Journal of Information Processing Vol.25 528–536 (Aug. 2017) [DOI: 10.2197/ipsjjip.25.528] Invited Paper A recursive Algorithm for the k-face Numbers of Wythoffian-n-polytopes Constructed from Regular Polytopes Jin Akiyama1,a) Sin Hitotumatu2 Motonaga Ishii3,b) Akihiro Matsuura4,c) Ikuro Sato5,d) Shun Toyoshima6,e) Received: September 6, 2016, Accepted: May 25, 2017 Abstract: In this paper, an n-dimensional polytope is called Wythoffian if it is derived by the Wythoff construction from an n-dimensional regular polytope whose finite reflection group belongs to An,Bn,Cn,F4,G2,H3,H4 or I2(p). Based on combinatorial and topological arguments, we give a matrix-form recursive algorithm that calculates the num- ber of k-faces (k = 0, 1,...,n) of all the Wythoffian-n-polytopes using Schlafli-Wytho¨ ff symbols. The correctness of the algorithm is reconfirmed by the method of exhaustion using a computer. Keywords: Wythoff construction, Wythoffian polytopes, k-face numbers, recursive algorithm, Coxeter group ( c ) volumes and surface areas. 1. Introduction Our purpose has been completed and already implemented High-dimensional polytopes have been studied mainly by the as computer programs. We will introduce a new methodology following methodologies: to study volumes, surface areas, number of k-faces sharing a (1) Combinatorics and Topology, (2) Graph Theory, (3) Group vertex, etc. of Wythoffian-n-polytopes constructed from regular Theory and (4) Metric Geometry. See [1], [2], [8], [9], [10], [11], polytopes in the forthcoming papers. In this paper, as a first [14], [15], [18], [19] for some background researches on tilings, step, we develop a recursive algorithm that computes the f-vector lattices and polytopes. Coxeter [4], [5], [6] described polytopes ( f0, f1,..., fn), where fk (k = 0, 1,...,n) is the number of k-faces in terms of the Coxeter groups. The finite and infinite discrete of a Wythoffian-n-polytope. On f -vectors, so far it is not known groups of reflections systematically explain polytopes and honey- except for some of the components f0, f1, fn−1 and fn [12], [16], combs (tessellations), respectively, giving one of the most elegant [17], Our algorithm is elementary but powerful in that f -vectors theoretical backgrounds to the subject. It is noteworthy that he of all of the Wythoffian-n-polytopes can be recursively com- used the term “Wythoff construction” for some construction of puted using matrix multiplications. For example, among all of uniform polytopes obtained by using abundant symmetry of reg- the 98 Wythoffian-n-polytopes, f -vectors of the 6-polytope with ular polytopes. the Schlafli-Wytho¨ ff symbol {3, 3, 3, 3, 4}(0, 1, 0, 1, 1, 0) are ob- The subject of our study is to find the efficient methods to ob- tained as tain the following properties, especially fundamental quantities (1) f = ( f0, f1, f2, f3, f4, f5, f6) = (5760, 23040, 32160, 19680, of Wythoffian polytopes obtained by Wythoff construction: 5276, 476, 1) globally, and ( a ) Global numbers and shapes of k-faces, (2) f = ( f0, f1, f2, f3, f4, f5, f6) = (1, 8, 22, 29, 20, 7, 1) around ( b ) local numbers and shapes of k-faces sharing a vertex, and each vertex, by matrix multiplications without requiring any knowledge of 1 Research Center for Math Education, Tokyo University of Science, Shin- group theory. juku, Tokyo 162–8601, Japan The algorithm given in this paper is based on the classical com- 2 Research Institute for Mathematical Sciences, Kyoto University, Sakyo, Kyoto 606–8502, Japan binatorics, but has given new results and insights on the subject, 3 BANDAI NAMCO Studios Inc., Koto, Tokyo 135–0034, Japan including new classes of polytopes with tessellability and some 4 Division of Information System Design, Tokyo Denki University, Hiki, other interesting properties. High-dimensional discrete geometry Saitama 350–0394, Japan 5 Department of Pathology, Research Institute, Miyagi Cancer Center, Na- has many applications in the real world. For example, sphere- tori, Miyagi 981–1293, Japan packing problems in the 8-dimensional and the 24-dimensional 6 Systems Design Co. Ltd., Suginami, Tokyo, 168–0063, Japan Euclidean spaces play an important role in information theory. a) [email protected] b) [email protected] The authors are expecting that this program gives somewhat c) [email protected] polytope-version of this success. d) [email protected] e) [email protected] c 2017 Information Processing Society of Japan 528 Journal of Information Processing Vol.25 528–536 (Aug. 2017) Fig. 1 Fundamental simplices for the square and the cube. Fig. 2 Spherical tiling obtained by the symmetry of fundamental simplices of regular polyhedra (a) {3, 3} = {3, 3}∗,(b){3, 4} = {4, 3}∗,and(c){3, 5} = {5, 3}∗, respectively. Regular polyhedra which are dual to each other have identical spherical tiling although labels of fundamental simpli- cies are inverted. duals of each other, we have 2nn! fundamental simplices (Fig. 2). 