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Friday, November 18, 2016

Removed posts

Five posts analyzing Olivier Messiaen's Île de feu II (thread title: 'Broken Symmetries') have been temporarily removed from this blog while I rewrite and consolidate them with additional material for an article to be submitted for publication.

Tuesday, September 20, 2016

Ode on a Cretan Maze

This one is just for fun.........

It may be that universal history is the history of the different intonations given a handful of metaphors.
– Borges, 'The Fearful Sphere of Pascal'

The Pylos Maze (tablet pre-1200 BCE)*



(a)                                                   (b)
Illustration 1.
Sketch for analysis of the Pylos Maze,
rotated clockwise 90 degrees from photo above
(Prof. Tony Phillips)


Illustration 1 is taken from the article 'Hidden Symmetries of Labyrinths from Antiquity and the Middle Ages' by Tony Phillips appearing in the American Mathematical Society's October 2015 Feature Column: Monthly essays on mathematical topics. As Phillips describes it:
The oldest securely dated labyrinth design appears on the back of an accounting tablet found in the ruins of King Nestor's palace in Pylos, on the western shore of Greece. The tablet (7 X 5.7 cm) was baked when the palace burned in 1200 BC. The design on this tablet is often called the Cretan Maze. It appears on coins minted on Knossos on Crete during the 5th century BC and later, and represented the legendary labyrinth where King Minos kept the Minotaur.
The aim of the article as a whole is 'to point out that the great majority of labyrinth designs share a topological symmetry which, while not obvious, cannot be accidental.' Phillips works through  examples of several labyrinths (the reader is encouraged to go to the entire article), but I'd like to concentrate on the Cretan Maze as another illustration of hidden or 'deep' form – one of the main themes in Essays & Endnotes.

If you stare at the labyrinth on the left as it appears on the tablet (Illustration 1a), you realize that, considered as a 2-dimensional rigid figure, it's asymmetric  – you can't rotate or flip it to get an identical figure. So first, Phillips labels the levels of the labyrinth in red, starting on the outside with 0 and ending at the goal with 8 (Illustration 1b). Now enter the labyrinth at 0 (at the opening on the right) and follow the path. Each time you cross the line of red numbers, record the number you cross. The 'level sequence' you come up with will be 032147658. Phillips reveals the hidden symmetry by first noting that 'if each number n is eight's complement 8–n [in music theory terminology this is "inversion" I8 (mod 9)] the sequence becomes 856741230, which is the original sequence read backwards [retrograde]' – the way to back out of the labyrinth. He then "lifts" the labyrinth into three dimensions by wrapping it around four faces of a cube (see his illustrations in the article) which ultimately reveals its hidden rotational symmetry. 

Using the pictorial representation of permutations that we've used previously (beginning with spiral representations), after some unscrambling of the original labyrinth using Phillips' labelling of the levels, we can see that Illustration 1b is topologically identical to Illustration 2.


Illustration 2.

So the labyrinth path can be analyzed as a concatenation of two identical paths,

(nn+3, n+2, n+1, n+4)                                                         (1)

with n=0,4. Tracing these paths in K9 (mod 9 circle space) with red for n=0 and blue for n=4 (Illustration 3) gives us another view of labyrinth permutation.

Illustration 3.

Now if we let n take the values 0,4,8 we get the obvious extension of the Pylos labyrinth permutation (Illustration 4).


Illustration 4.

And then, working backward, we can translate this into a maze of three 'levels' by surrounding the original two-level maze with another level:
+

=

Illustration 5.

We can indefinitely extend expression (1) that generates the permutation (and the labyrinth) by letting n=4k with k=0,1,2,3,.... Do we then have a labyrinth that has a center but no outside? If we begin at the center and some demon keeps adding to k it will take us an infinite amount of time to get out; that much is clear. But, with a nod to Zeno and pace Cantor, exactly where could one enter such a labyrinth?




__________________________

* Photograph from Weisstein, Eric W. "Maze." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Maze.html