How to Draw a Labyrinth

In the meantime, I have developed for myself a method to construct a labyrinth by only using the path sequence. I do not apply the seed pattern to do that.

I would like to show this here for a 7 circuit classical labyrinth with the path sequence  0-3-2-1-4-7-6-5-8. It is from an other shape as the usual classical labyrinth and was to be seen in a previous post.

First I draw freehand the path of the labyrinth (Ariadne’s thread) according to the path sequence. To get an other shape, I cross the main axis with the 4th and the 7th circuit. From the sketch I derive the number of the turning points and the construction lines. Then I count the number of the ways between the central point and the turning points which are also centers in the further construction.

The freehand sketch

The freehand sketch

The broad for the paths and the walls is the same with 1 m, this makes a dimension between axes of 2 m. The diameter of the middle is the fourfold dimension between axes. The lengths of the different construction lines are calculated from this details.
I begin with a horizontal line (M2 – M3) and I fix the central point M1 by applying the two distances from M2 and M3. In the same way the other points are defined. (Fig. 1)

The construction lines

The construction lines

Starting from the center M1 I draw auxiliary circles in an interval  of 1 m from the inner to the external diameter. (Fig. 2)

The auxiliary circles

The auxiliary circles

The lines M1 – M2 and M1 – M3 are extended up to the external diameter, also the lines M2 – M4 and M3 – M5. (Fig. 3) They are limiting the circular arcs.

The external arcs

The external arcs

Then the different curved sections are drawn with the help of the path sequence in the centers M2 to M5.
In fig. 4 this are the semicircles around the turning points of the ways 5/6 in M2 and the ways 6/7 in M3. The construction is made through connection with the end points of the bigger external arcs.
In fig. 5 the curved sections of the remaining external arcs are formed around the centers M2 and M3.
In fig. 6 follows the connection of the ways 1/2 around M4 and the ways 2/3 around M5.
In fig. 7 the open, innermost pieces of the way are connected by strictly watching the path sequence. These are curved sections which joins each other without sharp bends.

The inner arcs

The inner arcs

In fig. 8 I turn all curves and lines around the center M1 in such a way that the entry axis for the middle is strictly vertical.

Fig. 9 shows the completed labyrinth. The paths are highlighted in color. The walls have the same broad and are left white. The left blank fontanel is good to recognise. It has another shape as the usual classical labyrinth. It appears when the paths are drawn in constant width.

The 7 circuit Knidos labyrinth

The 7 circuit Knidos labyrinth

The drawing is rather a sort of model or template. The labyrinth can still have got different shapes, can be angular or completely round, right- or left-handed. The paths and the walls can be of different widths, also the middle can be bigger or smaller. It is only important to maintain the alignment of the labyrinth.

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The Knidos Labyrinth Prototype

Here the design drawing of the prototype of a Knidos labyrinth.

Prototype

Prototype

The dimension between axes is 1 m. The center has a diameter of 4 m because it is the quadruple of the way’s width. The overall diameter amounts thereby to 18 m (= 2 x 7 ways + 4 m center).

The dimension between axes of 1 m is the distance from the middle of the border line of the way up to the middle of the border line on the other side of the way. If the delimitation is 20 cm broad, the way will be 80 cm broad. That is to be considered when building the labyrinth.

All dimensions and measurements are scalable. If the labyrinth should have an overall diameter of only 9 m, I must multiply all measurements by 0.5 (the radii, the distances, the lines length, the diagonal measurements etc.). So I can build a labyrinth in any size.

If the ways should have 1.20 m dimension between axes instead of 1 m, I multiply all data by 1.2. The overall diameter becomes then 21.60 m (= 18 x 1.2).

If you want to change the direction of ways, thus e.g. the first way (3) leads to the right, the drawing must be reflected along the vertical axis or seen from the reverse side.

Here you can see/print/store/copy the design drawing as a PDF file  …

How to Make a Knidos Labyrinth

After the discovery of the Knidos labyrinth the geometry and the exact construction is asked.

There are classical labyrinths with a larger center since longer, also as walkable labyrinths. However I noticed that were certain artistic freedoms to design them. For instance that the turning points are not in a square, or that the central cross is displaced, or the ways are differently broad.

After longer trying I found out how one can design best (at least in my opinion) geometrically and mathematically exactly a classical labyrinth with a larger center (that I call from now on Knidos labyrinth).

In practice the quadruple of the way’s width is a good measure for the center. That is also the measure for the four turning points of the internal square, on which these are situated. The sizes of the circle of the center and the square have thus a good purchase within the labyrinth.

All lines of the labyrinth ways and the way axes are circular arcs, which are clashing without break. The segments, within those the circular arcs with the same center are lying, result from the lines through the 5 centers of the labyrinth.

Maybe the drawings shows more simply what in words sounds so complicated.

Figure 1

Figure 1

That is the basic structure of the construction. In the classical labyrinth the center for the upper arcs is between the two upper corner points of the square. In the Knidos labyrinth this center (marked with M1) goes upwards. The distance M2-M1 is fourfold the way’s width (2 ways + the half center); the distance M3-M1 is threefold the way’s width (1 way + the half center).

Figure 2

Figure 2

The lengthenings of the aforementioned distances form at the same time the delimitation of the upper 8 circular arcs, which have all their center in M1. The further segments are built by lengthenings of the distances M2-M4 and M3-M5. The points M2 to M5 are both the turning points of the labyrinth as well as the centers of the further circular arcs.

Figure 3

Figure 3

With M2 as center all free ends of the preceding arcs on the left side are connected. The three lower arcs end at the line M2-M4.

Figure 4

Figure 4

The same happens on the right side, whereby the lower 4 arcs end at the line M3-M5.

Figure 5

Figure 5

Around the center M4 a semi-circle connects two further open ends of the preceding arcs. A quarter circle around M4 connects the lower free end of the central cross with the left end of the central cross.

Figure 6

Figure 6

Four further semi-circles around the center M5 connect the three open ends on the right side and the right end of the central cross. Thus the construction of the labyrinth is finished.

Figure 7

Figure 7

Here again the whole labyrinth without the auxiliary lines. The central lozenge is the result when all ways are designed in same width.

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