate the reference to the book. There is also the combinatory table of the possible keys, with each of the 6 families corresponding to the 6 general templates.