The 5 Circuit Labyrinth on Rubik’s Cube

I started with the 3 circuit labyrinth on Rubik’s Cube (see Related Posts below). However, quite a few areas have remained empty. Therefore I have now dealt with a 5 circuit labyrinth.

First of all again the scheme for Rubik’s Cube with the notation of the different sides of the unfolded cube.

The unfolded, empty Rubik's Cube

The unfolded, empty Rubik’s Cube

From the 8 possible combinations for a 5 circuit labyrinth, I chose the variant with the path sequence: 0-3-4-5-2-1-6.
Ariadne's thread on the unfolded Rubik's Cube

Ariadne’s thread on the unfolded Rubik’s Cube

The path begins on the “front” side on the bottom left corner stone, then leads upwards (circuit 3), jumps around the corner to “left” and runs over “back” to “right” around the cube. From here it goes up (circuit 4) and back again left around to the “front” side. Then follow the circuits 5, 2, 1 and finally the jump to the center (6).
The number of the circuits begins on the bottom (the “down” side) of the cube, not on the “front” side. The goal, the center of the labyrinth is in “6” on the “up” side. And only one stone remains empty on the underside (the “down” side).

There is also a template that can be used to build a model. Here you can see, print or download it as a PDF file.

I tried that myself and made a more or less successful model out of paper:

Ariadne's thread on Rubik's Cube

Ariadne’s thread on Rubik’s Cube

In the large picture above, you can see the top (“up”), left and front sides from above. The three smaller pictures are each rotated by 90 degrees.

Here in a tilted representation:

Ariadne's thread on Rubik's Cube

Ariadne’s thread on Rubik’s Cube

In the large picture above, you can see the front, left and lower (“down”) side from above. The three smaller pictures are rotated by 90 degrees again.

I cannot judge how difficult that would be to solve. Especially if you do not say beforehand what that should be and you should find the beginning and end of the thread yourself.

Related Posts

Further Link

New 5 Circuit Labyrinths with Double Barriers

In dealing with the double-barrier technique in recent posts, I found this installation of Mark Wallinger’s Labyrinths on the London Underground:

The labyrinth 233/270 at the station Hyde Park Corner, Photo: credit © Jack Gordon

The labyrinth 233/270 at the station Hyde Park Corner, Photo: credit © Jack Gordon

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

The special feature of this is that two double barriers are located next to each other in the upper part of the central axis. In the routing chosen by him you move at the transition from the 2nd to the 3rd quadrant first away from the center.

I’ve changed that so much that you would “experience” a movement to the center in a walkable labyrinth.

This is what it looks like:

A new labyrinth in concentric style

A new labyrinth in concentric style

I have also moved the side double barriers and this makes the routing in all quadrants also different. So a new type of labyrinth is born.

Here in Knidos style:

A new centered sector labyrinth in Knidos style

A new centered sector labyrinth in Knidos style

Why not as a two-parted labyrinth?

A new two-parted 5 circuit labyrinth

A new two-parted 5 circuit labyrinth

The left part has the path sequence: 3-4-5-2-1 and the right part: 5-4-1-2-3, so there are two 5 circuit labyrinths in it.

And here again in Knidos style:

A new two-parted and centered 5 circuit labyrinth in Knidos style

A new two-parted and centered 5 circuit labyrinth in Knidos style

The remarkable thing about this type is that both the entry into the labyrinth in the 3rd lane takes place, as well as the entry into the center.

Related Posts

How to Draw the Eight 5 Circuit Labyrinths

There are, as now everybody knows, 8 different possibilities for a 5 circuit, one-arm labyrinth (where the axis is not crossed).

Tony Phillips has furnished evidence for this on his website. He is a Professor of Mathematics at the State University of New York and he approaches mathematically to the question.

Some of these versions are known and were already built as labyrinths to walk. But some just not. Andreas Frei has already shown these 8 types. Tony Phillips describes the curves of the Russian mathematician Arnol’d which lead exactly to these variations.

Now I would like to introduce once again all 8 variations. In the meantime, I have developed a method to construct a labyrinth according to the path sequence. This path sequence is also a good name and a differentiation sign for a certain labyrinth type.

The different labyrinths are not developed from the well-known pattern. Even not from the pattern for Ariadne’s thread.

The figures of the eight labyrinths, first in round shape and with a bigger middle, then in square form. The contained pattern is emphasised in colour. Indeed, it has been determined after the construction of the labyrinth. The ways and the boundary lines are equally wide. The path sequence should serve as a marking of the type.

0-5-2-3-4-1-6

0-5-2-3-4-1-6

0-5-2-3-4-1-6

0-5-2-3-4-1-6

0-3-4-5-2-1-6

0-3-4-5-2-1-6

0-3-4-5-2-1-6

0-3-4-5-2-1-6

0-1-2-5-4-3-6

0-1-2-5-4-3-6

0-1-2-5-4-3-6

0-1-2-5-4-3-6

0-3-2-1-4-5-6

0-3-2-1-4-5-6

0-3-2-1-4-5-6

0-3-2-1-4-5-6

0-5-4-3-2-1-6

0-5-4-3-2-1-6

0-5-4-3-2-1-6

0-5-4-3-2-1-6

0-5-4-1-2-3-6

0-5-4-1-2-3-6

0-5-4-1-2-3-6

0-5-4-1-2-3-6

0-1-2-3-4-5-6

0-1-2-3-4-5-6

0-1-2-3-4-5-6

0-1-2-3-4-5-6

0-1-4-3-2-5-6

0-1-4-3-2-5-6

0-1-4-3-2-5-6

0-1-4-3-2-5-6

Here some more explanations to the path sequence: The paths inside a labyrinth are numbered from the outside inwards, to the center. The order in which the paths are walked from the entrance up to the middle displays the path sequence. The digit “0” stands for outside (or beginning) and the last number marks the center itself.
The first number after “0” must always be an odd number, so 1, 3, or 5 (and so on). Otherwise it does not work.
The row of numbers must be composed of alternating even and odd digits. Otherwise it will not be a labyrinth. The mathematician expresses this differently, but for us this information should be enough.

