How to Make the Related Labyrinths

I use a different method to generate the related labyrinths than Andreas. But I’ll get the same result. This is how we complement each other.

Essentially, I am using the level or path (= circuit) sequence to get the version of a particular labyrinth I want. Also, I am taking the path sequence to construct the labyrinth, not the seed pattern.

The basic labyrinth
Fig.1: The basic labyrinth

I usually number from the outside in (the left digits in blue), additionally here from the inside to the outside (the right digits in green).
The level sequence for the basic labyrinth is here: 0-1-2-5-4-3-6. “0” stands for outside, “6” stands for the center. We have a 5 circuit labyrinth in front of us. “1” to “5” are the numbers of the circuits (paths), hence the path sequence 1-2-5-4-3 (fig. 1).

To create the dual labyrinth, I just use the green numbers on the right side of the basic labyrinth. I determine the level sequence by going outwards from the center. I get 6-3-2-1-4-5-0. Now I draw a labyrinth using this row of digits, going from the outside to the center. But first I replace “6” with “0” and “0” with “6”, I swap inside and outside as it were. The new level sequence is then: 0-3-2-1-4-5-6 (fig. 2).

The dual labyrinth
Fig. 2: The dual labyrinth to the basic labyrinth

The left numbers now indicate the level sequence: 0-3-2-1-4-5-6. If I now read the green numbers on the right side, I of course get the basic labyrinth again.

Now I use another technique to get the transposed labyrinth. I take the path sequence of the dual labyrinth, here: 3-2-1-4-5 and complement all numbers to “6”.
3-2-1-4-5 dual
3-4-5-2-1 transpose
The second line, completed by “0” for the outside and “6” for the center, gives the level sequence for the transposed labyrinth: 0-3-4-5-2-1-6 (fig. 3).

But there is still a different technique to get there: I can read the path sequence from the basic labyrinth backwards and again complete with “0” and “6”.
1-2-5-4-3 base
3-4-5-2-1 transpose
The second line, completed by “0” for the outside and “6” for the center, also gives the level sequence for the transposed labyrinth: 0-3-4-5-2-1-6 (fig.3).

The transpose labyrinth
Fig.3: The transposed labyrinth to the base labyrinth

If I now take the green numbers of the right side, I’ll get the dual of this transposed labyrinth, which is the next, the complementary labyrinth with the level sequence: 0-5-4-1-2-3-6 (fig. 4).

But again there is also the above described technique to get the complementary labyrinth. I take the basic labyrinth and complement the numbers of its path sequence to “6”.
1-2-5-4-3 base
5-4-1-2-3 complement
The second line, completed by “0” for the outside and “6” for the center, gives the level sequence for the complementary labyrinth: 0-5-4-1-2-3-6 (fig. 4).

I can also take the dual labyrinth and read the path sequence backwards, and again add “0” and “6”.
3-2-1-4-5 dual
5-4-1-2-3 transpose
The second line, added with outside and center gives the level sequence for the complementary labyrinth: 0-5-4-1-2-3-6 (fig. 4).

The complement labyrinth
Fig.4: The complementary labyrinth to the base labyrinth

If I now take the green numbers on the right side, I’ll get the dual labyrinth to this complementary labyrinth, namely the transposed labyrinth with the level sequence: 0-3-4-5-2-1-6 (fig. 3).

So we have seen three different ways to transform one labyrinth into another by using the path or level sequence.

However, it only takes two methods to create the appropriate labyrinths. I personally prefer the “transposing” technique and the “complementing” technique.

First we have the basic labyrinth (fig.1). Through transposing the path sequence of the basic labyrinth 1-2-5-4-3 into 3-4-5-2-1 I’ll then get the transposed labyrinth (fig.3).
This transposed labyrinth with the path sequence 3-4-5-2-1 I transform to the dual labyrinth by complementing the path sequence to 3-2-1-4-5 (fig.2).
This dual labyrinth I then transform to the complementary labyrinth by transposing its path sequence 3-2-1-4-5 into 5-4-1-2-3 for the complementary labyrinth (fig.4).
For control purposes, I can transform the basic labyrinth into the complementary by complementing the path sequence 1-2-5-4-3 of the basic into 5-4-1-2-3 (fig. 4) for the complementary.

