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## A “Form-Fitting” Knidos Labyrinth

Sometimes you will get a diamond-shaped, empty element in the middle part of a Knidos labyrinth which is formed normally by a cross. This happens when all the paths have the same width and the walls are aligned to them. This form arises because the four turning points form a square.
A shift of the rhomb may also result if one brings into line the entrance axis (of the path into the labyrinth) and the entry axis into the center of the labyrinth with the main axis of the labyrinth figure. In the “twisted labyrinth” I have demonstrated this already once (see related posts below).

If one wants to give a certain shape to this “empty form”, one can play with the position of the turning points. I have done this to get the “form-fitting” labyrinth. All elements are arcs, however, the four turning points do not lie any more in a square.

The suggestion for this labyrinth dates to the logo sketched by the Swiss artist Agnes Barmettler with the woman in the labyrinth for the public women’s places.

The logo for the public women’s places

Such a labyrinth can be drawn nicely, but is hard to build, above all as a big labyrinth. Hence, I have tried to develop the shape for this labyrinth with geometrical elements only. Some imagination is asked of course. Anyway, the “empty space” offers creative leeway.

The “form-fitting” Knidos Labyrinth

The following layout drawing for a sort of prototype shows the geometrical qualities in detail.

Who looks accurately and compares to the original Knidos labyrinth, recognises that one segment less arises. The turning point below on the right is laid in the lengthening of the line from the midpoint of the center and the upper right turning point. The usually narrow “cake piece” is thereby lost in this area.

The drawing

Who would like to build such a labyrinth, is invited warmly to it. The drawing contains all that is neccessary. Indeed, one can also use other parametres, because the labyrinth is scaleable. The underlying values are based on the dimension between axes of 1 m. This means that all details changes proportionally (are reduced or extended). If one wants, e.g., a half as big labyrinth, one multiplies all measures by 0.5. One can even proceed in such a way with “crooked” numbers to get the desired result.

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## The Two Seed Patterns of the Labyrinth

After you have seen several times the different seed patterns in the labyrinth, now a common consideration should follow: There is the seed pattern for the walls (limiting lines) and the seed pattern for the path, the so-called Ariadne’s thread.

While I tried to build a geometrically exact labyrinth from the different patterns, it has struck me that both seed patterns are not so different at all. And thus I would like to show both together.

Here first the square which four sides are divided into eight constant parts. In a drawing one can take squared paper and make every side 4 cm long. In  reality this would be four metres and the drawing would be on a scale of 1:100.

The scaled square

The mark and name of the different points already states something about the later use within the construction. “A” is the starting point; “Z” the goal or the centre, at the same time, however, also a centre of different arcs. Hence, marked with a bigger symbol of a circle. As well as the four corner points M1 to M4, also centres of arcs. And at same time delimiters for the walls.
The path axes (Ariadne’s thread) are marked with a small cross and are numbered from 1 to 7. In between are the walls which are marked with small circles.

The angular seed pattern for the walls

Here the well-known seed pattern with the isosceles cross, the four angles and the four dots.

The round seed pattern for the walls

However, the lines must not be angular, they can also be rounded and then the pattern looks like above.

The seed pattern for Ariadne’s thread looks in the limiting square like on top.

The two seed patterns in the square

If both patterns are shown together, one recognises the relationship and resemblance between them. And also that the centres of the different arcs are same. No surprise, because the lines are parallel and the red thread is, finally, the middle between the black boundary lines, so to speak the path axis.

Afterwards I would like to point out which arcs are constructed from a total of five centres. There are quarter circles and semicircles which run in each case in different sectors. The order is as you like, since basically it makes no difference which curve is drawn or constructed first. This arises by itself, if one applies the principle for the drawing of a labyrinth properly. As it was described in the older posts on this blog.

Tip: The following, as well as all remaining drawings, can be clicked to enlarge. Then a new window is opened.

