The pattern is a transformation of the Ariadnes’s Thread from the closed form into the rectangular form. This is not a labyrinth any more. The exterior of the labyrinth is represented by the area above the pattern, the interior by the area below the pattern. The circuits have transformed to horizontal lines. The arms are all oriented vertically.

Fig. 1 shows the Ariadne’s Thread of my demonstration labyrinth. In accordance with the direction into the labyrinth, I number the circuits from the outside in. Thus 1 is the outermost, 5 the innermost circuit.

Fig 2 shows how this is represented in the pattern. There, 1 is the uppermost, 5 the lowest horizontal line. In the transformation, the axis is split in a left and right half. These halves come to lie on the left and right outer verticals of the pattern. As long as the pathway follows a circuit, in the pattern this is represented by a horizontal course. When it changes to another circuit, it moves axially and by this forms the axis. In the pattern this is represented vertically. However, this can only be seen clearly in labyrinths with multiple arms.

Fig. 3 shows the Ariadne’s Thread of the Compiègne labyrinth. This labyrinth has also 5 circuits, but 4 arms. Labyrinths with multiple arms generally are composed of a main axis. This is where the pathway enters the labyrinth and from which it also reaches the center. In addition these labyrinths have one or more side-arms.

First, it has to be made clear, how we want to refer to the arms of the labyrinth. I give the number that corresponds with the number of arms of a labyrinth to its main axis. So, in four-arm labyrinths, I number the main axis with 4. I start the enumeration of the arms with the first side-arm next to the main axis in clockwise direction. The reason for this is, that I always start with the labyrinth orientated such that the entrance is at the bottom and the labyrinth in clockwise rotation when I transform it into the rectangular form.

Fig. 4 shows how this affects the pattern. The main axis is split. This is the same as with the only axis in one-arm labyrinths. Both halves of the main axis come to lie at the left and right outer verticals. The side-arms are not split for the transformation. In a four-arm labyrinth, therefore, we can find five vertical lines in the pattern. Two for both halves of the main axis and one for each side-arm. (By the way: I have shifted the labellings of the arms. In the pattern these should lie on top according to the positioning of the labels on the outside of the Ariadne’s Thread, however it reads better this way.)

So, the pattern can be thought as lying on a grid of horizontal and vertical lines. The horizontal lines indicate the circuits, and the path in the pattern follows on these horizontal lines. The vertical lines indicate the arms of the labyrinth. These lie between or aside the turns of the pathway.

What these considerations also show is that we here have read the pattern in one direction. Keep this in mind, it is important.

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Funny, dear Andreas, almost at the eve of Xmas, you made me aware of a strange flip of my mind that occurred eleven years ago, and did perdure until now !

When I discovered labyrinths through the Chartres labyrinth design (eleven years ago, when Charles Fontaine, then 90, sent me a document showing it on the floor of the cathedral better known for its stained glasswork), I was seduced by the apparently symmetric circular ring form generated from a white Ariadne thread winding in between black ‘walls’, and being an engineer, I wanted immediately to understand its workings.

I first discovered there was only one straight line directly connecting the entry of the path at the outside and its end at the inside, and this was the ‘vertical’ black line (indeed the ‘wall’) at the underside of the ring, the dividing line between the entry and end regions… I imagined I could cut this ring along that line with a pair of scissors, then unfolding symmetrically upwards both sides, like wings, the ring being held in place at the upper side. The annulus transformed so a horizontal ‘bar’ extending both sides from the fixed middle upper part, a bar I then could squeeze/compress to that ‘fixed’ location into a squarish figure, a figure that I then could understand : drawing now in black the Ariadne thread, I got my ‘pattern’, in fact yours !

But in the particular case of the Chartres labyrinth, indeed for all labyrinths derived from the Cretan/Cnossos/Classic labyrinth, the pattern has a rotational symmetry, i.e. you can turn it 180° and it still looks the same, entry and end playing the same role.

(By the way, this reveals the nice ‘pilgrim step’ imbedded in such labyrinths, as also visible in your further pattern picture : 2 steps forward, one backward, like 3-2 in the Echternach’s procession. Amazing when remembering the church labyrinths were strolled on knees by pilgrims…)

Sure enough, it didn’t take long, I displaced the squarish pattern onto the left under side of the original labyrinth ‘ring’, matching its right side to the labyrinth entry region. In other words, in fact I unfolded the ring this time keeping the entry ‘lip’ of the ring cut in place, the other lip being brought all the way to the left anticlockwise, so obtaining again the ‘bar’extended this time to the left from the under side, which in this ‘dual’ case produces the same pattern !

Your article thus made me aware of this as you develop your theory from a more general labyrinth that doesn’t possess this peculiar (and beautiful) duality feature. For 11 years, thus, I told the story of the ring cut underneath and déployed upwards like wings to the ‘bar’, and not realizing that so doing in the squarish pattern the entry is now above left and the end under right. After squeezing it into the sqarish pattern, I unconsciously flipped it 180°, to match the familiar entry ‘from below’ and ending upwards, which does no longer work for a non-dual labyrinth…

Grateful thanks for this, and happy Xmast !

Sam

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Dear Samuel

Thank you for sharing your interesting experience. This, in fact, is the reason, why I always recommend not to use only such perfect labyrinths as Chartres or Reims or the “Cretan”. You won’t be able to discover that each labyrinth has a dual. Because these excellent labyrinths are as we call it: self dual labyrinths, i.e. the dual is the same as the original (that’s what you probably mean by “dual”). Of course once you have understood that there is a dual to each labyrinth you can then notice self-duality is a special feature of some particular labyrinths.

Best wishes and a happy New Year.

Andreas

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