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Posts Tagged ‘Knidos’

A Brief Stylology

In my last two posts I have described six styles.  Of course, these can also be used to order labyrinths. I will show this here in an illustrative manner. I do not take the effort to group all or as many as possible labyrinths into the different styles. I will just use a few examples of each style to illustrate how such a grouping would work.

klassisch

 

Labyrinth Examples in the Classical Style

 

 

konzentrisch

 

Labyrinth Examples in the Concentric Style

 

 

MiM

 

Labyrinth Examples in the Man-in-the-Maze Style

 

chartres

 

Labyrinth Examples in the Chartres Style

 

reims

 

Labyrinth Examples in the Reims or Bastion Style

 

knidos_stil

 

Labyrinth Examples in the Knidos Style

 

Labyrinth Examples in other styles

Of course, with the six styles described above it is not possible to cover the entire spectrum of all labyrinths. Therefore I have added another group to capture other styles and attributed some examples of labyrinths to it. Among the many labyrinths that cannot be attributed to one of the six styles above, it is possible to identify other styles. This particularly applies to labyrinths of which several examples exist in the same style. This, for instance, applies to the last two examples shown in the other styles group (Other 8, Other 9).

We have now ordered the individual labyrinth examples by styles. The result is also a typology or at least an approach to a typology. The only criteriun we have used for the definition of the types is the style. We thus have defined: type = style.

Because style cannot be defined clearly and unambigously, to me it is not well suited as a criterion for the constitution of a typology. Based on style it is not possible to form a complete range of mutually exclusive groups or types of labyrinths. Furthermore, style does not show the essential of a labyrinth.

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Type or Style / 8

Knidos is a Style

Knidos is not a type but a style. Just as well as for instance the classical, concentric or MiM-style.

knidos_stil

Figure 1. The Knidos Style

This becomes already evident from the fact that Erwin has published several types of one arm labyrinths in the Knidos style, among others the Knossos type, the Cretan, the Otfrid tye (see below, related posts: “How to build a Labyrinth in Meander Technique”).

Erwin was inspired by a labyrinth example on a rock carving from Knidos, hence the name. This labyrinth is an early attempt of a Cretan type on a concentric layout.

knidos_graff

Figure 2. The Labyrinth of Knidos

In his post “How to Make a Knidos Labyrinth” (see related posts), Erwin describes how he developed a geometrically exact labyrinth based on this example. The result is a Cretan type labyrinth in the Knidos style (fig. 3).

knidos_konstr_7

Figure 3. The Cretan Type in the Knidos Style

In a later even more interesting post “How to Draw a Labyrinth” (see related posts), Erwin has generalized the method and described it more in detail. For this he has used a non-alternating labyrinth with the level sequence 3 2 1 4 7 6 5, and thus did not use the Cretan type labyrinth. First, the Ariadne’s Thread is drawn freehand. Also from this it follows, that “Knidos” is a style and not a type. With the Ariadne’s Thread the type of labyrinth is already given, before it has been designed in the appropriate style. This is then performed in the following steps.

Here I will show the Knidos style using the same figure I have used in my last post for the presentation of the styles (see related posts). I will abbreviate the description of the method and highlight the central elements of this style.

Knidos_Achsmass

Figure 4. The Ariadne’s Thread and Auxiliary Circles

In fig. 4 the Ariadne’s Thread “drawn freehand” can be seen in red. By this, the type to be drawn in the Knidos style is already defined. Now we add three auxiliary circles with radii of 1, 2, and 3 units around a point in the center. The unit 1 is the equivalent of the width of the pathway including one of the walls delimiting the pathway. The distance between the point in the middle and the innermost auxiliary circle and then between subsequent circles is always one unit.

Next, we have to determine the situation of the turning points. Our simple figure has only two turning points (W1 and W2).

Knidos_WP

Figure 5. Situation of the Turning Points

The situation of the turning points is determined by means of construction lines. One lies at the end of the inner (W2), the other at the end of the outer (W1) wall delimiting the pathway. The center is four units wide. Therefore the innermost wall delimiting the pathway lies on the second inner auxiliary circle. And, as the figure has only one circuit, the outermost wall lies on the third auxiliary circle. The distance between the center (M) and turning point W2 is 2 units, and 3 units between he center and W1. This is determined by their situation on the auxiliary circles. But what is the distance between W1 and W2?

Here the special property of the Knidos Style becomes apparent. All construction lines are integral multiples of the unit long. Their length is determined by the number of segments of the Ariadne’s Thread between two turning points. Between W1 and W2 there are two such segments. Therefore the distance between W1 and W2 is two units. We thus have to move both turning points along their auxiliary circles until they have reached an exact distance of two units.

