A Recursive Algorithm for the K-Face Numbers of Wythoffian-N-Polytopes Constructed from Regular Polytopes

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 number 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 obtained 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. In the case that n = 3, 4, some self. The orbit of P with respect to the symmetry group of the Wythoffian polytopes have more than one original and are multi- underlying regular polytope A (See Fig. 2) gives the vertices of ply counted. In the case that n ≥ 5, the total number is counted by

A(q0, q1,...,qn−1). adding the number of Wythoffian polytopes from self-dual regular The shape of A(q0, q1,...,qn−1) is naturally determined as polytope αn and the number from non-self-dual regular polytopes the convex hull of this vertex set. Thus a pair {p1, p2,..., βn and γn, adjusted by the duplications. In this case, contrast to pn−1}(q0, q1,...,qn−1)ofaSchlafli¨ symbol and a Wythoff symbol, the case that n = 3, 4, there are no Wythoffian polytopes with abbreviated as a Schlafli-Wythoff¨ symbol, determines a Wythoff- more than one original. See Section 9 for the details. ian polytope. This construction of a Wythoffian-n-polytope from Summing up the above discussion, we have the following a regular polytope is called a Wythoff construction [3], [13]. Note theorem: that any Wythoffian-n-polytope is circumscribed by a sphere. Theorem 4.2 For n > 3, the numbers of Wythoffian-n- Figure 3 gives an example of Wythoff construction for polytopes are counted as in the following table:

{3, 4}(1, 1, 1). Given a fundamental simplex Conv(P0, P1, P2, P3) of the original octahedron {3, 4},A= P0,B= P1,C= P2 and the Wythoﬀ symbol (1, 1, 1) corresponds to the fact that the vertex

P for the simplex is in the interior of the triangle P0P1P2.The whole vertex set is simply obtained by collecting corresponding point from each fundamental simplex. The points Q, R,andS n n−1 n−1 in Fig. 3 are obtained as mirror images of P with respect to the Here, a = 2 − 1, b = 2 + 2 2 − 1andc = duplication of depicted planes. a and b. ff Note that a fundamental simplex in the Wytho construc- 5. Direct Product Architecture of Wythoffian tion for the dual {4, 3} of the original can be constructed as Polytopes Conv(P2, P1, P0), preserving the triangular diagram on a facet of the original and inverting labels P0, P1, P2 to P2, P1, P0. Wythoffian polytopes constructed from β3 = {3, 4} have the 4. How Many Wythoffian Polytopes Are following direct product architecture (Fig. 4): We define ( ) = {}(0) to be a point and {}(1) to be a line seg- There? ment for the sake of convenience. The Wythoffian polyhedron There are 11 kinds of Wythoffian polyhedra, excluding the {3, 4}(1, 1, 0) has a square {4}(1, 0) at the vertices, a line segment snub cube and the snub dodecahedron from the standard list of {}(0) ×{}(1) at the edges, and a regular hexagon {3}(1, 1) at its 13 kinds of semi-regular polyhedra [7]. faces. In this section, we generalize this result to describe the total Note that five Wythoffian polytopes {3, 4}(1, 0, 0), number of distinct Wythoffian-n-polytopes. {3, 4}(1, 1, 0), {3, 4}(0, 1, 0), {3, 4}(0, 1, 1), and {3, 4}(0, 0, 1) n − ffi Given a regular polytope A, there are 2 1 Wytho an poly- are made by vertex-truncating β3 around each vertex, whereas topes each of which corresponds to a Wythoff symbol. Note that {3, 4}(1, 0, 1) and {3, 4}(1, 1, 1) are made by vertex-edge- = = ∗ = A(1, 0,...,0) A, A(0, 0,...,1) A ,andA(q0, q1,...,qn−1) truncating β3 around each vertex and edge. ∗ ∗ A (qn−1, qn−2,...,q0), where A is the dual of A. In general, consider a regular polytope A = {p1, p2,...,pn−1} ∗ n − Proposition 4.1 If A A , i.e., A is not self-dual, the 2 1 and a Wythoffian-n-polytope B = A(q0, q1,...,qn−1). Wythoffian polytopes constructed from A are distinct (not combi- For the k-th vertex Pk (k = 0, 1,...,n − 1) of each funda- = ∗ natorially isomorphic to each other).IfA A , i.e., A is self- mental simplex Conv{P0, P1,...,Pn} of A, we have a (possibly dual, Wythoffian polytopes for Wythoff symbol A(q0, q1,...,qn−1) degenerated) facet of B which is combinatorially {pk+2, pk+3,..., ffi and A(qn−1, qn−2,...,q0) are identical. Wytho an polytopes con- pn−1}(qk+1, qk+2,...,qn−1) ×{p1, p2,...,pk−1}(q0, q1,...,qk−1), structed from A are distinct except for this case. Thus, we have i.e., direct product of a (n − 1 − k)-face {pk+2, pk+3,..., n−1 n−1 ∗ 2 + 2 2 − 1 distinct Wythoffian polytopes when A = A . pn−1}(qk+1, qk+2,...,qn−1)oftheflagofB and a k-face {p1,