All of these transformation methods have the same effect as the rotating and mirroring techniques by Andreas.

Related Post

How to Make (new) 11 Circuit Labyrinths, Part 3

In the previous articles on this topic, I have already explained the method of the stamp-folding calculation brought into play by Tony Phillips.

Now it should go on here. Namely, it is possible to generate further variants of labyrinths by simply rotating the polygon used.

I take again the net with the polygon from the last post on this topic (part 2).

The net with the polygon

This diagram can be used to create four different labyrinths. Two directly (line 2 and 3), the other two by a simple calculation.

Other constellations can be gained by rotating the network 12 times by 30 degrees. Or in other words, it’s just like changing the clock for the summer or winter time.
Since only interesting labyrinths are of interest here, I omit all positions where the lines would point to the first and / or last circuit. So from the 12 you should not reach the 1 or the 11. Only the “times” are interesting, which point farther away, that is, run more sharply.
That would be in the above net the 1, 5 and 6. So I turn only to these times. In other words, I bring the 1, 5, and 6 into alignment with the 12. I turn the net by 30, 150, and 180 degrees. To rotate is the net with the polygon, the numbers stay in place.

Here’s the first turn:

rotated by 30 degrees

rotated by 30 degrees

I get four completely different path sequences than in the original above.

The second rotation:

rotated by 150 degrees

rotated by 150 degrees

I get four new variants again.

The last rotation:

rotated by 180 degrees

rotated by 180 degrees

Here I just get a different order of the sequences than in the original polygon. So there are no new variants, just another arrangement. This is because the rotation of 180 degrees corresponds to a symmetrical reflection.

It is not always possible to find new variants. With the help of this net I have generated a total of 12 different path sequences for 12 new labyrinths.

The path sequences can be directly converted into a labyrinth drawing.
Here only one (again in concentric style) is to be shown (the 2nd path sequence from the first polygon above):

A new 11 circuit labyrinth

A new 11 circuit labyrinth

Related Posts

How to Make (new) 11 Circuit Labyrinths, Part 1

In my last posts I had shown the method of transforming the Medieval labyrinth by leaving out the barriers.

The first possibility to generate a labyrinth is of course the use of the seed pattern. Thus most of the Scandinavian Troy Towns with 7, 11, or 15 circuits were created.

Some years ago I wrote about the meander technique. Thereby many new, up to now unknown labyrinths have already originated.

Andreas still has demonstrated another possibility in his posts to the dual and complementary labyrinths. New versions of already known types therein can be generated by rotating and mirroring.

Now I want to use this technology to introduce some new variations.

I refer to simple, alternating transit mazes (labyrinths). Tony Phillips as a Mathematician uses this designation to explore the labyrinth. He also states the number of the theoretically possible variations of 11 circuit interesting labyrinths: 1014 examples.

The theoretically possible interesting variations of the 3 up to 7 circuit labyrinths once already appeared in this blog.

I construct the examples shown here in the concentric style. One can relatively simply effect this on the basis of the path sequence (= circuit sequence or level sequence). There is no pattern necessary.  The path sequence is also the distinguishing mark of the different variations.

I begin with the well known 11circuit classical labyrinth which can be generated from the seed pattern:

The 11 circuit labyrinth from the seed pattern

The 11 circuit labyrinth from the seed pattern

To create the dual version of it, I number the different circuits from the inside to the outside, then I walk from the inside to the outside and write down the number of the circuits in the order in which I walk one after the other. This is the new path sequence. The result is: 5-2-3-4-1-6-11-8-9-10-7- (12).
In this case it is identical to the original, so there no new labyrinth arises. Therefore, this labyrinth is self-dual. This in turn testifies to a special quality of this type.

