On my own behalf

Featured

Welcome to the Labyrinth

The topic of this blog is the labyrinth. Under nearly all aspects, I would like to arouse your interest on the fascinating lines and the meaning of this old object. Being an old surveyor I put my focus on the geometrical shape.
A new post should be published about twice a month. Meanwhile I am accompanied by Andreas Frei as coauthor.

Contents

In a blog the single posts (articles) are disposed in reverse order: the latest posts first, the older ones following. The display of the content is thus different from a website where it is always permanent.

Anyone who is looking for something special about labyrinths or just wants to know what can be found on this blog, maybe would like to have an overview.

I can provide this now and offer it as an own page titled Contents.

The register with the table of Contents is on top of the blog under the header image next to About us.

For a better view

For a better view

Subscribe

If you would like to be constantly informed about new posts, you can also follow this blog by subscribing.
The widget: SUBSCRIBE TO BLOGMYMAZE  is on the sidebar between IN SEARCH OF … and BLOGROLL.
You only need to enter your e-mail address to receive a mail when a new article is posted.

Advertising

This blog uses a free and unlimited offer from WordPress.com. Therefore, advertising is displayed at various points. We ask for your understanding.

Rights of use

Most of the pictures and graphics were created by Andreas Frei and me (Erwin Reißmann), unless stated otherwise, and are provided under license CC BY-NC-SA 4.0.

Tutorial on How to Draw a Classical 7 Circuit Labyrinth in Knidos Style

Here are detailed step-by-step drawing instructions for the construction of a geometrically-mathematically correct labyrinth.

The specifications are as follows: The unit of measurement for the distance between the lines in the axis is 1 m. The diameter of the center should be four times this distance, hence 4 m. The entrance to the labyrinth and the center are aligned with the central axis.

Details on the Knidos style can be found in this article.

Figure 1: First, the center point M1 of the labyrinth is determined. Starting from here, the main axis (vertical line) to the entrance of the labyrinth below is drawn. Then a parallel line is drawn as an auxiliary line at a distance of 1.50 m and an auxiliary circle with a radius of 3 m is drawn in M1. Using an arc, the midpoint M2 is then constructed at the intersection of these auxiliary lines on the right side below.

Figure 2: The midpoint M3 is constructed by cutting two radii with a radius of 4 m around M1 and M2 to the left of the central main axis.

Figure 3: First the straight lines M1-M2 and M1-M3 are lengthened, then seven circular arcs are drawn around M1 as the center with the radii 2.5 m to 8.5 m. This is Ariadne’s thread, the axis of the pathways, for the labyrinth.

Figure 4: Circular arcs with the radii 0.5 m and 1.5 m are drawn around M2 and M3 up to the ends of the corresponding previously constructed circular arcs. The right circular arc with a radius of 1.5 m only goes up to the intersection with the horizontal construction line and then leads as a straight line to the center M1.

Figure 5: A parallel line is drawn as an auxiliary line at a distance of 1.5 m to the left of the central axis. An auxiliary circle with a radius of 4 m is drawn around M3 as the center point and intersected with the vertical auxiliary line. This creates the center point M4.

Figure 6: The three open arches to the left of the extended line M1 – M3 are connected with the radii 2.5 m, 3.5 m and 4.5 m to the line M3 – M4.

Figure 7: Around M4 as the center point, two curved pieces with the radii 0.5 m and 1.5 m are drawn, the radius 1.5 m only up to the horizontal construction line to M4. From here a straight line connects to the entrance of the labyrinth at the bottom.

Figure 8: Two auxiliary circles with a radius of 4 m are drawn around the center points M2 and M4 and the new center point M5 is constructed to the right of the central axis at the intersection of the same.

Figure 9: In the new sector, the free curved end pieces on the right side are connected with a radius of 2.5 m to 5.5 m to the line M2 – M5 or its extension.

Figure 10: Around M5 as the center, two semicircles with a radius of 0.5 m and 1.5 m are constructed. The complete Ariadne thread for the labyrinth is now drawn.

Figure 11: Parallel to all previous arches, the boundary lines of the labyrinth are now constructed at intervals of 0.5 m. Starting with R 1 m up to R 9 m for the outermost ring. With this all lines for the labyrinth are complete and can be used for different representations of the labyrinth in different variants.

For example here with the same widths for the boundary lines. The Ariadne thread is the free space between these lines:

The Classical 7 circuit labyrinth aligned to the central axis in Knidos style

The Classical 7 circuit labyrinth aligned to the central axis in Knidos style

Here again the previous drawing steps are summarized in a single design drawing, which can be scaled as required.

The design drawing

The design drawing

Here you might view, print or download it as a PDF file.

Related Posts

Sector Labyrinths with Triple Barriers

Labyrinths with triple barriers must have at least seven circuits. With seven circuits these are sector labyrinths in any case. A triple barrier covers six circuits side by side. Thus, there remains one circuit for the passage from one sector to the next. This can only be the innermost or the outermost circuit.
For the seven circuit labyrinths with triple barriers we can find a situation comparable with the sector labyrinths with double barriers. Except for the first and the last sector, only two different sector patterns can be applied to the sectors that lie between them.

Figure 1. Sector Patterns for the Inner Sectors

In the first sector, only sector patterns can be used that have three nested turns at their right side. 

