Kepler Towers, Catalan Numbers and Strahler Distribution
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Kepler towers, Catalan numbers and Strahler distribution (extended abstract) dedicated to my friend Adriano Garsia for his 75th birthday Xavier Viennot LaBRI, CNRS and Université Bordeaux 1, 33405 Talence, France viennot (à) labri.fr abstract We introduce a new class of combinatorial objects called “Kepler towers”, counted by Catalan numbers. This objects are related to the combinatorics of Strahler numbers of binary trees and to the theory of heaps of pieces. résumé Nous introduisons une nouvelle classe d’objets combinatoires appelés “Tours de Kepler” et dénombrée par les nombres de Catalan. Ces objets sont liés à la combinatoire des nombres de Strahler des arbres binaires et à la théorie des empilements de pièces. Some historical background Another title for this talk could have been “From Kepler to computer graphics of trees”. Adriano loved synthetic images of trees and landscapes we have produced some years ago with D.Arques, G.Eyrolles and N.Janey, see [11]. These images, such as the one displayed on Figure 1, are given by some algorithms based on some extensions of the so-called Strahler analysis of binary trees. Figure 1. Synthetic images of trees ASIA 1989 1 In fact Strahler was an hydrogeologist and, together with Horton, was studying the morphology of rivers networks. They indroduced in [5], [7] the notion of order of a river in a river network. Each river takes birth with an order 0. When two rivers of order i and j meet, they form a river of order max(i, j) whenever i ≠ j ; if i = j , then the resulting river has order i+1. In the same way, this notion can also be defined for any complete binary tree. A complete binary tree has n internal vertices and n+1 external vertices (or leaves). Each internal vertices has€ two sons: a left and€ a right son.€ Starting with each leaf labeled 0, one can recursively label the vertices with the same rule: a father is labeled max(i, j) (resp. i+1) whenever the two sons are labeled i and j with i ≠ j (resp. i = j ). The so-called Strahler number of the binary tree is the label of its root (which is also the largest label of its vertices). This defines a parameter on the set of binary trees . The distribution of this parameter among all trees with n internal vertices (counted by Catalan numbers) is called the Strahler€ distribution. This distribution has some very nice and deep properties.€ € Figure 2. Strahler number of a binary tree Strahler analysis and combinatorics of Strahler numbers appear in many completely different contexts such as theoretical computer science (minimum number of registers needed to compute an arithmetical expression) [3], molecular biology (secondary structure of RNA molecules [9]), analysis of fractal patterns in physics [8] with application to radiology (digital mammography) [2], visualisation of information (see for example [1] and references therein). A survey paper (in 1990) is [12]. One may ask the question: what is doing Kepler in this Catalan-Strahler story ? Kepler’s Mysterium cosmographicum During my stay in Mittag-Leffler Institute in January-February 2005, under passionate and fruitful discussions with Don Knuth, I introduced the concept of “Kepler towers”. Why such a reference to Kepler ? Johannes Kepler is well known for his three laws describing the elliptic movement of planets around the sun, published in his Astronomia nova 1609 and Harmonice mundi 1619. Since the very beginning of his scientific life, Kepler adopted Copernic model of the solar system described in De revolutionibis 1543, and he was looking for an explanation of the distribution of the speed and distance of the planets to the Sun. He was also looking for an explanation of the number of planets (6 at that time). In his first book, the Mysterium cosmographicum, published in 1596, Kepler invented a model of the solar system, where the orbit of each planet is represented by a sphere and where the five regular 3D polyhedrons (known since centuries as platonic solids) would be embedded between each sphere in a certain precise way. Of course nowadays this model seems absurd, together with the question of finding an explanation for the number of planets. But let’s go back to original ideas and tentatives of Kepler: at the beginning he was considering 2D embedding of circles into regular polygons. This is our starting point for defining Kepler towers. 