The centre M1 top left

The centre M2 bottom left

The centre M3 bottom right

The centre M4 top right

The centre Z on top

The classical seven circuit labyrinth

Here the finished labyrinth with the walls in black and Ariadne’s thread in red. It is a classical seven circuit, left handed labyrinth.

Prototype

If you would like to build such a labyrinth, you will find all specifications and all radii in this design drawing on a scale of 1:100. It is a sort of prototype for a dimension between axes of  1 m and scalable.

## The 3-Circuit Classical Labyrinth

The labyrinth in its simplest form is one with 3 circuits. For some people it is not a “real” labyrinth because the path is leading directly into the centre without being closer and then farther away from it.
As there is no universally valid definition for the labyrinth, we may nevertheless consider this labyrinth as a real one.

The labyrinth with 3 circuits

How do we get one?

The basic pattern to make a 7-circuit classical labyrinth is known, in the meantime, probably by all readers of this blog. (If not, please take a look here.)

No 3-circuit historical labyrinths are known, it is made from a reduction of the basic pattern. If one omits the four angles, only the cross and the four dots are remaining.

The pattern

This reminds a little: Dot, dot, comma, dash – smiley face in a flash :-). However, it is really so simple to make a labyrinth, and this is why it is a child’s play to draw one this way.

The first arc

The second arc

The third arc

The fourth arc

However, there are still other methods to draw the labyrinth: In two lines, from one end of the line to the other end. Try to draw it on a sheet of paper. So often that you can do it by heart.
Tip for right hander: Begin at the left end. Left-handed persons are beginning at the right end. The lines may become crooked.

With two lines

An other variation would be to begin in the central intersection point and to draw to all four directions. This has practical meaning if one would build a labyrinth with different material for example.

With four lines

The most elegant method is to draw the labyrinth in one line. Therefore we take the path, the famous thread of Ariadne. We can begin on the inside or from the outside.

Who realises this by heart, maybe even for the 7-circuit labyrinth, may be called labyrinth expert.

Here a few examples of 3-circuit labyrinths:

Ceramic

Artwork

Graphic art

The first picture shows a gem of Alexander Lautenbacher.

The central picture shows the shoe labyrinth from Schwäbisch Hall. The four “shoe lines” are beginning in the central intersection point.

The last picture shows the graphic on the invitation card from the Labyrinth Society for the Gathering this year.

## The Lines in a Labyrinth

There are many possibilities to design a labyrinth .

Some of it were already introduced here. Still more are following. In addition it is however good to look more exactly to the lines  of a classical labyrinth.

The Knidos labyrinth in 2 colours

In this two-coloured drawing one can see very good that the labyrinth consists of 2 crossing lines. The one (here green here) line leads from the upper, left turning point to the lower right turning point. The other (here brown) line leads from the lower, left turning point to the upper, right turning point.

We look at the border lines of a classical labyrinth with large centre and 7 circuits. The path (Ariadne’s Thread) runs between the lines.

The Knidos labyrinth in 4 colours

In this four-coloured drawing the lines begin/end in each case in the centre and end/begin at the 4 turning points.

It is more difficult to draw a labyrinth according to these specifications; the well-known basic pattern is much simpler. But it is more practical, if I want to place a labyrinth from ropes e.g.. How to do that exactly, we will see in a later post.

Here we have only one line before us. That is the way, or also the so-called thread of Ariadne  or the proverbial red thread. It is a uninterrupted line, which is not crossing anywhere and which leads on a devoured, but direct way from one end (at the beginning) to the other end (in the centre); and vice versa.

It is worth to remember the layout of this line to incorporate the labyrinth. The practical application is to step a labyrinth into the snow or to mow one into a meadow. Because then I build the path and not the delimitation.

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## It is a Child’s Play to Draw a Labyrinth

… if one knows how

Now Paula and Gernot Candolini are showing us in YouTube how to do that.

## The Knidos Labyrinth Prototype

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

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.

## 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

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

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

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

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

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

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

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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