Figure 2. Sector Patterns for the First Sector

In the last sector, only sector patterns can be used that have three nested turns at their left side.

Figure 3. Sector Patterns for the Last Sector

The opposite sides of these sector patterns form the main axis. At these sides, the sector patterns may have different turns of the pathway. For the time being it remains open how the pathway may enter the labyrinth and how it may reach the center. 

However, what happens between the first and the last sector can already be determined here. Independent of the number of axes there exist four different possibilities for a course of the pathway through the sectors in between. Two possibilities for labyrinths with an even number of axes and also two for labyrinths with an odd number of axes. 

One of the possible courses in labyrinths with an even number of axes is shown in fig. 4. In labyrinths with 2, 4, 6 asf axes, the path may traverse from the first to the second sector on the outermost circuit. This may just be the last sector (upper row). But also with 4 (2. row), 6 (lower row) or any other even number of axes, the pathway will also traverse into the last sector on the outermost circuit. 

Figure 4. First Possible Course in Labyrinths with an Even Number of Axes

The second possible course in labyrinths with an even number of axes ist shown in fig. 5. This is complementary to the course shown in fig. 4. The pathway traverses from the first to the second sector on the innermost circuit and it also traverses from the previous to the last sector on the innermost circuit. 

Figure 5. Second Possible Course in Labyrinths with an Even Number of Axes

The first possibility for a course of the path in labyrinths with an odd number of axes (3, 5, 7 asf.) is presented in fig. 6. Here, the path changes from the first to the second sector on the outermost circuit. However, in labyrinths with an odd number of axes, the path does not reach the last sector on the same (outermost) circuit as in labyirnths with an even number, but on the opposite (innermost) circuit. As shown for labyrinths with 3, 5, and 7 axes, this applies to all labyrinths with an odd number of axes. 

Figure 6. First Possible Course in Labyrinths with an Odd Number of Axes

The second possibility for a course of the pathway in labyrinths with an odd number of arms is shown in fig. 7. This is complementary to the first possibility shown in fig 6. 

Figure 7. Second Possible Course in Labyrinths with an Odd Number of Axes

These are the only four possibilities for the course a pathway can take hrough all side arms, two each for labyrinths with an even and with an odd number of axes. Now it remains to design the main axis. For this, the halves of the sector patterns for the first and last sectors have still to be completed. The sector patterns that are eligible for this purpose shall be identified in the next post. 

Related Posts

The Labyrinths with 3 Double Barriers, 4 Arms and 5 Circuits

The Complementary (transposed) Classical 7 Circuit Labyrinth in Knidos Style

I described this type in the concentric style in my last post (see related posts below). Today it is about the representation of the transposed labyrinth in Knidos style.

The path sequence is: 5-6-7-4-1-2-3-8. The special thing about it is that one enters the labyrinth on the 5th circuit, and the center on the 3rd. circuit.

The walls and Ariadne's thread

The walls and Ariadne’s thread

And yet this type can be aligned to the central axis. This is only possible by editing in the Knidos style.

I come back to the original labyrinth using the same method that I used to get to the complementary type: I add the difference to the last digit (the goal) to the row of numbers in the path sequence. So:
5-6-7-4-1-2-3-8
3-2-1-4-7-6-5-8
8-8-8-8-8-8-8-8
This is then the original, well-known classical (Cretan) labyrinth.

What does the Knidos style actually mean?
By this I mean, above all, that the labyrinth has a larger center than just the width of a path, that it is as compact as possible and, above all, that it is developed from the path sequence and not from the basic pattern for the boundary lines (the walls). So it is Ariadne’s thread, the path in the labyrinth, that determines the construction. And this must be geometrically correct with constant path widths, elements that are as round as possible and as few “spaces” as possible.

Here in another graphic:

The transposed labyrinth in Knidos style

The transposed labyrinth in Knidos style

Here are the drawing instructions for a kind of prototype to be scaled for the axis dimension of 1 m.

The design drawing

The design drawing

Here you might view, print or download it as a PDF file.

Related Post

A Second Self-dual Pattern based on Gossembrot Type 51r

In the last post I have shown how the pattern of the Seven Times Seven labyrinth was generated (see: related posts, below). Of course, the same approach can also be taken with the part on the left side of the meander. This then results in a self-dual labyrinth with three axes.

In figure 1 I show the preparatory steps. The pattern of the Gossembrot type labyrinth forms the basis (a) and is presented in grey. Again, I then isolate the segment with the meander (b). In a second step, however, I select the figure from segment I and place it to the left of segment II (c).

Figure 1. Preparatory Steps

In fig. 2 the generation of the pattern is shown in an abbreviated way. Here we begin in the row in the middle with the figures shown in grey (c). In a third step, the figure from segment I is duplicated d). This duplicate is then, fourth, rotated by 180 degrees. This produces the dual figure of it (e). This is attached to the right side of the figure with the meander, and the three parts are connected to each other again (f).

Figure 2. Generating the Pattern

Figure 3 shows the labyrinth in base form.

Figure 3. The Labyrinth in Base Form

Again, a very well balanced labyrinth. The main axis is similar to the basic type labyrinth. Segment II opposite to the main axis contains the meander.

Relatled posts: