Many are surprised how long the way in a labyrinth can be, especially if they walking labyrinth for the first time. And many who want to build a labyrinth, e.g., with stones or with candles, are astonished how much material they need.

Under the heading “Principles of Form” in his book **Labyrinths** Hermann Kern writes about the >tortuous path principle<:

– if the path fills the entire interior space by wending its way in the most circuitous fashion possible

If I stand ahead of a labyrinth, the middle, the goal is within my reach. However, only when walking in I get to know how winding and complicated the way is in reality. But yet, this way, the red thread or Ariadne’s Thread is the continuous line in the labyrinth, without crossroads or junctions.

From A to Z: The long and the short path

In the drawing I call “A” the beginning of the path and “Z” the goal, the center or middle. In many labyrinths I could reach directly the middle with a few steps across all limitations. But this is not really what is intended with a labyrinth.

Now I compare for a 7 circuit labyrinth with a diameter of about 15 m the short way (direct connection between A and Z) with the long way along the Ariadne’s Thread. The length of the short way amounts to 6.33 m, the long way has a length of 154.62 m. Or differently expressed: The long way is 24.4 times longer than the short way (154.62: 6.33 = 24.4).

One could also see in this a formula for the labyrinth. To calculate how powerful is the layout for example. Or how wended is the way? Or from what minimal surface area I can extract which maximal length?

Maybe one could call this value in honour of Hermann Kern “detour factor” 24.4?

If I handle this thread at the beginning and at the end and pull it apart, I will get a straight line which reaches from “A” to” Z” and is as long as the way inside the labyrinth, i.e. 154.62 m.

I can arrange this to a circle. The perimeter corresponds to the straight line of 154.62 m. The resulting diameter would be 49.22 m.

I can also make a square with the same extent from it. Then this would have four side lengths of 38.65 m.

The following drawing, yet not true to scale, illustrates the different figures and the true ratios among each other:

Ariadne’s Thread unrolled

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Dear Erwin,

This is an interesting post that nicely complements the explanations I just gave today in front of children at the current Art&Math exhibition at Brussels University (ULB) where I present a history of labyrinths from the Cretan to Chartres and beyond in the Chartres ‘philosopy’, which probably quite departs from the reality, yet constitutes a logically credible story…

I like starting by asking what is a labyrinth. Generally the kids rather bring in the ‘maze’ concept, i.e. a winding set of trails with choices and dead-ends.

I then explain that the word ‘labyrinth’ generally refers to a very long path winding from the edge of a rather small place to its center, indeed a dead end.

Then I ask what would be the simplest figure that packs a straight line say 150m long (that would require a long stretch of land) into a place say 10m across), and generally it doesn’t take long to hear, after the word circle or square (that would still be too lage to fit in) the word (Archimedes or

arithmetic) ‘spiral’ which indeed is a ‘protolabyrinth’ much older than about the 5000 years ago that saw the birth of the Cretan labyrinth. The length of a spiral being a good approximation of an equally densely packed Cretan.

Then follows the story of the Cretan labyrinth : I ask the children to close their eyes and imagine to be at the sea shore at low tide, when the beach is an ideal place to draw labyrinths. Starting with the usual +-sign (two perpendicular lines 2m long crossing in the middle), 4 points in the corners (of a square built on the +-sign as its medians) and a (diameter 1m) circle divided in four where each quarter fits between a point and the closest edges of the +, the obtained figure being the ‘seed’ of the Cretan labyrinth.

I select 8 children ranked by size, the little 3 years old is placed on the upper end of the + and the others are placed to the left; half a meter apart from each other to the point close to the bottom edge of the +. I show there are exactly 8 points awaiting them at the other side.

Then I first help the little kid drawing a small half circle from the top of the + where she stands to the end of the quarter circle to the right. After that all the other kids have to do is to follow the previous kid at exactly half a meter distance, and soon enough the labyrinth is in place and all the children can get inside, between the ‘walls’ they have just drawn, discovering the center is too small to hold them all… Etc…

Kind regards,

Sam

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Dear Sam,

it is fine that you share your labyrinth experiences with children here on the blog.

They generally have a good feeling for what a labyrinth can be.

Thank you

Erwin

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