The construction lines fit together to a polygon (in our case a triangle). This connects the center with the turning points. For the length of the construction lines between the turning points and the center, two units have to be added to the number calculated as described above, due to the enlarged area around the center. The resulting triangle has sides with lenghts of 3 (M – W1), 2 (W1 – W2) and 2 (W2 – M) units. By this, all points are determined relative to each other. This also works in labyrinth types with more than two turning points.

That is ingenious!

This polygon can now be rotated around the center. By this it is possible to align the axis as desired. Erwin aligns the segment of the pathway, that leads to the center, centrally on the central axis.

Knidos_Achse

Figure 6. Alignment of the Axis

For this, as shown in fig. 6, the triangle must be rotated around the center in such a way that the turning point W2, where the pathway turns off to the center, comes to lie ½ unit to the right of the central axis. The situation of W1 is then given as a result of this position.

Knidos_fertig

Figure 7. Finalization

Fig. 7 finally shows how the figure is finalized. For this, from the turning points, auxiliary circles with a radius of 1 unit are inscribed. Then on these auxiliary circles, arcs of circles are drawn that connect the walls delimiting the pathway with the construction line between the turning points.

With “Knidos”, Erwin has developed a extraordinary way of the gaphical design of labyrinths. This can be justifiably classified as a style. While I have stated in my last post, that style cannot be determined as exactly as type, this is true for style in general. However, individual styles may be excepted. This applies to the Knidos style. This style is geometrically exact, fully determined by the relations of multiple integrals of the width of the pathway and can be designed using only compass and ruler.

Related Posts:
Type or Style / 7
How to Draw a Labyrinth
How to Draw / Build a Labyrinth with Meander Technique
How to Make a Knidos Labyrinth
The Knidos Labyrinth

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There is one more labyrinth in Würzburg, in the monastery gardens behind the church Saint Alfons. It is a Knidos labyrinth for which I could make the design. So we have a classical 7-circuit labyrinth with a larger centre.

The Knidos Labyrinth at St. Alfons

The Knidos Labyrinth at St. Alfons

Here the technical data which are not the most important things for a labyrinth, but which are in the area of my competence as a geometer: The dimension between axes (paths) amounts to 70 cm, the centre has the fourfold of the path width, therefore 2.80 m. The square over the four turning points has also 2.80 m. The whole diameter amounts to 12.60 m, calculated with 18 x 0.70 m. The way into the middle is 166 m long, the boundary lines are 188 m long.

The labyrinth lies in a nice flower meadow which is separated by a beech hedge from the surrounding garden. Soon it will have an own access from the adjoining small street. Now one still has to pass by the gate to the garages.
On Sunday 24th of July, 2011 was a parochial party in Saint Alfons and there the labyrinth was introduced for the first time publicly. There should be an official inauguration later.

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The “construction phase” began last year in autumn with a lawn-mowed labyrinth. Because I heard nothing more from the project, I already thought that the labyrinth could not be realised; until I got the invitation for the parochial party last week.
There I found out that the keen initiators, Mrs. Dreier and Mrs. Heilmann, had all the time mown the labyrinth and thus got it through the winter and the spring without I once again had to mark it out.
In the meantime the then projected beech hedge was also planted and thus this wonderful meadow labyrinth was created. At the entrance stands a bank donated by the friendly society of settlers and invites to stay at the labyrinth.

Congratulations to the labyrinth!
With the wish that it may be visited and walked by many people.

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The preparations for this labyrinth are known. They were object of the previous article with the title: How to build a Knidos Labyrinth with Bark Mulch.

This was as it were the test run or a specimen for the final version.

In the meantime, the meadow had been mowed once again and the bark mulch had disappeared to a great extent. Hence, for the preparation of the construction the lines had to be marked out again. Following the still recommended method the 5 mean points were marked again with iron bars. Beginning from the inside, the limitations of the labyrinth were sprayed on the ground outwardly with white spray paint. A double line was chosen at about 14 cm width. This was the mark for the pits to be dug up with the spade.

Here pictures of the preparation:

The 5 mean points

The 5 mean points

The inner parts

The inner parts

The inner piece of cake

The inner piece of cake

The whole marking

The whole marking

The inner square

The inner square

The complete labyrinth

The complete labyrinth

A total of 10 to 12 people were at work. Digging up the lawn could be done at the same time and at different places. The overrun lawn was taken away, on the way back the concrete paving stones (20 cm long, 10 cm wide) were brought, as well as the grit for the paving.
The laying of the paving stones began with the laying of the foundation stone in the middle of the central cross. Then from here were laid, beginning at the four ends of the cross, the 8 arcs of the labyrinth up to the respective inflection point (which at the same time are also centres). Through that no interpieces or passport pieces were necessary. Also this could already happen during the time of digging.

Here pictures of the construction:

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Everybody has searched his work on its own, the teamwork arose by itself. A praise and a thank-you to all participants.
The labyrinth was ready with the laying of the “keystone” after 5 hours construction time.