c 2017 Information Processing Society of Japan 530 Journal of Information Processing Vol.25 528–536 (Aug. 2017)

Fig. 4 Direct product and truncation architectures of Wythoﬃan polytopes constructed from β3 = {3, 4}.

Fig. 5 The ﬂag and the reverse ﬂag of {3, 3, 3, 3, 4}(0, 1, 0, 1, 1, 0).

} p2,...,pk−1 (q0, q1,...,qk−1) of the reverse flag of B. We note ()×{3, 3, 3, 3}(0, 1, 0, 1, 1) at P5 that as shown in Fig. 5, they have nested structure; each polytope includes polytopes on its left side as its faces. 6. Sequential Truncation Architecture of Example 5.1 Every facet of the Wythoffian-6-polytope {3, 3, Wythoffian-n-polytopes 3, 3, 4}(0, 1, 0, 1, 1, 0) is listed as follows: 6 The numbers of faces depend on the above-mentioned arbi- For each of the 2 6! fundamental simplices Conv{P0, P1,...,P6} trary initial point P (Fig. 3) and the truncation as its sequelae. of {3, 3, 3, 3, 4} = β6, we have the following recursive direct prod- = − uct architecture: Given a regular-n-polytope A,thek-truncation (k 0, 1,...,n 2) at the depth 0 ≤ d ≤ 1 is the function that corresponds each ffi {3, 3, 3, 4}(1, 0, 1, 1, 0) × () at P0, Wytho an-n-polytope A(q0, q1,...,qn−1) to the intersection of n A(q0, q1,...,qn−1) with the half-spaces {x ∈ R | (x − Pn) < (1 − {3, 3, 4}(0, 1, 1, 0) ×{}(0) at P1 (degenerated), d)(P − P )} for each fundamental simplex Conv{P , P ,...,P } {3, 4}(1, 1, 0) ×{3}(0, 1) at P , k n 0 1 n 2 of A. Each half-space in the intersection is bounded by the hyper- {4}(1, 0) ×{3, 3}(0, 1, 0) at P , 3 plane perpendicular to the vector Pk − Pn. {}(0) ×{3, 3, 3}(0, 1, 0, 1) at P4 (degenerated), and Note that k-truncations at different depths dare classified into

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some combinatorial equivalence classes based on the range of duplicates nor leaks), we use the inclusion-exclusion principle. + i(q) − i d. We should specify the range of depth when we describe a Then, we have i(q) 1 terms i=0( 1) fi(A) fk(an−1−i) where i(q) k-truncation. is the index of the ﬁrst 1 in the Wythoﬀ symbol. Note that our