Now I generate the complementary version. For that to happen I complement the single digits of the path sequence to the digit of the centre, here “12”.
If I add the single values of the row on top to the values of the row below, I will get “12” for every addition.

Or, I read the path sequence in reverse order. This amounts to the same new path sequence. But this is only possible with self-dual labyrinths.

I now draw a labyrinth to this path sequence 7-10-9-8-11-6-1-4-3-2-5-12.
Thus it looks:

The complementary 11 circuit labyrinth from the seed pattern

The complementary 11 circuit labyrinth from the seed pattern

This new labyrinth is hardly known up to now.

Now I take another labyrinth already shown in the blog which was generated with meander technique, however, a not self-dual one.

The original 11 circuit labyrinth from meander technique

The original 11 circuit labyrinth from meander technique

First, I determine the path sequence for the dual labyrinth by going inside out. And will get: 7-2-5-4-3-6-1-8-11-10-9- (12).

Then I construct the dual labyrinth after this path sequence.
This is how it looks like:

The dual 11 circuit labyrinth

The dual 11 circuit labyrinth

Now I can generate the complementary specimens for each of the two aforementioned labyrinths.

Upper row the original. Bottom row the complementary one.
The bottom row is created by adding the upper row to “12”.

The complementary labyrinth looks like this:

The complementary labyrinth of the original

The complementary labyrinth of the original

Now the path sequence of the dual in the upper row. The complementary in the lower one.
Again calculated by addition to “12”.

This looks thus:

The complementary labyrinth of the dual

The complementary labyrinth of the dual

I have gained three new labyrinths to the already known one. For a self-dual labyrinth I will only receive one new.

Now I can continue playing the game. For the newly created complementary labyrinths I could generate dual labyrinths by numbering from the inside to the outside.

The dual of the complementary to the original results in the complementary of the dual labyrinth. And the dual of the complementary to the dual one results in the complementary one of the original.

The path sequences written side by side makes it clear. In the upper row the original is on the left, the dual on the right.
In the row below are the complementary path sequences. On the left the complementary to the original. And on the right the  complementary to the dual one.

3-2-1-4-11-6-9-8-7-10-5  *  7-2-5-4-3-6-1-8-11-10-9
9-10-11-8-1-6-3-4-5-2-7  *  5-10-7-8-9-6-11-4-1-2-3

The upper and lower individual digits added together, gives “12”.

It can also be seen that the sequences of paths read crosswise are backwards to each other.

I can also use these properties if I want to create new labyrinths. By interpreting the path sequences of the original and the dual backwards, I create for the original the complementary of the dual, and for the dual the complementary of the original. And vice versa.

If I have a single path sequence, I can calculate the remaining three others purely mathematically.

Sounds confusing, it is too, because we are talking about labyrinths.

For a better understanding you should try it yourself or study carefully the post from Andreas on this topic (Sequences … see below).

Related Posts

Variations on the Babylonian Visceral Labyrinths in Knidos Style

By rotating or mirroring one will get dual and complementary labyrinths of existing labyrinths. Or differently expressed: Other, new labyrinths can be thereby be generated.
So I have three more new labyrinths as I can make a complementary one from a new dual labyrinth and I can make a dual one from a new complementary, which are identical. (For more see the Related Posts below).

Seen from this angle I have examined the still introduced 21 Babylonian Visceral Labyrinths in Knidos style and present here the variations most interesting for me. Since not each of the possible dual or complementary examples seems noteworthy.

Many, above all complementary ones, would begin on the first circuit and lead to the center on the last, which is yet undesirable.

Leaving out trivial circuits also will generate new labyrinths. This applies to the last two ones. If you compare the first and the last example you see two remarkable labyrinths: The first with 12 circuits and the last with 8 circuits, but using the same pattern.

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