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A mathematical notation method for Medieval labyrinthsJune 14th 2004
This paper has been published in Jeff Saward's review
Object of that paperMy intention is to propose a new mathematical method for describing the Medieval labyrinths. This method is bi-univocal (each description refers to only one labyrinth design and describes it completely: it is therefore its "signature"). It can also be readily used for rhythmical and other formal analyses. In my book on the rhythmical structure of the Medieval labyrinth (of which the text and some other sections are on my site), I have already proposed a descriptive-analytical method for the notation of the canonical medieval labyrinths. That method is static-structural: it is more related to the geometric structure of the labyrinth drawings. The method proposed here is dynamic and sequential: it is descriptive of the path of the labyrinth. Some of the ideas behind the development of this method come from Robert Ferré, Craig Wright and Tony Phillips. The impetus to really do it came from my mathematician friend Guy Simard, who also helped me in certain details of the formalization. It should be said here that, unless otherwise stated or clear from the context, I always refer to what I call the "script" version of the labyrinths I am discussing, that is, the way it would have been drawn on a medieval manuscript: Chartres being without decorative motifs ("lunations", "labryses", "petals"), Reims being round and without "bastions", and the lines representing the walls having no measurable width. For simplicity, I call "Chartres" the design which eventually became known as the Chartres floor labyrinth, even if it first existed on much older manuscripts having nothing to do with Chartres. I do the same with the Cretan labyrinth. References are made to Kern's English edition.
Preliminary considerationsA simple numerical notation method is already in use for the Cretan and other full-circle (non-quadrant) labyrinths. The lanes are usually numbered starting from the exterior. 0 is used to designate the spatial region exterior to the labyrinth. The center of the labyrinth (and end of the path) is designated by the next number greater than the number of lanes. Each lane visited by the path is enumerated sequentially, according to the sequence of visit. The Cretan classical 7-lane labyrinth is notated thus: 0 3 2 1 4 7 6 5 8; its 11-lane version is notated thus: 0 5 2 3 4 1 6 11 8 9 10 7 12. The notation of the Medieval labyrinth has to take into account the division into quadrants and the resulting variation (in number of angular length units) of the path segments. Robert Ferré and Willem Kuipers have made original graphical analyses of the path of different Medieval labyrinths. These graphical analyses need the labyrinth to be first described in numerical terms, but these numerical descriptions have not been used as a notation method independently of the graphical analyses. I have developed a simple rhythmical notation method using dots and dashes to represent one- and two-quadrant segments, with bars to isolate rhythmical motifs. Craig Wright has also developed a similar notation method using the letters Q and H to designate quarter- and half-circle segments, but without indication of rhythmical motifs (except for the center of symmetry, which is indicated by a bold letter). These two simple notation methods do not refer to the lanes being visited by the path, and therefore do not describe completely the labyrinth or the structure of its path. My first notation of the Chartres labyrinth: . . - - . . | - . - . - | . . | - . - . - | . . | - . - . - | . . - - . . Craig Wright's notation of the Chartres labyrinth: QQHHQQHQHQHQQHQHQHQQHQHQHQQHHQQ A complete notation method for the Medieval labyrinth should describe both the length of each successive segment and the sequence of lanes being visited. Accessorily, the description could include the main direction of rotation of the path of the labyrinth, if it is counterclockwise. In general, Medieval-type labyrinths are clockwise, one notable exception being Villard de Honnecourt's Chartres labyrinth (K. 344, see my site: particular topics).
Description of the method proposedThe numbering of the lanes of the labyrinths is done starting from the exterior. Therefore the sequence of lanes visited by the path of the Chartres labyrinth is: 0 5 6 11 10 9 8 7 8 9 10 11 10 9 8 7 6 5 4 3 2 1 2 3 4 5 4 3 2 1 6 7 12 One way of indicating the length of the segments is by repeating the lane number for the two-quadrant segments. Thus Chartres becomes: 0 5 6 11 11 10 10 9 8 7 7 8 9 9 10 11 11 10 9 8 8 7 6 6 5 4 4 3 2 1 1 2 3 3 4 5 5 4 3 2 2 1 1 6 7 12 which seems to be the notation used by Willem Kuipers in constructing his "cadence" graphs (Caerdroia No 31). I would call this the "expanded" form of the notation. One can also use in front of each lane number, a prefix which would function both as an indicator of the length of the segment and as a separator between two subsequent numbers or segments. I propose "-" and "+" for one- and two-quadrant respectively. The separator before 12 has no length signification. Thus Chartres becomes: 0-5-6+11+10-9-8+7-8+9-10+11-10-9+8-7+6-5+4-3-2+1-2+3-4+5-4-3+2+1-6-7-12 This is a complete description of the Chartres labyrinth: indeed, given a set of 12 concentric circles divided into 4 quadrants, one can trace the path of the labyrinth. It can therefore be considered as the "signature" of the Chartres labyrinth.