Consider a regular polytope A = {p1, p2,...,pn−1} and a counting methods use flags of a Wythoffian-n-polytope but no re- Wythoffian-n-polytope B = A(q0, q1,...,qn−1). Wythoffian poly- verse flags. Reverse flags play an important role in calculating topes have the sequential truncation architecture which is a Tur- volumes. ing machine interpretation of the Wythoff symbol, analogous to For vertex-truncated types, we simply add the number fk(A) DNA. The starting code (TAG) and the stopping code (TAA, of k-faces of the original regular polytope, which correspond + TAG, TGA) in DNA correspond to the first 1 and the last 1 in to k-faces remaining after truncation, to the i(q) 1 terms i(q) i the Wythoff symbol, respectively. A Code (e.g., AAA) between (−1) f (A) f (a − − ) for the k-faces on the overlaps; f (P) = i=0 i k n 1 i k i(q) − i + them in DNA has information (e.g., add Lysine) and so does the i=0( 1) fi(A) fk(an−1−i) fk(A). Wythoff symbol between the first and the last 1’s. For vertex-others-truncated types, we should consider addi- ff − The index of the first 1 in the Wytho symbol (q0, q1,...,qn−1) tional k-faces made by truncations. Thus we add k i(q) terms k is called the truncation initiator and is denoted by i(q). The in- f (A) f − (a − − ) for the k-faces by original faces and di- i=i(q)+1 i k 1 n 1 i i(q) i dex of the last 1 in the Wythoff symbol (q0, q1,...,qn−1) is called rect products to the i(q) + 1 terms (−1) f (A) f − (a − − ) for i=0 i k i n 1 i i(q) i the truncation terminator and is denoted by t(q). Every index k the k-faces on the overlaps; f (P) = (−1) f (A) f (a − − ) + k i=0 i k n 1 i k greater than the initiator i(q) and less than the terminator t(q)is i=i(q)+1 fi(A) fk−i(an−1−i). called the truncation informer. Example 7.1 The face-vector of the direct product α2 × α3 ffi Now the Wytho an polytope B can be obtained by succes- can be calculated using f (α2) = (3, 3, 1, 0, 0) and f (α3) = (4, 6, sively truncating A as follows: 4, 1, 0) as follows: (1) 0-truncation (vertex truncation) = × = The first truncation is always a 0-truncation. Since the hyper- f0 3 4 12 = × + × = plane of truncation is perpendicular to P0Pn and passes through f1 3 6 3 4 30 P on the interior of the face Conv {P | q = 1}, the depth of i i f2 = 3 × 4 + 3 × 6 + 1 × 4 = 34 truncation is deeper (if i(q) t(q)) or just passing through (if f3 = 3 × 1 + 3 × 4 + 1 × 6 + 0 × 4 = 21 i(q) = t(q)) the vertex P and is shallower than P + , where i(q) i(q) 1 = × + × + × + × + × = i(q) is the truncation initiator. f4 3 0 3 1 1 4 0 6 0 4 7 (2) k-truncations for each truncation informer k Now, an arbitrary initial point P (Fig. 3) does not a matter for For each truncation informer k, the depth of the k-truncation counting the numbers of faces. In the next section, we describe a passing through P is deeper than (the hyperplane perpendicular to simple algorithm to count k-faces by working on Schlafli-Wytho¨ ff PkPn passing through) the vertex Pk and is shallower than Pk+1. symbols. Example 6.1 ( 1 ) The Wythoff symbol (0, 1, 0, 0, 0, 0) is read

as “start and truncate vertex passing through P1, and stop.” 8. A Matrix-form Recursive Algorithm for the ( 2 ) The Wythoff symbol (0, 1, 1, 0, 0, 0) is read as “start and k-face Numbers of Wythoffian Polytopes truncate vertex deeper than P1, and stop.” In this section, we introduce a matrix-form algorithm which ff ( 3 ) The Wytho symbol (0, 1, 0, 1, 1, 0) is read as “start and makes it possible to calculate the k-face numbers of Wythoff- truncate vertex deeper than P1, then truncate 2-face shal- ian polytopes using Schlafli-Wytho¨ ff symbols without consider- lower than P3, truncate 3-face shallower than P4, and stop.” ing specific shapes of the faces. The first two Wythoffian polytopes are vertex-truncated types We calculate the n + 1-dimensional row-vector f (B) = f (an) whereas the third one is vertex-others-truncated type. by induction on f (ai)(i = 0, 1,...,n − 1). + + ¯ ≡ ¯ ¯ 7. Combinatorial Counting Methods for k- We construct an n 1-by-n 1 matrix f (an) ( f (an), f (a1),..., ¯ ¯ ¯ face Numbers of Wythoffian-n-polytopes f (an−1), f (an)), where f (ai)isgivenby

