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Using the Cayley graph description, I proved a nice little characterization Christoffel words that I will be using in an upcoming paper. I have been looking in the literature to see if I can just reference the result, but have so far turned up nothing. So this is really a reference request.

The result is as follows: suppose you take a binary word on the symbols $A$ and $B$ and arrange the characters around a circle, equally spaced. The result says that the word is a Christoffel word if and only if

  1. The diagram has a line of "almost" mirror symmetry, where the only symmetry break is around one of the points where the line meets the circle -- the two adjacent characters don't match.
  2. There is an additional line of mirror symmetry.

For example, AAABAABAAB is a Christoffel word, which produces the diagram

enter image description here

where in this depiction we start at the bottom and move clockwise to form the given Christoffel word. The dotted pink line is a line (1) of "almost" symmetry, while the dotted green line is a line of symmetry.

(In fact, the restriction that we begin with a word on two symbols is unneeded. The existence of lines satisfying 1 and 2 is enough to force the word to only have two distinct symbols.)

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    That looks a lot like palindromization, but for sure you know the reference: http://www-igm.univ-mlv.fr/~berstel/LivreCombinatoireDesMots/2008wordsbookMtlUltimate.pdf – 2017-01-04
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    Yes, I've seen it. Definitely (1) in my post is a standard property, mentioned for instance in that book. But the section on palindromization characterizes the palindrome obtained by omitting the initial and final characters. It does this in terms of periods, and also in terms of the palindromic closure operator. But neither of these seems to me in any obvious way to be equivalent to the existence of an additional line of symmetry for the entire necklace. – 2017-01-04
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    This is a really nice result - how does it work for words of odd period? – 2018-03-29
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    Do you mean odd length? A word with such a mirror symmetry and an "almost" symmetry can be shown to be primitive, i.e., the necklace cannot have a nontrivial rotational symmetry. So the corresponding word cannot be completely periodic. But if you lop off the two letters that are breaking the symmetry, the shorter necklace that results will be periodic. – 2018-03-30

1 Answers 1

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Palindromization, as basic concept in Combinatorics on Words, is not helpful to understand the symmetry indicated by the green dotted line. The central place of this concept in that discipline is related to a view on Christoffel sequences in which these are conceived as finite letter strings. In such finite letter strings the symmetry represented by the green dotted line doesn't show up. Al least not universally. I guess this is the reason why, as far as I can see, there are no publications regarding this type of symmetry within this branche of mathematics. To understand the symmetry represented by the green dotted line, a different conceptual approach is needed, in which symmetry in letter ordering is more suitable as a central concept than palindrome. When you apply this different approach, in an indirect way there are some links to existing literature within the area of Combinatorics.

One of the two letters traversed by the green dotted line, dependent on which of the two you choose as starting point of the period, belongs to two different periods of the respective Christoffel sequence. At least, this applies when you assume that this line represents a period-covering-symmetry-structure within a Christoffel sequence which is two sided-infinite and periodic. To bring this type of symmetry within the borders of one period, no matter if you present a period on a circle or demarcate it within a bi-infinite serial presentation of the sequence, you need to distinguish reversal-point-particles (rpp's) within the respective period-covering-symmetry: one in the mid and one divided over the two ends of the respective letter string.These rpp's have an intrinsic mirror symmetry, so each can be represented by two letter halves and by consequence one of the two can be divided over the two ends of the letter string. In the example above the letters A and B traversed by the green dotted line are the rpp's within this symmetry. The period covering symmetrical structure can be written as A/2 ABAA B/2 B/2 AABA A/2 or B/2 AABA A/2 A/2 ABAA B/2. In cases in which either the number of A's or the number of B's in a period is even, one rpp is empty. In that case the respective rpp doesn't coincide with a letter and can be neglected (the other is placed in the centre).

The discipline 'Combinatorics on Words' only in a specific categorie of Christoffel sequences acknowledges the presence of the symmetry represented by the green dotted line. Within this discipline it is well known [1] that every period in a Christoffel sequence can be seen as the product of two palindromes:w=qp.(I got this notion from Sebastien Labbe in a discussion via email in 2015. I am not a mathematician myself and don't master the concept of palindromization in all its nuance). Reasoning from this mathematical law it is evident that, in the cases where the number of letters in either q or p is even,for examble in aabab, there is a conjugate of this word (ababa) in which the symmetry in the green dotted line shows up. These are the cases in which one of the two rpp's in the period-covering-symmetry is empty. In the example ababa this empty rrp is divided over the two ends of the letter string. However, when the number of letters in both p and q are odd, for example in aabaabab, according to the discipline of Combinatorics there's no palindrome that covers the whole period. So in their view this type of symmetry is not universally present in every Christoffel sequence.

That the symmetry type represented by the green dotted line according to the 'Combinatorics' is not universally present,as I already stated, is due to the fact that they focus on one period in a Christoffel sequence, which they call a Christoffel word. A Christoffel word in their approach is the discretization of a line segment by a path in an integer lattice. But working with such a geometrical definition, they could also have conceived that line as running two-sided infinitely through a lattice of squares. Then the resulting sequence is a two-sided-infinite Christoffel sequence in which the symmetry represented by the green dotted line in all cases shows up. So it is a quenstion of how you define a Christoffel sequence: as finite or as infinite and when as infinite, in two directions infinite or only in one. A circular presentation of one period, as in your diagram, represents a two-sided-infinite Christoffel sequence. Christoffel himself only recognized the nearly-symmetry represented by the pink dotted line [2]. That Christoffel didn't observe the other type is logical. The algorithm he used generated one-sided infinite sequences. In these sequences the symmetry represented by the green dotted line doesn't exist.

Christoffel in his original obeservations, different from later mathematicians in the area of Combinatorics, did not speak of palindromes but of symmetry. And he distinguished an rpp in the symmetry he recognized, judging from the fact that he used a notification system to demarcate that rpp. When an rpp coincides with a letter, for example A, he noted that letter as |A|. An empty rpp he noted as |.So Christoffel already delivered the basic terms for the appoach we described above.

Because in the discipline Combinatorics on Words the Christoffel sequences are restricted to one period and conceived as finite sequences instead of as two-sided-infinte periodic sequences, representatives of this discipline, reviewing their publications, never have realized the intrinsic beauty of the type of symmetry represented by the green dotted line. The symmetry in the period covering letter string is built up of lower order building blocks (letter strings) that show the same type of symmetry as the period covering string. The different layers are related to the partial quotiënts in the continued fraction of b/a (b = number of letters B; a = number of letters A).

[1] Jean Berstel,Christophe Reuteneauer, Aaron Lauve, Franco Saliola. Combinatorics on Words: Christoffel Words and Repetition in Words. [2] Christoffel, E.B., Observatio Arithmetica. Annali di Matematica Pura ed Applicata 6 (1875),1, pp. 148 -152.

Harrie Welles

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    Thanks Harrie. The result about w=qp, with q and p both palindromes, does though imply directly that there will always exist a line of true symmetry in the circular arrangement. What I cannot find in the literature is the converse statement: if you start with ANY binary word ApB on characters A and B with p a palindrome, and if you arrange around a circle and find a line of true symmetry, then the word you started with must have been a Christoffel word. – 2017-05-27