2. Regular Polytopes and Schlafli¨ Symbols Notice that the shifting operation is important. Let us take a A Platonic solid is defined to be a convex polyhedron which regular polytope P which is expressed as {p1, p2,...,pn−1} by is regular in the sense that the faces are congruent regular poly- the Schlafli¨ symbol. The vertex figure and the facet of P are ex- gons and the number of faces meeting at each vertex is uniform. pressed as {p2, p3,...,pn−1} and {p1, p2,...,pn−2}, respectively. There exist 5 Platonic solids. Extending the concept to higher di- For the 4-dimensional polytopes, the following inclusion prop- mensions, call a convex polytope regular if facets are congruent erty holds; {3}⊂{3, 3}⊂{3, 3, 4}⊃{3, 4}⊃{4}, which is called regular polytopes and every vertex figure is uniform. In the 4- double flag architecture. dimensional Euclidean space, there exists 6 types of regular poly- 3. Wythoffian Polytopes and Wythoff Symbols topes. In the n-dimensional (n ≥ 5) space, however, there exist only 3 types of regular polytopes; namely, the regular simplex, The term of semi-regular polytope is often used for some al- the cross-polytope and the hypercube. They are labeled αn, βn, ternative classes of polytopes with the most abundant symmetry and γn, respectively, by Coxeter, H.S.M. [7]. after regular polytopes. To avoid confusion, in this paper, we = − ffi The numbers of k-faces (k 0, 1,...,n 1) of αn, βn,andγn would only consider Wytho an-n-polytopes. n+1 k+1 n n−k n n ff are k+1 ,2 k+1 , and 2 k , respectively. Here, k means a Di erent from the Schlafli¨ symbol, there is no definitive no- binomial coefficient and is 0 if k > n. tation for approaching to Wythoffian polytopes. For the sake Along with the convention that the number of n-faces of an of computer programs, we introduce the notation for Wythoff- n-polytope is 1, we have the f -vectors of αn, βn,andγn. ian polytopes based on Ref. [12]. Note that the notation we use in It is well-known [7] that an n-dimensional regular polytope can this paper is slightly different from the Coxeter’s original one. For be described by a unique Schlafli¨ symbol {p1, p2,...,pn−1}, con- example, the Wythoffian-4-polytope {3, 3, 3}(1, 0, 0, 1) is denoted sisting of n−1 components of natural numbers each of which is 3, by eα4 in Coxeter [5]. However, {3, 3, 3}(1, 0, 0, 1) corresponds 4, or 5 if n ≤ 4 and each of which is 3 or 4 if n ≥ 5. For example, naturally to the Coxeter-Dynkin diagram in the following table. we have α3 = {3, 3}, β4 = {3, 3, 4}, γ5 = {4, 3, 3, 3}. The polytope {p1, p2,...,pn−1} is defined recursively to be the polytope consisting of pn−1 facets p1, p2,...,pn−2 meeting at each (n − 3)-dimensional faces. Note that the dual of A = ∗ {p1, p2,...,pn−1} is A = {pn−1, pn−2,...,p1}. A regular polytope A is divided into fundamental simplices, each of which is given as the convex hull Conv (P0, P1,...,Pn) An n-dimensional polytope is called Wythoffian if it is de- where P0 = a0 is a vertex of A, P1 is the center of an edge a1 of A rived by the Wythoff construction from an n-dimensional regu- including a0, P2 is the center of a face a2 of A including a1,..., lar polytope. Given a regular polytope A = {p1, p2,...,pn−1} n n and Pn is the center of A (Fig. 1). Note that the distances PkPn and given a Wythoff symbol (q0, q1,...,qn−1) ∈{0, 1} \{0} , ··· are ordered as Pn−1Pn < Pn−2Pn < < P0Pn. In the case of the Wythoffian-n-polytope A(q0, q1,...,qn−1) is defined as fol- the regular simplex αn,wehave(n + 1)! fundamental simplices. lows: Choose an arbitrary initial point P on the interior of the face In the cases of the hypercube and the cross-polytope which are Conv{Pi | qi = 1} of the surface facet Conv{P0, P1,...,Pn−1} of c 2017 Information Processing Society of Japan 529 Journal of Information Processing Vol.25 528–536 (Aug. 2017) Fig. 3 Wythoff construction for {3, 4}(1, 1, 1) and its corresponding reflection planes. a fundamental simplex Conv{P0, P1,...,Pn} of A. For the sake We should count the total number of Wythoffian polytopes of notation, the interior of a point is defined to be the point it- from all of the regular polytopes.