The concept of truncation gives not only a way to visualize f¯(ai) = f (ai)(i = 0, 1,...,i(q)) ffi Wytho an-n-polytopes but also a way to count the number of itimes its k-faces. By counting 1 facet for each fundamental simplex of f¯(ai) = (0,...,0, f0(ai), f0(ai),..., fn−1−i(ai)) Abased on the sequential truncation architecture, we obtain the (i = i(q) + 1, i(q) + 2,...,n − 1) list of all the (possibly degenerated) facets of B. t Consider a Wythoffian-n-polytope B with a Schlafli-Wytho¨ ff Note that f¯(an) = (0, 0,...,1) always holds and thus it does symbol {p1, p2,...,pn−1}(q0, q2,...,qn−1) and choose a flag a = not matter that we do not have f (an) in this step of induction. We m (a0, a1,...,an) where ai has a Schlafli-Wytho¨ ff symbol {pn+1−i, also define an n+1-dimensional row-vectorg ¯ withg ¯m = (−1) gm pn−i+1,...,pn−1}(qn−i, qn−i,...,qn−1)anda0 ⊂ a1 ⊂···⊂an. if m ≤ i(q) andg ¯m = gm if m > i(q), where gm is the number of For vertex-truncations deeper than P1, facets obtained by trun- m-dimensional faces of the original {p1, p2,...,pn−1} and i(q)is cations intersect each other with overlaps (the deeper we truncate, the truncation initiator of the Wythoff symbol. ¯ the more they overlap). In order to count the number of k-faces on Now f (an) can be written as the matrix product f¯(an) × g¯ and the intersection of neighboring truncations correctly (i.e., without we reach the value of f (B), concluding the induction step. To

c 2017 Information Processing Society of Japan 532 Journal of Information Processing Vol.25 528–536 (Aug. 2017) sum up, we have the following procedure: Using the notation de- i(q) = 1 scribed above, the number of k-dimensional faces of Bcan be cal- = { } culated by the following recurrence relation and the initial data: (c) TO 3, 3 (1, 1, 1) ¯ f = 4 × 6 = 24 f (an) = f¯(an) × g¯ 0 = × + × = f ({}(0)) = t(1, 0, 0,...) (point) f1 4 6 6 2 36 = × + × + × = f ({}(1)) = t(2, 1, 0,...) (line segment) f2 4 1 6 1 4 1 14

