� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 34, Num. 1 (2019), 99 – 122 doi:10.17398/2605-5686.34.1.99 Available online March 13, 2019 Tetrahedral chains and a curious semigroup Ian Stewart Mathematics Institute, University of Warwick Coventry CV4 7AL, United Kingdom i.n.stewart@warwick.ac.uk Received January 7, 2019 Presented by Juan A. Navarro Accepted February 27, 2019 and Alexandre Turull Abstract: In 1957 Steinhaus asked for a proof that a chain of identical regular tetrahedra joined face to face cannot be closed. Świerczkowski gave a proof in 1959. Several other proofs are known, based on showing that the four reflections in planes though the origin parallel to the faces of the tetrahedron generate a group R isomorphic to the free product Z2 ∗ Z2 ∗ Z2 ∗ Z2. We relate the reflections to elements of a semigroup of 3 × 3 matrices over the finite field Z3, whose structure provides a simple and transparent new proof that R is a free product. We deduce the non-existence of a closed tetrahedral chain, prove that R is dense in the orthogonal group O(3), and show that every R-orbit on the 2-sphere is equidistributed. Key words: tetrahedral chain, free product, semigroup, density, equidistribution, spherical harmonic, Cayley graph. AMS Subject Class. (2010): 20F55, 20M05, 22E46, 37A30, 42C10, 43A15. 1. Introduction In 1957 Hugo Steinhaus contributed two related questions to the Problems section of Colloquium Mathematicum [21]. In loose translation from the French, they were: P 175. The image of a regular tetrahedron T1 (fixed in Euclidean space of 3 dimensions) under reflection in one of its faces gives the tetrahedron T2. Iteration gives rise to a sequence of pairwise congruent tetrahedra {Tn}. Supposing that each face serves as a mirror only once, demonstrate that: (1) m 6= n implies Tm 6= Tn, (2) Whatever the region R may be, there exists a sequence of tetrahedra {Tn} such that the set of vertices is dense in R. Steinhaus indicated that this problem is from the New Scottish Book, Problem 290 1.III.1956. The original Scottish Book was a notebook of open ISSN: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.1.99 mailto:i.n.stewart@warwick.ac.uk https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 100 i. stewart mathematical problems compiled by regular visitors to the famous Scottish Café in what is now Lviv, Ukraine. An English typescript is available [28], and the original can be viewed online as Volume 0 at [30]. When World War II ended, Steinhaus revived the book as the New Scottish Book, Volumes 1 and 2 at [30]. Figure 1: Closed chains of cubes, octahedra, dodecahedra, and icosa- hedra. Adapted from [10]. As far as we are aware, statement (2) is still open. Its analogue for cubes is clearly false, but its analogue for the other regular solids is a plausible conjecture. In particular, their dihedral angles are not rational multiples of π. Statement (1) implies that a chain of identical regular tetrahedra, joined face to face, cannot be closed. In contrast, it is easy to find closed chains for the other four regular polyhedra, Figure 1. (One key difference is that unlike the tetrahedron, opposite faces of these polyhedra are parallel.) Stanis law Świerczkowski subsequently proved that no closed chain of tetrahedra exists, aside from trivial examples where consecutive tetrahedra coincide. In [24] he proved that two particular rotations in R3 generate a free group on two generators, and stated as a corollary that this result disproves the existence tetrahedral chains and a curious semigroup 101 of a closed chain of regular tetrahedra. He wrote: “This corollary gives a positive answer to a question of H. Steinhaus ... . However we shall not prove it here.” In [25] he completed the proof by explaining the connection with chains of tetrahedra. The rotations concerned are  1 3 −2 √ 2 3 0 2 √ 2 3 1 3 0 0 0 1   ,   1 0 0 0 1 3 −2 √ 2 3 0 2 √ 2 3 1 3   . Their axes are at right angles to each other and both are rotations through cos−1( 1 3 ). Świerczkowski’s proof that these matrices generate a free group uses induction on a sequence of integers determined by the two matrices, and his main aim is to prove that these are not divisible by 3. In passing, we mention that this group-theoretic result can also be used as the basis of a proof of the famous Banach-Tarski paradox [27]: a solid ball in R3 can be dissected into finitely many disjoint subsets, which can be fitted together via rigid motions to create two solid balls, each congruent to the original one. The free product group discussed below can also be used in this manner. Dekker [8] and Mason [19] sketched new proofs that no closed tetrahedral chain exists, based on the idea that the group generated by reflections in the four planes through the origin parallel to the faces of the tetrahedron is isomorphic to a free product Z2 ∗Z2 ∗Z2 ∗Z2. (Without loss of generality, the barycentre of the first tetrahedron in the chain is the origin. It is important to distinguish these linear reflections from the affine reflections in the faces of the tetrahedron, which do not fix the origin; see Subsection 2.2.) Tomkowicz and Wagon [27, Theorem 3.10] represents the four reflections as 4×4 matrices using barycentric coordinates, and analyse an arbitrarily long product of these matrices. The entries of such a product are polynomials, evaluated at ±1 3 , ±2 3 . As in [24], the key step in an inductive proof again involves divisibility by a power of 3. Say that a chain of tetrahedra is embedded if distinct tetrahedra are dis- joint except for the common face of consecutive members of the chain. All of the above proofs rule out the existence of nontrivial closed chains, embedded or not. These proofs are relatively short and simple, but none is particu- larly transparent. In Section 3 we present a new proof, with a clear storyline that emphasises the role of the integer 3. We use Cartesian coordinates, but it is possible to recast the discussion using the more traditional barycentric coordinates. With a convenient choice of the initial tetrahedron in R3, with 102 i. stewart vertices at four corners of the cube [−1, 1]3, the 3×3 matrices representing the four reflections have rational entries with denominator 1 or 3; see Section 2. That they generate Z2 ∗Z2 ∗Z2 ∗Z2 is a reformulation of the statement that the product of any nontrivial sequence of the four matrices (that is, avoiding consecutive repetitions of the same matrix), other than the empty sequence, can never be the identity. For a contradiction, suppose such a sequence exists. Let the group gen- erated by the four reflections be R, which is a subgroup of the orthogonal group O(3) acting on R3. If each matrix is multiplied by 3 it has integer entries, and the corresponding product must be the identity multiplied by 3k where k ≥ 1 is the length of the sequence. These products no longer form a group, but together with the zero matrix they form a semigroup. Reducing modulo 3, a nontrivial product of the corresponding matrices over Z3 must be the zero matrix. Theorem 3.2 proves that the four reflection matrices (mod 3) generate an order-33 semigroup. This contains the zero matrix 0, but we prove that no nontrivial product of its nonzero elements is zero. (As before, ‘nontrivial’ means no generator appears twice consecutively.) Indeed, every matrix in the semigroup other than 0 has all entries equal to ±1 (mod 3). This contradiction proves that R is