Here the result:

Foundation stone

At work

At work

At work

Cap stone

Cap stone

The centre

The centre

The completed labyrinth

The completed labyrinth

View back

View back

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How to do that should be clear theoretically. It was posted  earlier on this blog. But if you do not remember any more or would like to look once again: the posts below are recommended.

The labyrinth here has a dimension between axes of 50 cm, the centre should get a diameter of 2 m, the central square also has sides of 2 m length, the whole diameter will thereby be 9 m.

We first stake off the 5 mean points on which later literally everything turns.

The 5 mean points

The 5 mean points

The marked 5 points

The marked 5 points

Then the internal and a little more complicated parts are “built”: strewn with bark mulch. These are all parts which are lying within a triangle formed by the 5 marked points. Thus a cake piece is built through the the inner ways of the labyrinth which are all specified by 4 different centres and different radii.
It is advisable to start with the internal cross, the only straight lines in the labyrinth; and then to form the remaining bends.
Doing so, the basis is created for the 8 following in parallel distance running arcs.
The inner arcs

The inner arcs

The inner arcs

The inner arcs

Now it is only a matter of finishing the remaining bends around the centre in the middle. Every arc has the same centre and a 50 cm bigger radius than the preceding arc, if one works from the inside outwardly. The arcs are running from the right side completely to the left and vice versa. For this work many people can be occupied. One could put on even 8 curves at the same time. Or work from two sides as it was done here.
The complete labyrinth

The complete labyrinth

The completed labyrinth

The completed labyrinth

The exact dimensions of the used radii can be taken from the following drawing, also the other necessary measures.
Here you can see/print/store/copy the drawing as a PDF file  …
 
The drawing

The drawing

Here the dimension between axes of 50 cm was used. A radius of 50 cm for the smallest arc and for the biggest one a radius of 4.50 m. Therefore the whole diameter amounts to 9 m. For the size of the centre the 4-fold dimension between axes was applied. Thus the diameter of the centre is the same as the side length of the square formed by the 4 inflection points.
The labyrinth is scaleable in the size. If one wants to build, e.g., one with a dimension between axes of 1 m (what results 18 m for the whole diameter), one multiplies all measurements by 2.

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Here you will see another way to build a labyrinth.

The original task was to make  a temporary labyrinth out of ropes in a large room. That seemed to be quite simple, because one knows how a labyrinth looks like, and how to make one from the basic pattern with the square, the four angles and the points in every corner. It should be a classical 7-paths labyrinth with a larger centre. But the question was: How many ropes does one need and where does one begin best?

A rope can easily be moved in curves and switched. Thus finally after longer trying a labyrinth resulted, even if the centre was somewhat small.

After that experience I wanted to know it more exactly and developed a kind of instruction, which shall be presented here.

First I locate some starting points, at which I can orient myself easily and to which I can take reference.

Figure 1

Figure 1

In this example the ways have a width of 50 cm, the centre has a diameter of 2 meters and the entire labyrinth a largest diameter of 9 meters. The points E1 to E4 are the end points of the lines and lie in a square, which can be very simply designed with the diagonal of 2.83 m.

Figure 2

Figure 2

On the basis of the two upper points E1 and E2 I can determine the middle of the centre M through the masses 2.00 m and 1.50 m. This point is temporarily marked, just as the middle of the square. Because from here and around the point M the ropes must be placed.

Figure 3

Figure 3

From the central point I go straight upwards and put then the rope in an even distance of 1 m (= radius 1m) around point M; when I am near E2, I go in a distance of 0.5 m (=Radius 0.5m) around point E2 and put the rope in the even distance of 1 m parallel to the part already lying until point E1. So I have done the red line 8 and 6.

Figure 4

Figure 4

Now the blue line 5 and 7 shall be constructed. I start again in the central square point, goes a piece straight to the right until the points E2 and E4 and turn then to the left and above, by following parallel to and in the distance of 0.5 m the red line 6 already put. At point E1 I circle round this point, then moving between the red line 6 and 8 until point E2. I have put the blue line 5 and 7.

Figure 5

Figure 5

Now it is the turn of the brown line. Again from the central square point I go  first straight until to the points E1 and E3. Then in the distance of 0.5 m parallel to the blue line 5 until the points E2 and E4. The point E4 is circled in a distance of 0.5 m outwards and then I lay the rope parallel to the still lying part of line 4 in the distance of 1 m until point E3. The brown line 2 and 4 is finished.

Figure 6

Figure 6

For completion still the green line is missing. From the central square point I go straight downwards to the points E3 and E4; then I turn to the right in the distance of 0.5 m parallel to the brown line 2 around point E4 and onward until point E3. This point is circled inward in 0.5 m distance and the green line 3 is moved between the existing brown line 2 and 4 up to the point E4. The green line 1 and 3 is done. The labyrinth is finished.

Here you can get the instruction as PDF file to see/print/store/copy it …

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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  …

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