2 Figure 3. Kepler’s model from Mysterium cosmographicum Kepler towers Let’s consider an infinite sequences of regular polygons, starting with a polygon with only 2 edges, then with 4 edges (a square), 8 edges (octogon), ...., each polygon is centered in the th n n 1 + origin and the n polygon Pn has 2 edges and is embedded in the polygon Pn+1 having 2 edges, see Figure 4. € € € € € Figure 4. Polygons P2 , P4 , P8 3 € € € Now we refer to the theory of heaps of pieces introduced in [10]. For each polygon Pn , we can define the set of heaps of dimers over this polygon. The polygon Pn is considered to be at level 0. Above the polygon Pn , we imagine a cylinder or tower, i.e. a sequence of the same polygon, each polygon of the sequence being placed at level 1, 2, ..., k, .. etc. € € € Figure 5. A system of Kepler towers with 22 dimers and 3 towers A heap of dimers above Pn is a finite set of disjoint edges (or dimers) choosen in the sequence of polygons above Pn (including Pn itself) satifying the following condition: - if a dimer is at level i > 0, then there exist a dimer at level i-1 such that their projections on the “ground floor” (i.e. polygon Pn at level 0) intersects. Recall that a dimer€ is a set of two consecutive vertices in the polygon. They are visualized on Figure€ 5 by red edges.€ Suppose we color alternatively the edges of each polygon Pn black and white. € € Definition 1. A system of Kepler towers is a finite sequence of heaps of dimers H1,...,Hk , each th i heap Hi (called the i tower) is a heap of dimers above the polygon Pi (with 2 edges)€ satisfying the following condition: - at level 0, each heap H , 1 ≤ i ≤ k , contains all the 2 i−1 black edges of the polygon P . i € i € € € Thus, for any system of Kepler towers with k towers, the€ total number of dimers is at least 1+ 2 + ...+ 2 k = 2k+1 −1. These dimers are visualized by the red edges displayed on Figure 4. € € € € Another love of Adriano are the mysterious (q,t)-Catalan numbers. We do not have (q,t), but at least we have here a remarkable fact. It is probably not obvious to recognize in these € Kepler towers another incarnation of the very classical Catalan numbers: Proposition 2. The number of system of Kepler towers having n dimers is the Catalan number 1 2n Cn = . n +1 n In fact, we have more: € 4 Proposition 3. The distribution of systems of Kepler towers according to the number of towers is the Strahler distribution, i.e. the number of systems of Kepler towers having n dimers and k towers is equal to the number of (complete) binary trees having n internal vertices and Strahler number equal to k. We will give two proofs, bijective and analytic of this fact. Before that, let’s give another equivalent definition of Kepler towers, as considered by Don Knuth, by taking a 2D projection of the 3D sequence of Kepler towers H1,...,Hk , and replacing each regular polygons Pi by circles (or rings) divided into 2 i segments of equal length. Kepler towers viewed as circles and walls € € € Figure 6. Kepler disk as a 2D projection of 3D Kepler towers A Kepler disk (system of Kepler towers) is formed by a certain number of walls (towers) th W1,...,W k .The i wall of the Kepler disk consists of one or more rings (or circles) (the polygons i above each polygon Pi ) , where each ring is divided into 2 segments (edges) of equal length. Inside each ring, we choose certain segment called bricks. Each brick is slightly longer than the length of the segment. € € th i Brick positions in the rings of the €i wall are numbered 1, 2, ..., 2 , going counterclockwise€ and starting from the segment due east of the center. For example the Kepler disk of Figure 6 can be specified by the following segment-number sequences: 1; 2; 2; (first wall) 1,3 ; 4; 1,3; (second wall) € € 1,3,5,7; 1,4,7; 3,8; 2,4,7; 1,7 (third wall) This Kepler disk is the projection of the system of Kepler towers displayed on Figure 5. With this point of view, the condition defining Kepler towers becomes the following conditions on rings and walls: (i) The positions of the bricks in the first ring (the innermost = level 0) of the ith wall are 1, 3, ..., 2 i -1 , for 1 ≤ i ≤ k . 5 € € € (ii) Bricks cannot occupy adjacent segments of a ring. (iii) Bricks in each non-innermost ring of a wall must be in contact with bricks in the ring just inside (=just below), that is if the brick is at position j, 1 ≤ j ≤ 2 i , then the ring just inside must contain at least one brick in position j-1, j or j+1 (modulo 2 i ). Note that in his memoir Mysterium cosmographicum, Kepler was also considering the orbits of planets symbolized by circles, the sun€ being in the center of the circles.