f3 = 4 × 0 + 6 × 0 + 4 × 0 + 1 × 1 = 1 The k-face numbers of 2-dimensional Wythoffian polygons are as f ({3, 3}(1, 0, 0)) = t(4, 6, 4, 1) follows: f ({3}(1, 1)) = t(6, 6, 1, 0) f ({3}(0, 1)) = t(3, 3, 1, 0,...) (regular triangle) f ({}(1)) = t(2, 1, 0, 0) f ({3}(1, 0)) = t(3, 3, 1, 0,...) (regular triangle) f (( )) = t(1, 0, 0, 0) f ({3}(1, 1)) = t(6, 6, 1, 0,...) (regular hexagon) i(q) = 0 f ({4}(0, 1)) = t(4, 4, 1, 0,...) (square) f ({4}(1, 0)) = t(4, 4, 1, 0,...) (square) 9. Appendix on the Numbers of Wythoffian-n- f ({4}(1, 1)) = t(8, 8, 1, 0,...) (regular octagon) polytopes t f ({5}(0, 1)) = (5, 5, 1, 0,...) (regular pentagon) When n = 3, we have following duplications due to the fact f ({5}(1, 0)) = t(5, 5, 1, 0,...) (regular pentagon) that two expressions 2n (the cross-polytope) and 2(n + 1) (trun- f ({5}(1, 1)) = t(10, 10, 1, 0,...) (regular decagon) cated simplex) coincide for numbers of facets. { } = { } These results can be calculated from the matrix-form algo- 3, 3 (0, 1, 0) 3, 4 (1, 0, 0), rithm. In general, we need the k-face numbers of m-dimensional {3, 3}(1, 0, 1) = {3, 4}(0, 1, 0), Wythoffian polytopes for m < n in order to calculate the {3, 3}(1, 1, 1) = {3, 4}(1, 1, 0), k-face numbers of n-dimensional Wythoffian polytopes. By = using the matrix-form algorithm along with these results for When n 4, we have following duplications due to the fact n + = the 2-dimension case, we can calculate the k-face numbers of that 2 2n 24 (for truncated cross-polytope and 24-cell) coin- 3-dimensional Wythoffian polyhedra. cide for numbers of facets. First we calculate the f -vector of the truncated octahedron TO {3, 3, 4}(0, 1, 0, 0) = {3, 4, 3}(1, 0, 0, 0), using three equivalent Schlafli-Wytho¨ ff symbols. We omit some { , , } , , , = { , , } , , , , calculations which are meaningless in a hand calculation. 3 3 4 (1 0 1 0) 3 4 3 (0 1 0 0) (a) TO = {3, 4}(1, 1, 0) {3, 3, 4}(1, 1, 1, 0) = {3, 4, 3}(1, 1, 0, 0). ≥ f0 = 6 × 4 = 24 When n 5, however, there are no duplications. This is proved by mathematical induction on n. f1 = 6 × 4 + 12 × 1 = 36 When n = 5, β5(q0, q1,...,q4)hasβ4(q1,...,q4) for each f2 = 6 × 1 + 12 × 0 + 8 × 1 = 14 vertex truncation which is distinct from α4(r0, r1, r2, r3) for any f = 6 × 0 + 12 × 0 + 8 × 0 + 1 × 1 = 1 3 Wythoff symbol (r0, r1, r2, r3). t f ({3, 4}(1, 0, 0)) = (6, 12, 8, 1) When n ≥ 6, βn(q0, q1,...,qn−1)hasβn−1(q1,...,qn−1) for f ({4}(1, 0)) = t(4, 4, 1, 0) each vertex truncation which is distinct from αn−1(r0,...,rn−2) for any Wythoff symbol (r ,...,r − ). f ({}(0)) = t(1, 0, 0, 0) 0 n 2 f (( )) = t(1, 0, 0, 0) 10. Appendix on Calculation of f-vectors by i(q) = 0 the Matrix-form Algorithm We illustrate how the matrix-form recursive algorithm works (b) TO = {4, 3}(0, 1, 1) for two kinds of Wythoffian-6-polytopes. ffi = { } f0 = 8 × 6 − 12 × 2 = 24 (1) Wytho an polytope B 3, 3, 3, 3, 4 (0, 1, 1, 0, 0, 0) with 0- truncations only f1 = 8 × 6 − 12 × 1 = 36 [1] f (B) is obtained as follows: f2 = 8 × 1 − 12 × 0 + 6 × 1 = 14 t f3 = 8 × 0 − 12 × 0 + 6 × 0 + 1 × 1 = 1 f ({3, 3, 3, 4}(1, 1, 0, 0, 0)) = (80, 280, 400, 240, 42, 1, 0) f ({4, 3}(1, 0, 0)) = t(8, 12, 6, 1) f ({3, 3, 4}(1, 0, 0, 0)) = t(8, 24, 32, 16, 1, 0, 0) f ({3}(1, 1)) = t(6, 6, 1, 0) f ({3, 4}(0, 0, 0)) = t(1, 0, 0, 0, 0, 0, 0) f ({}(1)) = t(2, 1, 0, 0) f ({4}(0, 0)) = t(1, 0, 0, 0, 0, 0, 0) f (( )) = t(1, 0, 0, 0) f ({}(0)) = t(1, 0, 0, 0, 0, 0, 0)

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t f (( )) = (1, 0, 0, 0, 0, 0, 0) f0(B) = 12 × 960 − 60 × 96 = 5760

i(q) = 1 f1(B) = 12 × 2880 − 60 × 192 = 23040 ⎛ ⎞ ⎜ 808 00000⎟ f2(B) = 12 × 2960 − 60 × 120 + 160 × 24 = 32160 ⎜ ⎟ ⎜2802400000⎟ = × − × + × + × = ⎜ ⎟ f3(B) 12 1200 60 24 60 36 240 4 19680 ⎜ ⎟ ⎜4003210000⎟ = × − × + × + × + × ⎜ ⎟ f4(B) 12 162 60 1 160 14 240 4 192 1 f¯(B) = ⎜2401601000⎟ ⎜ ⎟ = ⎜ ⎟ 5276 ⎜ 421 00100⎟ ⎜ ⎟ f (B) = 12 × 1 − 60 × 0 + 160 × 1 + 240 × 1 + 192 × 0 + 60 × 1 ⎝⎜ 1 0 00010⎠⎟ 5 0 0 00001 = 476 = × − × + × + × + × Note that the square matrix appearing in the lower-right cor- f6(B) 12 0 60 0 160 0 240 0 192 0 ner of the matrix f¯(B) above is the identity matrix (5 × 5inthe + 60 × 0 + 1 × 1 = 1 case above), which yields a simpler expression of face numbers The numbers of k (0 ≤ k ≤ n)-faces of the n (n ≤ 6)-dimensional for Wythoffian polytopes with 0-truncations only. Wythoffian polytopes obtained by the matrix-form algorithm [2] By using g = t(12, 60, 160, 240, 192, 64, 1) from the origi- have been confirmed to coincide with the results obtained by nal {3, 3, 3, 3, 4} and i(q) = 1, we haveg ¯ = t(12, −60, 160, 240, the method of exhaustion using a computer with CPU i7-4770, 192, 64, 1). 3.40 GHz, thereby reconfirming the accuracy of the results. The [3] The number fk(B)ofk-dimensional faces of B is obtained by ¯ implemented computer program returns the answer for the class f (B) = f¯(B) × g¯ as follows: − of n-dimensional Wythoffian polytopes {3n 1}(1n) within a sec- ond up to n = 20. We note that some of the values of f -vectors f0(B) = 12 × 80 − 60 × 8 = 480 exceed the limit of 64-bit integers when n ≥ 17. In this appendix, f B = × − × = 1( ) 12 280 60 24 1920 we show the 4- and 5-dimensional values out of whole data list. f2(B) = 12 × 400 − 60 × 32 + 160 × 1 = 3040

f3(B) = 12 × 240 − 60 × 16 + 160 × 0 + 240 × 1 = 2160

f4(B) = 12 × 42 − 60 × 1 + 160 × 0 + 240 × 0 + 192 × 1 = 636

f5(B) = 12 × 1 − 60 × 0 + 160 × 0 + 240 × 0 + 192 × 0 + 60 × 1 = 76

f6(B) = 12 × 0 − 60 × 0 + 160 × 0 + 240 × 0 + 192 × 0 + 60 × 0 + 1 × 1 = 1

(2) Wythoﬃan polytope Q = {3, 3, 3, 3, 4}(0, 1, 0, 1, 1, 0) with some k (k ≥ 1)-truncations [1] f¯(B) is obtained as follows:

f ({3, 3, 3, 4}(1, 0, 1, 1, 0)) = t(960, 2880, 2960, 1200, 162, 1, 0) f ({3, 3, 4}(0, 1, 1, 0)) = t(96, 192, 120, 24, 1, 0, 0) f ({3, 4}(1, 1, 0)) = t(24, 36, 14, 1, 0, 0, 0) f ({4}(1, 0)) = t(4, 4, 1, 0, 0, 0, 0) f ({}(0)) = t(1, 0, 0, 0, 0, 0, 0) f (( )) = t(1, 0, 0, 0, 0, 0, 0) i(q) = 1 ⎛ ⎞ ⎜ 960960 0000⎟ ⎜ ⎟ ⎜2880 192 0 0000⎟ ⎜ ⎟ ⎜2960 120 24 0000⎟ F(Q) = ⎜ ⎟ ⎜ ⎟ ⎜1200 24 36 4000⎟ ⎜ ⎟ ⎝⎜ 1621 144100⎠⎟ 0 0 0 0001

[2] By using g = t(12, 60, 160, 240, 192, 64, 1) from the original {3, 3, 3, 3, 4} and i(q) = 1, we haveg ¯ = t(12, −60, 160, 240, 192, 64, 1).

