Study Session:

Analysis of a Simple Maze

Introduction:

There are many different categories of mazes, most of them often complex and confusing, be it from the mathematical or artistic point of view. There are 3 types of unicursal mazes (otherwise known as labyrinths) that will be introduced here: the simple, alternating, transitive (s.a.t) labyrinth, the Roman labyrinth and the Church labyrinth, where more emphasis will be placed on a s.a.t. labyrinth. I chose to introduce these categories of labyrinths because I feel that it is necessary to start off with simple, unicursal mazes that follow a standard design and can be analyzed mathematically, in the topological point of view.

General properties of the 3 labyrinths:

1. The path is entered from the outside and ends in the center. Hence, there are an equal number of levels equidistant from the center.

(This figure shows the path taken to walk a typical s.a.t. labyrinth. The numbers indicate the levels that this labyrinth has from the outside to the center. In this case, the labyrinth has 8 levels)

Figure 1 Path of a labyrinth

1. The labyrinths have one path from the outside to the center, never crossing itself.
2. There is a finite number of levels.

Some basic definitions to know:

Fundamental form (FF) - Rectangular shape/design obtained when a labyrinth is unrolled.

Fundamental element (FE) - Minimal building block/element of a fundamental form of a labyrinth.

Figure 2 Elements of a labyrinth

Simple, Alternating, Transitive Labyrinth

The main characteristic of this labyrinth, other than the properties mentioned before, is the law of alternation that guides the formation of this labyrinth. The direction of the path will change whenever it changes level (eg. the paths travels from left to right in level 3 but moves from right to left in level 1) Also, the path makes one complete circle around the labyrinth before changing level. A famous example of the s.a.t. labyrinth is the Cretan labyrinth. The diagram below shows how a Cretan labyrinth can be drawn easily:

Figure 3 How to draw a cretan labyrinth

There is a topological view on s.a.t. labyrinths that can determine the level sequence in which an n-level labyrinth can be created. Level sequence refers to how the path is being walked from the outside before reaching the center. For example, the cretan labyrinth has a level sequence of 032147658, where the path starts with level 3 of the labyrinth, then proceeds to level 2, before going to level 1 and so on. Since a s.a.t. labyrinth is made of up of fundamental elements (FE), the level sequence which forms a FE will hence determine the level sequence that forms the labyrinth.

It is interesting to note that there are only 3 main FE that defines the topology of the s.a.t.  labyrinth:

§         0(n-1)…321n  (eg. 0543216 for a 6-level FE)

§         01234…(n-1)(n-1)n  (eg. 0123456 for a 6-level FE)

§         0(n-1)2(n-3)4(n-5)6…1n  (eg. 72543618 for a 8-level FE)

Using the Cretan labyrinth (refer to fig 2) as an example, it can be seen that this labyrinth is made up of 2 fundamental elements and when attached together, they form a FF that has a rotational symmetry. Taking the top FE as level 1-4 and the bottom FE as level 5-8, the level sequence for the top FE is 3214 and that of the bottom FE is 7658. They followed the 1st topology definition of the s.a.t. labyrinth. Attaching them together will form the level sequence of the Cretan labyrinth as 032147658.

Apart from using a topology to determine level sequences, a graphical method using chords of a circle can also be used to determine the level sequences of a s.a.t. labyrinth. This method makes the creation of a s.a.t. labyrinth much easier since different level sequences can be easily found.

Figure 4 Graphical method in finding level sequence

In general, a n-level labyrinth constitutes to n-equally-spaced points on a circle, labeled 1, 2, 3…n. Using 2 different colours or 2 different lines to represent the different directions of moving in a labyrinth (left to right, or right to left), draw a chord starting from point n to link another point, keeping to the condition defining the s.a.t. labyrinth (ie. odd and even integers must alternate). Draw another chord using a different colour or line to link to the next point. Continue drawing the chords, alternating the colours or lines until all the points are being linked up. Chords of the same colour of line cannot intersect each other. In figure 4, the red line represents moving from right to left, while the green line represents moving from left to right. A level sequence is then created by tracing back the points. Different level sequences can then be found by drawing chords that link up different points.

Looking at the level sequence of a s.a.t. labyrinth, it is easy to note that:

-         the sequence starts with a 0 and ends with a n

-         odd and even integers alternate in the sequence

-         the pairs of consecutive numbers in a level sequence corresponds to the vertical path segments on the right or left side of the labyrinth, depending on whether the number starts with an even number or an odd number

It is interesting to examine the structure of a simple, alternating, transitive labyrinth. Being the basis upon which other labyrinths are constructed, the s.a.t. labyrinth becomes the  first step to understanding the more complex unicursal labyrinths.

Roman Labyrinth

One obvious layout of the Roman labyrinth is the four sectors that form the labyrinth, giving it a 4-fold rotational symmetry.

Figure 5 Roman mosaic labyrinth           Figure 6 Path of a FF of the Roman labyrinth

The FF (unrolled form) of the Roman labyrinth is also made up of 4 sectors. In fact, the first 3 sectors are isomorphic to the FF of the Cretan labyrinth and the last sector is a form of FE, where one FE is nested in another different FE.

Figure 7 The last sector of FF of Roman labyrinth

In general, the study of the Roman labyrinth allows us to see the transference of one culture into another, where the Cretan labyrinth remains an integral part of the Roman culture.

Church Labyrinth

Just like the Roman labyrinth, the Church labyrinth is made up of 4 sectors that divide up the concentric levels.

Figure 8 Church labyrinth and its fundamental form

Again, like the Roman labyrinth, the FF of a Church labyrinth also consists of 4 sectors. However, the FFs of a Church labyrinth are all isomorphic to that of a s.a.t. labyrinth, making s.a.t. labyrinths the underlying design upon which Church labyrinths are constructed.

Conclusion

It can be easily seen that the 3 labyrinths, though different in structures, are fundamentally similar to each other. This may be due to the cultural connection they have with each other since ancient times. Despite having shown only a few fundamental elements here, it is possible to use these few FE to classify the overall topology of all unicursal labyrinths.