Meander and Labyrinth – Why Straighten-out?

In my last post I have pointed out the close relationship between the figures of Arnol’d and the patterns of labyrinths. These figures are very similar, but they are not equal. One has to straighten-out Arnol’d’s figures to get to the patterns of the labyrinths. But what does this mean: “straighten-out”? I want to show this with the example of one of Arnol’d’s figures. I have purposely chosen figure 8 for that.

Illustration 1: double-spiral

Illustration 1 shows figure 8 of Arnol’d in its original orientation (first image) and rotated by a quarter of a circle (second image). The third image shows the same figure as the second, although reproduced on the computer and a bit more rounded. As can be seen, the curve comes from above left and winds itself inwards anticlockwise with narrowing radii.  It intersects the straight line twice. At the point where it intersects the straight a third time, the curve changes to clockwise direction and winds itself outwards. This curve in fact presents as a double-spiral, that intersects the straight five times.

Now, what happens, if we straighten-out this figure?

This is shown in the next illustration.

Illustration 2: double-spiral-type meander

The double-spiral is dissected along the straight line and both halves are shifted horizontally. Next, the pairs of points that were generated by this separation are connected with horizontal lines (dashed in the figure). These lines in fact represent the circuits of the labyrinth. Separating the double-spiral and inserting connection lines leads to the pattern of the labyrinth. Here lies the relationship between the intersection points in Arnol’d’s curves and the circuits of the labyrinth.

In addition, this example illustrates very well the difference between a double-spiral and a double-spiral-type meander. Remember that Arnol’d’s figure 8 corresponds with the meander Erwin found best suited for labyrinths. This meander is in fact a double-spiral-type meander, and this holds for all of Erwin’s types of this meander (type 4, 6, 8, etc.). Separating Arnol’d’s figure and inserting connection lines transforms the continuous movement of the double-spiral into a stepwise progression from one circuit to the next. That is what makes the difference between the double-spiral and the double-spiral type meander that can be found in the patterns of labyrinths.