How can I draw / design a geometrically / mathematically exact labyrinth by applying the seed pattern?

Here I will explain the way step by step. Everybody can reproduce it itself. To do that you need paper (at best squared), circle, ruler and pencil; when required a rubber.

First once more the seed pattern of Gundula Thormaehlen-Friedman:

If one looks exactly at it, one recognises that the four centres of the internal arcs shape a square. Furthermore that 8 and 0 stand about one another. That means that the entrance axis is lying on the same line as the middle axis. This recalled me the article of the centred Knidos labyrinth about which I had written on 8/29/2010 in “The Twisted Labyrinth“. There the labyrinth is mirrored and has a bigger middle, however, the principle is the same.

If one wants to construct a geometrically correct labyrinth, one must follow the limitations by the square. Because the labyrinth is an order of arcs with different radii which are connected without sharp bends. This is also valid for Ariadne’s Thread.

To simplify the procedure I rotate the seed pattern at an angle of 90:8 = 11.25 degrees. Then the square is right-angled.

I divide every square side now into 8 equal sections. I start to number in the corners and jump over every second point. Z stands for the goal and replaces 8, A stands for the beginning and replaces 0. The order would be indifferent, actually; but if one makes it in such a way, the circuits are numbered from the outside inwards in the completed labyrinth.

Now I lengthen the right and left square sides downwards, the upper horizontal one I lengthen to the right and to the left.

The four corner points shapes the centres for the arcs, I call them M1 to M4; Z is the end of Ariadne’s Thread, but also the centre for the upper arcs.

I extend the arcs up to the lengthened square sides, so 1 on the left side around M2 to the left, 1 on the right side around M3 to the right, 2 also. The curves 4 and 5 on the left side around M1 upwards, the curves 6 and 5 on the right side around M4 upwards.

*(If the drawing is too small, click on it. Then it opens in a new window and becomes bigger).*

Now I complete the left part. I connect the curve ends 3, 2 and 1 upwards to the horizontal line in own arcs round M1. I do not lengthen the arcs, but I draw a new arc with another radius. The curve may even change the direction and form a S-bend (like 3 and 2).

I make the same one on the right side with 4, 3, 2 and 1, building arcs with M4 as centre.

Now I complete the thread in the upper part. I connect all pulled up curve ends from 1 to 7 with a semicircle around Z as centre. I extend the end of the arc from 3 to A in the lower left part with a straight line downwards, this is the beginning of Ariadne’s Thread.

The complete Thread of Ariadne without “decorative accessories”.