[3] The number fk(B)ofk-dimensional faces of B is obtained by f (B) = f¯(B) × g¯ as follows:

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vex, Bolyai Society Mathematical Studies, Vol.24, pp.1–21, Springer- Verlag (2013). [2] Conway, J.H. and Sloane, N.J.A.: Sphere packing, lattices and groups, Springer, New York (1988). [3] Coxeter, H.S.M.: Wythoff’s construction for uniform polytopes, Proc. London Math. Soc., Vol.38, pp.327–339 (1934). [4] Coxeter, H.S.M.: Regular and semi-regular polytopes. I, Mathematis- che Zeitschrift, Vol.46, pp.380–407 (1940). [5] Coxeter, H.S.M.: Regular and semi-regular polytopes. II, Mathema- tische Zeitschrift, Vol.188, pp.559–591 (1985). [6] Coxeter, H.S.M.: Regular and semi-regular polytopes. III, Mathema- tische Zeitschrift, Vol.200, pp.3–45 (1988). [7] Coxeter, H.S.M.: Regular polytopes, Dover Publications Inc., New York (1973). [8] Delone, B.N.: Proof of the fundamental theorem in the theory of stereohedra, Soviet Math. Dokl., Vol.2, No.3, pp.812–817 (1961). [9] Dolbilin, N.P.: Parallelohedra: a retrospective and new results, Trans. Moscow Math. Soc., Vol.73, pp.207–220 (2013). [10] Engel, P.: Uber¨ Wirkungsbereichsteilungen von kubischer Symmetrie, Zeitschrift für Kristallographie, Vol.154, pp.199–215 (1981). [11] Gruber, P.M. and Lekkerkerker, C.G.: Geometry of numbers,North- Holland, Amsterdam (1987). [12] Ishii, M.: On a general method to calculate vertices of n-dimensional product-regular polytopes, Forma, Vol.14, pp.221–237 (1999). [13] Maxwell, G.: Wythoff’s construction for Coxeter groups, Journal of Algebra, Vol.123, pp.351–377 (1989). [14] McMullen, P.: The maximum number of faces of a convex polytope, Mathematika, Vol.17, pp.179–184 (1970). [15] McMullen, P.: Convex bodies which tile space by translations, Math- ematika, Vol.27, pp.113–121 (1980). [16] Moody, R. and Patera, J.: Voronoi domains and dual cells in the gener- alized kaleidoscope with applications to root and weight lattices, Can. J. Math, Vol.47, pp.573–605 (1985). [17] Moody, R. and Patera, J.: Voronoi and Delaunay cells of root lattices: classification of their faces and facets by Coxeter-Dynkin diagrams, J. Phys. A: Math. Gen., Vol.25, pp.5089–5134 (1992). [18] Sabariego P. and Santos, F.: On the number of facets of three- dimensional Dirichlet stereohedra IV: quarter cubic groups, Contri- butions to Algebra and Geometry, Vol.52, No.2, pp.237–263 (2011). [19] Ziegler, G.M.: Lectures on polytopes, 2nd ed., Springer, New York (1998).

Jin Akiyama received D.Sc. in 1982 from Tokyo University of Science. He is now Director of the Institute of Math Ed- ucation at TUS. His current interest is discrete and computational geometry.

Sin Hitotumatu graduated from the Uni- versity of Tokyo in 1947 and received D.Sc. from the same university in 1954. His research interests include applied math and computer science. He is now Professor Emeritus of Kyoto University.

References [1] Akiyama, J., Kobayashi, M., Nakagawa, H., Nakamura, G. and Sato, I.: Atoms for parallelohedra, Geometry/Intuitive, Discrete and Con-

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Motonaga Ishii received Ph.D. in Hu- man and Environmental Studies from Ky- oto University in 1999. He is currently with BANDAI NAMCO Studios Inc., Tokyo, as an engineer of computer graphics and computer vision.

Akihiro Matsuura received Ph.D. in In- formatics from Kyoto University in 2002. In 2003, he joined Tokyo Denki Univer- sity and is currently an associate professor. His research interests include discrete algorithms, combinatorial games, and en- tertainment computing.

Ikuro Sato received M.D. from Tohoku University, School of Medicine, in 1981. He works in Miyagi Cancer Center as a chief pathologist, and as a visiting professor in Tohoku University. His current interest is hyperspace geometry.

Shun Toyoshima received B.I. in Infor- matics from Tokyo Denki University in 2016. He is currently with Systems De- sign Co. Ltd. as a system engineer.

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