Who would like to build a new 5- circuit labyrinth, can get here suggestions.
I personally like the second variation with the path sequence 0-3-4-5-2-1-6 best of all. Who builds it?

Related Posts

Related Link

Considering Meanders and Labyrinths

In a series of recent posts Erwin has elaborated on meanders in labyrinths. He has found a meander best suited for labyrinths. This particular meander exists in various forms. Erwin refers to these as “type” followed by an even number, e.g. type 4, type 6 and so forth.

Indeed, these types of meanders can be often found in existing labyrinths. However other figures occur in labyrinths too that can be denoted as meanders. Generally a broad range of figures is referred to as meanders. And it seems not so clear what a meander is at all.

So, what is a meander?

On his website, Tony Phillips cites Arnol’d, a Russian mathematician. Independent of the labyrinth, Arnol’d wanted to investigate how many meanders there are. He defines a meander as follows:

  • Connected oriented curve,
  • that does not intersect itself
  • and intersects a fixed, oriented line in several points.

He illustrates this with the example of a curve that intersects the fixed line five times. He found that there exist eight different such curves. I have copied the illustration from Tony’s website, enumerated the 8 curves and display it below.

8 curves

8 curves

It can easily be recognized that these curves are very closely related with the labyrinth. One only has to rotate them by a quarter of a circle in clockwise direction and straighten them out somewhat. This results in the patterns of 8 one-arm labyrinths with 5 circuits each. Therefore, let us have a closer look at these figures.

figure 1

figure 1

This figure represents a serpentine that leads from the entrance to the center of the labyrinth. It is this the pattern of the historical Näpfchenstein labyrinth. This labyrinth is self-dual.

figure 2

figure 2

This curve depicts the pattern of the labyrinth type Löwenstein 5b. This is the dual of the labyrinth shown in figure 4. Dual labyrinths have the same pattern, although the pattern is rotated by half a circle, and the entrance and center are exchanged. The “Rockery Labyrinth“, designed by Erwin is also of this type.

figure 3

figure 3

This curve includes a pattern of the Knossos type (3 circuits) to which are attached an additional circuit on both, the outer and the inner side. I am not aware of any existing labyrinth of this type. This labyrinth is self-dual.

figure 4

figure 4

This is the pattern of the labyrinth type Löwenstein 5a. It is dual to the labyrinth shown in figure 2. The “Pilgrim Hospices” labyrinth, designed by The Labyrinth Builders is of this type.

figure 5

figure 5

This figure contains the pattern of the labyrinth I use for my investigations and presentations, so to speak my demonstration labyrinth. It has the following properties that are important for this purpose: the path does not enter on the first circuit, it does not reach the center from the last circuit and the labyrinth is not self-dual. The dual to this labyrinth is shown in figure 7.

figure 6

figure 6

This pattern corresponds with a serpentine from the inside out. The path enters along the axis and first encounters the innermost circuit. From there it winds itself out circuit by circuit until it reaches the first (outermost) circuit. Then it is directed axially to the center. Erwin has discovered this type of labyrinth (Chartres 5 classical) by omitting the side-arms of the Compiègne-type labyrinth. This labyrinth is self-dual.

figure 7

figure 7

This is the dual of my demonstration labyrinth shown in figure 5.

figure 8

figure 8

This curve represents Erwin’s meander best suited for labyrinths. It is a type 6 meander. This is the pattern of the core-labyrinth of the historical Rockcliffe Marsh labyrinth. Rockcliffe Marsh is a very special labyrinth. First it has an unusual layout. The figure is opened along the axis and unrolled to a segment of a circle. Second, it is made up of a core-labyrinth (the inner 5 circuits) that is enclosed by a spiral outside.

Conclusion

Arnol’d’s definition of a meander is closely related with the labyrinth. His curves correspond with the patterns of all one-arm labyrinths in which the pathway does not cross the axis. The number of intersections between the curve and the fixed line corresponds with the number of circuits in the labyrinth. This was demonstrated in detail for labyrinths with 5 circuits.

  • Thus there exist 8 different patterns for a labyrinth with one axis and five circuits with the pathway not crossing the axis.
  • With an increasing number of circuits, the number of different pattern increases dramatically. E.G. there are 14 possible patterns for a labyrinth with 6 and 42 for a labyrinth with 7 circuits.
  • According to Arnol’d’s definition, all 8 figures are meanders. If we follow Erwin’s definition, only figure 8 is a meander suited for labyrinths.
  • If we adopt the definition of Erwin, we will capture the most common and essential labyrinths. However, we will also miss a broad range of existing and potential patterns of labyrinths.
  • If we adopt the definition of Arnol’d, every pattern of a one-arm labyrinth, in which the way does not cross the axis is referred to as a meander. This definition seems too broad and can be further differentiated.