The Pattern in Labyrinths with Multiple Arms

I have shown with the example of my demonstration labyrinth how the pattern of a labyrinth can be obtained. This was a one-arm labyrinth. Of course, it is also possible to transform labyrinths with multiple arms into the rectangular form.

Compiegne

Figure 1. Compiègne

I will show this here with the Compiègne labyrinth as an example (fig. 1). This is a labyrinth with four arms. It is presented with the walls shown. Erwin has already used this type of labyrinth in this blog (see related posts). In order to obtain the rectangular form, I will use the Ariadne’s Thread and apply method 2, as usual.

Lage Achsen

Figure 2. The Ariadne’s Thread, situation of the arms

Fig. 2 shows the baseline situation. The labyrinth is represented by the Ariadne’s Thread with the entrance at the base and in clockwise rotation. The main axis and the side-arms are highlighted.

Muster Merhachs Meth2

Figure 3. Flipping the Arms

As can be seen in fig. 3, both side-arms on the left and right side are flipped upwards by about 1/4 of the arc of a circle. There they meet the upper side arm which remains unchanged. Only the main axis is split into two halves, and these are flipped upwards each by ca. Half the arc of a circle, where they attach to the side-arms.

Muster MA Erg2

Figure 4. Finalization of the Pattern

Fig. 4 shows the straightening-out process and, as a result of it, the pattern of the Compiègne labyrinth.

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4 thoughts on “The Pattern in Labyrinths with Multiple Arms

  1. Dear Erwin and Andreas,

    It is always a pleasure receiving new entries in your interesting blog, one of the very few places where consideration is given to the geometry of labyrinths.
    This blog is so deep and my time schedule so tight versus all the things I like digging into, that I’m still far from reaching its bottom !

    Yet, I started 11 years ago in labyrinths from an other direction than the nice and interesting way you did your own progression, but also in the realm of the sole geometric and plastic viewpoints. I have documented my work in five papers and several artworks distributed in the annual art-math Bridges Conferences and Exhibitions, building up some kind of a ‘logic’ construction of what could have been the historic evolution of labyrinths (and which most probably was not…) between the Archimedean spiral as a proto-labyrinth to the Classic/Cretan/Cnossos (I like to refer to as CCC in short) to the Chartres design and beyond, a story that, like your work, looks pleasing to the logical mind. I’ve produced hundreds of drawings of dozens of situations along and there are a number of points I discovered through other ways that do embrace yours (and those of other researchers).
    Both (and certainly more) appoaches have salient original points as well as unexplored paths, so a reconcile of both (with others’ work) would prove beneficial to consolidate the subject.

    Therefore I’d be much grateful if you both and other people would comment my Bridges material that is freely online (I think googling to ‘samuel verbiese labyrinths’ both in text and images gives a fairly complete account, by the way also paying largely a tribute to this blog !).

    Some of the salient things I like to put forward are these:
    – The titles of my papers are based on the words ‘Amazing Labyrinths’, that others also like to combine. I like doing it because beyond the play of words, in our jargon, ‘mazes’ and ‘labyrinths’ are mathematically different (unicursal vs. multicursal :
    see what I just wrote in http://www.diffen.com/difference/Labyrinth_vs_Maze ).
    – In the now Christian Middle-ages, after circularisation, monks replace CCC’s
    square ‘seed’ by a vertical rectangular one that dispays the replacement of the ‘+ sign’ by the typical set-back found in the middle of Chartres’ under-half vertical axis, and followed by the elegant turn-abouts with intermediate pass-throughs on the 3 other half axes, to get the now central ‘cross’ repacing the lost original CCC + sign (‘cross’, they now say), announcing Chartres. This is still a 7-circuit labyrinth.
    – Chartres design was finally obtained by a doubling of the 4 original points, to get an 11 circuits labyrinth, with more turn-arounds and pass-throughs. That doubling is the very reason why the inner and outer half Chartres can be infinitely repeated, the Saffron Walden labyrinth North-East of London being the sole example of this fractality featuring more than two identical parts, indeed three of them.
    – The ‘pilgrim step’ within Chartres easily seen in the ‘rectangularisation’ of the Chartres Ariadne thread, of which Andreas just showed another nice way of obtaining it. This pilgrim step resembles the Echternach Procession steps, there three steps forward and two backwards, here two forward (indeed half circles) and one backward (quart circles). Funny when you know Chartres in early times was strolled on the knees by pilgrims, while princes battered Jerusalem during the Middle Age Crusades…
    And there are more points you can find in the papers I’d be happy to see commented by you people…
    Thanks again for your work and kind regards,
    Sam

    PS: in the Art&Math exhibition at Brussels University this spring (see http://gatito.be/expo), I showed a ‘LabyLabo’ cubic box containing 46 explanation panels explaining my ‘logic’ would-be ‘labyrinth history’ which I later showed on two large panels (I can send you soon detailed pics if you tell me how, unfortunately in French texts.)

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    • Dear Samuel,
      thank you for this contribution. I appreciate your work and I try to follow your thoughts, wherever possible. (Even if they are a little bit hard to find).
      With my kind regards
      Erwin

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    • Dear Samuel,
      thank you for your comment. I had already glanced through some of your publications and intend to come back to it later. For the time being, due to other time constraints, my labyrinthine work is limited to the production of one post per month. So maybe next year.
      BR
      Andreas

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      • Dear Erwin and Andreas,
        Grateful thanks you for your interest. I contemplate we are all buried in activities: this is the very reason why I still have a fair backlog in studying your work …
        To ease finding on-line documents, I will provide an explicit listing for when you get some more time for this.
        In the meantime, I am always interested to read your new productions coming in on a regular basis : tx & congrats fot that !
        Kind regards,
        Sam

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