Reading the signatureUsing this "prefixed" form of the notation, it is relatively easy to "read" directly from the signature a certain number of sequential or rhythmical motifs just by examining the groupings and sequences of numbers and prefixes. Here are some examples pertaining to the Chartres labyrinth. Example 1: 0-5-6+11+10-9-8+7-8+9-10+11-10-9+8-7+6-5+4-3-2+1-2+3-4+5-4-3+2+1-6-7-12 The underlined section is a centrifugal (numbers decreasing) complete sequence covering successively all of the 11 lanes (idea from Willem Kuipers). Example 2: 0-5-6+11+10-9-8+7-8+9-10+11-10-9+8-7+6-5+4-3-2+1-2+3-4+5-4-3+2+1-6-7-12 The underlined pairs are long multiple-lane crossings being jumps between distant lanes, which, when traveling the labyrinth, produce the effect of a somewhat "rough ride" contrary to the idea of a long continuous sequence (idea from Willem Kuipers) Example 3: -5-6+11+10-9-8+7-8+9-10+11-10-9+8-7+6-5+4-3-2+1-2+3-4+5-4-3+2+1-6-7 The underlined sections are three-step "circuits" or "round courses", which are found in all Medieval-type labyrinths. In this case there are three, of a spiral form, and respectively centripetal, centrifugal, centripetal (my idea). Example 4: a)
These are what I call "bridges" or "detours", which are either a) long (++) or b) short (--). These different motifs or others may be easily indicated using parentheses, brackets, accolades. All of these indications are easy to use on the web, in HTML code. Underlining is slightly more awkward to use but is also interesting (used here to indicate the center of symmetry). -5-6 [+11+10] -9-8 (+7-8+9-10 {+11) -10-9 (+8-7+6-5+4) -3-2 (+1} -2+3-4+5) -4-3 [+2+1] -6-7
Signatures of some Medieval labyrinths: the canonical labyrinthsI have explained on my site (www.labyreims.com) what I mean by canonical. Among the historically known Medieval labyrinths, three are canonical: Chartres, Reims and Sens (new design recently discovered by Craig Wright). A particularity of canonical labyrinths is that they have only one- and two-quadrant segments, which makes their notation easier. The numbering refers to my repertory of canonical labyrinths (see my site or book). The circuits and some particularly long sequences are indicated (by ( ) and { } respectively). Drawings of labyrinths No 3 (1), 1 (2), 2 (3), 4 (4), 6 (5) and 9 (9) are on my site (my old numeration between parentheses). No 1 (2) (historical Sens): 0-3-2 (+1-2+3-4+5) -4-3+2+1-6-7 (+8-7+6-5+4) -5-6+11+10-9-8 (+7-8+9-10+11) -10-9-12 No 2 (3) (new): 0(+1-2+3-4+5) -4-3+2+1-6-7-10-9 (+8-7+6-5+4) -3-2-5-6+11+10-9-8 (+7-8+9-10+11)-12 No 3 (1) (historical Chartres): 0-5-6+11+10-9-8 (+7-8+9-10 {+11) -10-9 (+8-7+6-5+4) -3-2 (+1} -2+3-4+5) -4-3+2+1-6-7-12 No 4 (4) (historical Reims): 0-3-2 (+1-2+3-2 {+1)-2-3+4+5-6-7 (+8} -7+6-5 {+4) -5-6+7+8-9-10 (+11} -10+9-10+11) -10-9-12 No 6 (5) (new): 0 {-5-6+7+8-9-10 (+11} -10+9-10 {+11) -10-9 (+8-7+6-5+4) -3-2 (+1} -2+3-2 {+1) -2-3+4+5-6-7} -12 No 9 (9) (new): 0-3-2 (+1-2+3-2+1) -2-3+8+7-6-5 (+4-5+6-7+8) -7-6+5+4-9-10 (+11-10+9-10+11) -10-9-12 No 12 (11) (new): 0+1+2-3-4 {-7-6 (+5-4+3-2 {+1)} -2-3 (+4-5+6-7+8) -9-10 {(+11} -10+9-8+7) -6-5} -8-9+10+11-12 No 19 (18) (new): 0-1-2 (+3-2+1-2+3) -2-1+4+5-6-7 (+8-7+6-5+4) -5-6+7+8-11-10 (+9-10+11-10+9) -10-11-12
More signatures: the non-canonical labyrinthsFor the notation of non-canonical labyrinths, which sometimes have segments longer than two quadrants, and crossings without change of direction ("bayonet" crossings, as in the Bayeux cathedral labyrinth, between lanes 7 and 10), Note: The version sent for publication in Caerdroiaother symbols are necessary. I propose the following:
"+-" = 3 quadrants
Here are a few examples of signatures of non-canonical medieval labyrinths (with indication of interesting sequences): Solomon (K. 217): 0 {-5-6+7-8+9+-10} -9-8 {++11-10-9 (+8-7+6-5+4) -3-2 (+1} -2+3-4+5) -4-3+2+1-6-7-12 Liber Floridus (K. 191): 0+7+8 {-11-10++9 (+8-7+6-5+4) -3-2} -5-6+1-2+-3+4+5+2+1-6-7+-10+-11-12 Paduan miniaturist (K. 211):
0-7-6+1+2-3-4 (+5-4+3-2+1) -6-5 {-2-3 (+4-5+6-7+8) -9-10 (+11} -10+9-8+7) -8-9+10+11-12
Bayeux (10 lanes) (K. 255): 0++5-4-3 (+2-3+4-3+2) -3-4++1++6-7-8+9-8-7 !++10-11