isomorphic to Z2 ∗ Z2 ∗ Z2 ∗ Z2. (An isomorphic semigroup can be obtained using barycentric coordinates, and the proof can be also be expressed in that framework.) The non-existence of a nontrivial closed chain of regular tetrahedra follows easily, using essentially the argument of Świerczkowski [25], which constructs the chain using successive reflections in faces. The translations in the Eu- clidean group E(3) form a normal subgroup and can be factored out, reducing the geometric features required here to sequences of reflections. The four re- flections lie in the quotient group O(3). A sequence of reflections determines a unique chain of face-to-face tetrahedra, and a nontrivial sequence determines a nontrivial chain. If this chain closes up, the corresponding sequence fixes the initial tetrahedron. There is one subtlety, discussed briefly in Subsection 3.1: this sequence can fix the tetrahedron setwise rather than pointwise. That is, it belongs to the symmetry group of the tetrahedron, but need not be the identity. There are two ways to deal with this possibility. One is to observe that some power of the sequence must then be the identity (it is also necessary to deal with possible repetitions if the sequence starts and ends with the same reflection: this leads to a shorter chain, but after a series of such cancellations it turns out that the result must be nontrivial if the original chain is). The other, employed here, is to check that the semigroup proof remains valid if tetrahedral chains and a curious semigroup 103 we replace the identity by a symmetry of the tetrahedron, because all such symmetries correspond to integer matrices. After multiplying by 3 and reduc- ing modulo 3, these all become the zero matrix and the same proof works. Section 4 adds extra information about this semigroup. In the absence of a closed chain, a natural question, also asked by Świerczkowski [26], arises: can almost closed chains be formed, in the sense that the gap between the initial and final tetrahedra can be made as small as we please? This question is connected to Steinhaus’s statement (2), but it involves the faces of the tetrahedra, not just individual vertices. Elgersma and Wagon [9] give an affirmative answer for non-embedded chains, based on Kronecker’s Theorem [1, 17] that if θ is an irrational multiple of π, the set {eniθ : n ∈ Z} is dense in the unit circle S1 ⊆ C. The most interesting case arises when the chain is embedded. Elgersma and Wagon [10, 11] prove the existence of closed embedded chains with arbitrarily small gaps. Their con- struction begins with a Boerdijk-Coxeter helix [4, 7], also named the tetrahelix by Fuller [12]. This is a linear chain of identical regular tetrahedra, all of whose vertices lie on a cylinder. This chain is generated by periodically repeating reflections in four distinct faces. They then construct a ‘quadrahelix’ by join- ing four copies of a tetrahelix of length L + 1, overlapping them at a common end tetrahedron at the first and third joins, and attaching them face to face at the second join, so that the overall chain has reflectional symmetry about its midpoint. They prove that if L = q − 1 where p/q is a convergent of the continued fraction of θ = cos−1( 2 3 )/(2π), the quadrahelix has the approximate form of a rhombus, and is almost closed, with the size of the gap tending to zero as q increases. For example when L = 601,944 the gap has size 1.3×10−7. Here we prove two related results, which do not prove the existence of almost closed chains but have independent interest. Section 5 gives a simple proof that R is dense in O(3). Steinhaus’s statement (2) asks for more: the group generated by the affine reflections in the faces of a fixed regular