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Posts Tagged ‘detour factor’

Up to now we have examined the “detour factor” of the Classical and of the Chartres labyrinth. That’s why the Roman Labyrinth as own type in the long history of labyrinth may be considered today.
We choose a sort of prototype with 21 m for the side lengths and an axial distance of  1 m.

The path in the Roman Labyrinth

From A to Z: The long and the short way in the Roman Labyrinth

The direct way from “A” to “Z” straight across all boundary lines to the center amounts to 10.55 m.
The whole, long way from the entrance into the center amounts to 433.50 m if I follow all the twists through the four sectors. This proves a relation between the long and the short distance of 433.50: 10.55 = 41.1. This is a much higher “detour factor” than the value of 24.4 for the Classical labyrinth. However, it corresponds approximately to the Chartres labyrinth with 40.78.

If I handle the thread at the beginning and at the end and stretch it apart, I will get a straight line which reaches from “A” to “Z” and which corresponds to the way into the middle, that is 433.50 m.

If I join together the beginning and the end, I will get a circle. The circumference corresponds to the straight line of 433.50 m. The diameter would be 137.99 m.
I can also make a square with the same size from it. This would have four side lengths of 108.38 m.

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

Ariadne's Thread unrolled

Ariadne’s Thread unrolled

Don’t be surprised that the original labyrinth looks so tiny. This is due to the “detour factor” of 41.1.
The unrolled thread of Ariadne is much longer relative to the original labyrinth. The proportions in the drawing however are right.

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Recently you could read something about the long and winding path in a classical labyrinth.
Today we want to look more exactly at the path in the Chartres labyrinth. This is quite an other type of labyrinth. It has more circuits than the Cretan labyrinth, eleven instead of seven.
We orientate by the original, that is since about 800 years on the floor inside the cathedral of Chartres. The ways are much smaller than they should be for a “open land labyrinth”.
It depends not only on the type labyrinth, how long the ways are, but also from the constructive form. So: How wide the ways are, how wide the boundary lines are in between, how big the middle is etc.
In the Chartres labyrinth we have a distance of 42.6 cm from axis to axis.

From A to Z: The long and the short path

From A to Z: The long and the short path

The direct way from “A” to “Z” straight across all boundary lines to the center of the Chartres labyrinth amounts to 6.45 m. This corresponds to half a diameter of 12.90 m.
The whole, long way from the entrance into the center amounts to 263.05 m by following all the turns. This proves a relation between the long and the short distance of 263.05: 6.45 = 40.78. So a much higher “detour factor” than in the Cretan labyrinth.

If I handle the thread at the beginning and at the end and stretch it apart, I will get a straight line which reaches from “A” to “Z” and which corresponds to the way into the middle, that is 263.05 m.
If I join together the beginning and the end, I will get a circle. The circumference corresponds to the straight line of 263.05 m. The diameter would be 83.73 m.
I can also make a square with the same size from it. Then this would have four side lengths of 65.76 m.

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

Ariadne's Thread unrolled

Ariadne’s Thread unrolled

Don’t be surprised that the original labyrinth looks so tiny. This is due to the “detour factor” of 40.78. (To keep in memory: For the Cretan labyrinth this value amounts to 24.4).
The unrolled thread of Ariadne is much longer relative to the original labyrinth. The proportions in the drawing however are right.

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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.

Ariadne's Thread

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

Ariadne’s Thread unrolled

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