The round course or three-step circuit:The round course, or three-step circuit, which I seem to have discovered, consists in a dance-like three-step course around a circle. The three longer steps are separated by two shorter backward steps. Each one of our three historical canonical labyrinths (Chartres, Reims and Sens) contains three such "circuits", alternately clockwise and counterclockwise. Spatially, the circuit can be of two different forms: either spiral (the third long step being of the same thrust as the first two ones) or folded (the third long step folding back on the same lane as the first one). Thus the spiral circuit is laid on five lanes, the folded circuit is on three lanes. The Chartres labyrinth has three spiral circuits, the Reims labyrinth has two folded circuits and one spiral circuit. Both forms can be either centrifugal or centripetal. All canonical labyrinths contain three such circuits. The form and disposition of these circuits can be used as a means to identify models and families of labyrinths. Using the new notation method, the different circuit forms can be described in a general notation, "n" being the number of the lane where the circuit begins. centrifugal spiral circuit: +n-(n-1)+(n-2)-(n-3)+(n-4) centripetal spiral circuit: +n-(n+1)+(n+2)-(n+3)+(n+4) centrifugal folded circuit: +n-(n-1)+(n-2)-(n-1)+n centripetal folded circuit: +n-(n+1)+(n+2)-(n+1)+n
The two forms of the notation: the prefixed and the expandedThe prefixed form is the easier to read, especially in terms of rhythmical clarity. Many properties and peculiarities of a labyrinth are readily visible in its signature and can be read directly, indeed much more easily than on the drawing of the labyrinth itself or even on any graph of it, although some of these properties may be more visually explicit in specific graphs. The expanded form consists in repeating each lane number for every quadrant it spans, and using the separators (normally a simple space or a tab) only as separators, not as length indicators. This form is more readily used for certain kinds of mathematical analyses. Importing the expanded-form "signature" of a labyrinth in a spreadsheet with graphical capabilities will produce directly all kinds of analytical graphs, one of which happens to be Willem Kuipers' "cadence" diagrams.
The question of the numbering orderWorking on that notation method brought me back to the order in which the lanes should be numbered: from inside out or reverse. Most authors I know, including myself, number the lanes from the exterior towards the interior. Robert Ferré does the reverse, which I think makes a lot more sense than I had thought before. Indeed, the distance from the center may sometimes be more important than the depth from the outer "surface". But numbers are only numbers, and they can also be interpreted in the opposite way: like the sequence in which the lanes are visited when walking into the labyrinth, or like the representation of some gravitational force increasing towards the center. From this point of vue, I see no reason to choose a particular numbering order, other than the fact it is already being used by most people. But there may be another point of view. Let us look at the way historical labyrinths were drawn. Most seem to be drawn in a geometrically correct way. Floor labyrinths are indeed correctly drawn, which should not be surprising. But a large number of manuscript labyrinths are geometrically incorrect: the width of the lanes is inequal: lanes near the center are often narrower that those near the exterior. This is also the case with the Lucca (K. 269) and Pontremoli (K. 278) labyrinths, which were carved in stone. One might say that the intention of the designer was to keep some proportion between the radius of the lane and its width, in view of the general appearance of the design. That may be true for the carved labyrinths. But the low degree of graphic sophistication of the manuscript labyrinths does not warrant that interpretation. I think the labyrinths were usually drawn starting with the outermost circle, which makes sense in terms of filling the available space. Often, if his planning was insufficient, the draftsman would eventually find he lacks space to finish the last circles and would simply make them narrower. The fact that this geometrical irregularity was frequent seems to confirm that the labyrinths were indeed drawn from outside in. That may be a good reason to number the lanes following the same order.
References
Hébert, Jacques: the rhythmical structure of the Medieval labyrinth (2004)
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