tetra- hedron has a dense orbit in R3. Taking account of the translations is more difficult, in part because the Euclidean group E(3) in R3 is non-compact. Our density result is too weak to prove statement (2), and has no obvious consequences for almost-closed chains of tetrahedra, because it factors out translations. Finally, in Section 6, we use the O(3) density result to prove a stronger theorem: the R-orbit of any point of the unit 2-sphere S2 is equidistributed with respect to normalised Lebesgue surface measure on S2, where the density of R is defined using the limit of the proportion of words in the generating 104 i. stewart reflections that lie in a given open subset of S2, as the length of the words tends to infinity. The proof is an adaptation of a method of Arnold and Krylov [2], and can be viewed in the context of ergodic theory of non-abelian group actions; see Gorodnik and Nevo [13]. 2. Reflections in faces of the tetrahedron Elgersma and Wagon [9, 10, 11] pose the problem using barycentric coor- dinates and 4×4 matrices. Babiker and Janeczko [3] analyse chains of regular tetrahedra using a tensor product representation, combining the translations and reflections, to prove several new results. However, the reflections are again represented using barycentric coordinates. In this paper we use Cartesian co- ordinates and 3 × 3 matrices. We choose a fixed reference tetrahedron ∆ ⊆ R3 with vertices p0 : (1, 1, 1) , p1 : (1,−1,−1) , p2 : (−1, 1,−1) , p3 : (−1,−1, 1) . The barycentre is at the origin. See Figure 2. x y z Figure 2: The cube [−1, 1]3 and the reference tetrahedron ∆. tetrahedral chains and a curious semigroup 105 2.1. Symmetries of the tetrahedron. The eight points (±1,±1,±1) are the vertices of a cube. The symmetry group of the cube consists of all permutations of the coordinates (x,y,z) together with sign changes on any coordinate, so has order 48. The symmetry group Sym(∆) of the tetrahedron is the subgroup in which the number of minus signs is even, and has order 24. It is, of course, isomorphic to the symmetric group S4, which permutes the four vertices. Lemma 2.1. All matrices in the symmetry group Sym(∆) have integer entries. Proof. All 3 × 3 matrices in the symmetry group of the cube, hence also of the tetrahedron, are signed permutation matrices, with entries 0,±1. 2.2. The 3×3 matrices. Next, we compute the four reflection matrices and associated translations. The faces F [pipjpk] through vertices pi,pj,pk have equations: F[p1p2p3] : X + Y + Z = −1 , F[p0p1p2] : X + Y −Z = 1 , F[p0p1p3] : X −Y + Z = 1 , F[p0p2p3] : −X + Y + Z = 1 . Midpoints of centres of these faces are q0 : (−13,− 1 3 ,−1 3 ) , q1 : ( 1 3 , 1 3 ,−1 3 ) , q2 : ( 1 3 ,−1 3 , 1 3 ) , q3 : (−13, 1 3 , 1 3 ) . Let Mi be reflection in face i, where 0 ≤ i ≤ 3. Then it is the identity on that face, and reverses the line perpendicular to the face at its centre. This line joins the midpoint of the face to the remaining vertex. Begin with the plane P = F[p1p2p3]. The perpendicular is (a,a,a) : a ∈ R. A general point Y = (U,V,W) = (a,a,a) + Z where Z = (U −a,V −a,W −a) ∈ P . Therefore U + V + W − 3a = −1 106 i. stewart so a = U + V + W + 1 3 . Then, solving for Z, Z =   2 3 U − 1 3 V − 1 3 W − 1 3 2 3 V − 1 3 U − 1 3 W − 1 3 2 3 W − 1 3 U − 1 3 V − 1 3   . This maps via M0 to Z − (a,a,a), which is M0Y =   1 3 U − 2 3 V − 2 3 W − 2 3 1 3 V − 2 3 U − 2 3 W − 2 3 1 3 W − 2 3 U − 2 3 V − 2 3   = R0Y + T0 , where R0 =   1 3 −2 3 −2 3 −2 3 1 3 −2 3 −2 3 −2 3 1 3   , T0 =   − 2 3 −2 3 −2 3   . Repeating similar calculations for the other three faces, we obtain: For face F [p0p1p2]: R1 =   1 3 −2 3 2 3 −2 3 1 3 2 3 2 3 2 3 1 3   , T1 =   2 3 2 3 −2 3   . For face F [p0p1p3]: R2 =   1 3 2 3 −2 3 2 3 1 