# Classifying the Labyrinths with 3 Double-barriers, 4 Arms, and 5 Circuits

There exist 64 patterns of labyrinths with 3 real or pseudo double-barriers, 4 arms, and 5 circuits (see: related posts 1, below). Up to now, only a few of them have been published in any form at all, and among those probably most were published in this blog. The earliest one, however, is a Roman mosaic labyrinth. Now I am tempted to attribute all examples that have been shown last year in this blog to one of the 64 patterns.

Until now, I have used the following names for the patterns:

A – H for the 8 labyrinths with exclusively real double-barriers
A’ – H’ for the 8 labyrinths with exclusively pseudo double-barriers

Therefore, so far, only 16 of the 64 patterns have been given a name. These names have evolved during the elaboration of the previous posts. In order to attribute a name to each of the 64 patterns a more systematic approach is needed. For this, the naming has to be refined. This can be done by drawing upon the tree-diagram (related posts 1). I have given the name D to the uppermost pattern with exlusively real double-barriers, and D’ to the lowermost that uses exclusively pseudo double-barriers. Now we need a differentiation that provides unique names to all eight patterns, i.e also to the six remaining patterns.

Fig. 1 shows (still with the example of the basic labyrinth D) the kind and the sequence of connections between the sectors. Direct connections (real double-barriers) are represented with a horizontal (–), indirect (pseudo double-barriers) with a vertical (|) mark. This order of the combinations is not arbitrary, but systematically ordered. The uppermost combination consists of only direct connections and is represented by the sequence – – –. In the second combination, the last direct connection is replaced by an indirect connection, resulting in the sequence – – |. The third combination replaces the central direct by an indirect connection and results in the combination – | –. The fourth combination uses indirect instead of direct connections in the central and last connection (– | |). And so forth.

Figure 1. Sequence of Connections

If we substitute „–„ with „0“, and „|“ with „1“ , we can see, that the order of the combinations simply corresponds with the binary numbers from 000 to 111. Namely, these are the first eight numbers from Zero to Seven written in the binary system.

Figure 2. Order of the Combinations

With this we can give a unique denomination to each of the 8 patterns that have been derived in fig. 1 (starting from labyrinth D). And not only that. This denomination also provides insight into how the sectors are connected one with another. In this new denomination, I name the first pattern D – – –. It’s previous name was D. The second pattern previously had no name yet and it’s new name is D – – | and so on til the seventh pattern, all without previous names. The lowermost, eighth pattern prevously was named D’ and it’s new name is D | | |. This systematics is independent of the basic labyrinth. We can apply it to all labyrinths A – H. By this, we can give an unique denomination to each of the 64 patterns made-up of a uppercase letter followed by three horizontal or vertical marks.

Now I will attribute some real examples.

Three labyrinth examples can be attributed to one of the patterns from tree diagram D. The earliest one is Roman mosaic labyrinth of the Avenches type (related posts 5). This has the lowermost pattern D | | |.

The second example was introduced by Erwin in his post from August.2019 (related posts 4) and has the uppermost pattern D – – –.

The third example is the one by Mark Wallinger 233/270 shown in the post from October 2019. This has the third pattern D – | – (related posts 2).

Figure 3. Labyrinths of Group D

The labyrinths from fig. 4 cannot be attributed to any variant of the basic labyrinth D. These are all labyrinths with only real double-barriers. That means, they are the basic labyrinths and have the uppermost patterns of other tree diagrams.

The first labyrinth was shown in Erwin’s post from August 2019 and has the pattern G – – – (related posts 4).

The second labyrinth from the post of Erwin from September 2019 has the pattern F – – – (related posts 3).

Third is the labyrinth 10/270 by Mark Wallinger from the same post, and this has the pattern A – – –.

Figure 4. Classification of Other Labyrinths with Real Double-Barriers

The new sector labyrinth in fig. 5 from Erwin’s post from October 2019 has the pattern G – | – (related posts 2). Thus, it is one of the 48 patterns with mixed real and pseudo double-barriers.

Figure 5. Labyrinth from Group G with Real and Pseudo Double Barriers

The labyrinth shown in fig. 6, however, cannot be attributed to any one of the 64 patterns, because it does not have double-barriers in all side-arms. This labyrinth originates also from the post of Erwin from September 2019 (related posts 3). This example shows very well, that it is not possible to generate double-barriers using sector patterns no 1 and no 6.

Figure 6. Not a Labyrinth with Only Double Barriers

However, this was not the intention either. Erwin just wanted to once make use of all eight sector patterns in a four arm sector labyrinth.

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