3 2 3 −2 3 2 3 1 3   , T2 =   2 3 −2 3 2 3   . For face F [p0p2p3]: R3 =   1 3 2 3 2 3 2 3 1 3 −2 3 2 3 −2 3 1 3   , T3 =   − 2 3 2 3 2 3   . The Ri are orthogonal, and Ti points perpendicular to the face concerned. Since Sym(∆) permutes the faces of ∆ and fixes the origin, the Ri are conju- gate under symmetries of ∆: σRiσ −1 = Rσ(i) , σ ∈ Sym(∆) ≡ S4 . tetrahedral chains and a curious semigroup 107 2.3. Factoring out translations. The construction of chains of tetrahedra is intimately related to the subgroup of O(3) generated by the four reflection matrices, and for many purposes we can ignore the translations. Definition 2.2. The group R is the subgroup of O(3) generated by the Ri for 0 ≤ j ≤ 3. The corresponding affine maps are MiY = RiY + Ti , 0 ≤ i ≤ 3 . The group E(3) of all rigid motions of R3 is a semidirect product E(3) = R3 n O(3) where R3 is the normal subgroup of translations. The homomorphism E(3) → O(3) that factors out R3 maps (Ti,Ri) to Ri. Therefore a product of affine maps MikMik−1 · · ·Mi2Mi1 is a map of the form RikRik−1 · · ·Ri2Ri1 + Wikik−1...i2i1 (2.1) where Wikik−1...i2i1 is a translation determined by the semidirect product struc- ture, which we do not state explicitly. The product of the reflections Ri gives the orientation of the image tetra- hedron; the translation Wikik−1...i2i1 leaves the orientation invariant. The se- quence of translations is implicit in the sequence of reflections because exactly one pair of faces matches at each stage. 3. Free product structure We now give a simple, structural proof of the main result of Dekker [8] and Mason [19]: Theorem 3.1. The group R is isomorphic to the free product Z2 ∗ Z2 ∗ Z2 ∗Z2. Our proof is based on a rather curious semigroup, and we discuss this first. To get rid of fractions, define Qi = 3Ri. We can then reduce modulo 3, to get matrices Si = Qi (mod 3) , 0 ≤ i ≤ 3 , 108 i. stewart which are: S0 =   1 1 11 1 1 1 1 1   , S1 =   1 1 −11 1 −1 −1 −1 1   , S2 =   1 −1 1−1 1 −1 1 −1 1   , S3 =   1 −1 −1−1 1 1 −1 1 1   . Here the ±1 lie in Z3, but in fact the calculations reported below also apply in Z, except when the zero matrix arises and some entries may be multiples of 3. The twelve products SiSj (i 6= j) are: S0S1 =   1 1 −11 1 −1 1 1 −1   , S0S2 =   1 −1 11 −1 1 1 −1 1   , S0S3 =   −1 1 1−1 1 1 −1 1 1   , S1S0 =   1 1 11 1 1 −1 −1 −1   , S1S2 =   −1 1 −1−1 1 −1 1 −1 1   , S1S3 =   1 −1 −11 −1 −1 −1 1 1   , S2S0 =   1 1 1−1 −1 −1 1 1 1   , S2S1 =   −1 −1 11 1 −1 −1 −1 1   , S2S3 =   1 −1 −1−1 1 1 1 −1 −1   , S3S0 =   −1 −1 −11 1 1 1 1 1   , S3S1 =   1 1 −1−1 −1 1 −1 −1 1   , S3S2 =   1 −1 1−1 1 −1 −1 1 −1   . Clearly Q2i = 9R 2 i = 9I, so S 2 i = 0. Observe that the sixteen matrices Si and SiSj (i 6= j) are distinct, and distinct from their negatives −Si and −SiSj (i 6= j). Let S be the set of these 32 matrices. Case-by-case analysis tetrahedral chains and a curious semigroup 109 shows that the Si satisfy the following relations: SiSjSi = Si (i 6= j) , SiSjSk = −SiSk (i 6= j, i 6= k, j 6= k) . (3.1) (Using Sym(∆) we can reduce this calculation to the special case i = 0, j = 1, k = 2.) Let S be the set of all of the above 32 matrices together with the zero matrix, so that S = { 0, ±Si, ±SiSj : 0 ≤ i, j ≤ 3, i 6= j } . Theorem 3.2. (a) The set S is a semigroup. (b) The product of two nonzero members of S is nonzero, except for the trivial cases SiSi = 0 , Si(SiSj) = 0 , (SiSj)Sj = 0 , (SiSj)(SjSk) = 0 , and similar products involving minus signs. Proof. For (a) we must show that all products of nonzero elements of S lie in S. For (b) we must also show these products are nonzero. Both follow from a case-by-case check. For products SiSj this is clear. Products of the form Si(SjSk) and (SiSj)Sk are taken care of directly by the relations (3.1). Those relations also imply that when i 6= j, k 6= l we have (SiSj).(SkSl) =   0 if j = k , Si if j 6= k , k = i , −SiSl if j 6= k , k 6= i . When j = k the string SiSjSkSl = SiSjSjSl is trivial. Note in particular that when i 6= j (SiSj) 2 = (SiSjSi)Sj = SiSj which is nonzero, unlike squares of the Si. We are now ready to give the: 110 i. stewart Proof of Theorem 3.1. The four free factors Z2 are generated respectively by R0, R1, R2, R3. We claim that the only relations between these generators are R2i = I, where I is the identity. Using the relations R2i = I we can write any element γ ∈R in the form γ = RikRik−1 · · ·Ri2Ri1 where Rij 6= Rij+1 for all 1 ≤ j ≤ k − 1. We claim this representation as a word is unique. If not, some nontrivial word is equal to the identity I. The correspond- ing nontrivial word in the Qi of length l is equal to 3 lI. Modulo 3, this word becomes zero. Consider the corresponding word in the Si, which is also nontrivial: w = SikSik−1 · · ·Si2Si1 . By Theorem 3.2, w lies in S\{0}, so all of its entries are ±1 (mod 3). There- fore w 6= 0, so no nontrivial word in the Ri can be the identity. 3.1. Non-existence of a closed chain. It is well known that Theo- rem 3.1 implies the non-existence of a nontrivial closed chain of regular tetra- hedra. For completeness, we give a proof. Theorem 3.3. No nontrivial closed chain of tetrahedra exists. Proof. Suppose, for a contradiction, that there is such a chain. Consider the corresponding product of reflections Ri in R. Because each reflection Ri fixes the origin, the construction of the chain of tetrahedra corresponding to a given element RjkRjk−1 · · ·Rj2Rj1 ∈R does not add successive tetrahedra to an otherwise stationary chain. Instead, the chain corresponding to Rjk−1 · · ·Rj2Rj1 is reflected by Rjk , and then trans- lated by an appropriate amount so that it joins to the corresponding face of the reference tetrahedron ∆. Thus the chain at stage k has the structure ∆ → Rjk ∆ → RjkRjk−1 ∆ → RjkRjk−1Rjk−2 ∆ →··· · · ·→ RjkRjk−1 · · ·Rj2Rj1 ∆ where the arrow indicates ‘joins at a face’. (An alternative approach, in which each new face is added to a growing but otherwise static chain, is geometrically tetrahedral chains and a curious semigroup 111 more natural but involves conjugates of the Ri, so this convention is a little simpler algebraically.) A necessary condition for the chain to close up is then that there some nontrivial product of reflections is a symmetry of the tetrahedron: RjkRjk−1 · · ·Rj2Rj1 = A where A ∈ Sym(∆). Then the corresponding nontrivial product in S satisfies SjkSjk−1 · · ·Sj2Sj1 = 3 k+1A = 0 by Lemma 2.1. This yields the same contradiction as in the proof of Theorem 3.1. 4. Properties of the semigroup The semigroup S has a lot of structure, which the calculations do not explain. We briefly investigate some of its features. The results of this section are not used later, but they help to explain some aspects of the structure of S from a different point of view. The ±Si are symmetric matrices, whereas the ±SiSj are not symmetric. There are 29 = 512 matrices of size 3×3 with entries ±1. The 32 such matrices in S \{0} are distinguished by the following properties: (1) All entries are ±1. (2) The matrix has a repeated row, and the remaining row is either the same as the repeated row or the negative of the repeated row. (3) The same goes for columns. It is easy to prove that for matrices satisfying (1), condition (2) holds if and only if (3) does. We omit the proof. Proposition 4.1. The equivalent conditions (1)+(2) or (1)+(3) charac- terise the 32 nonzero elements of the semigroup S. Proof. This follows from the list of elements, but we now give an indepen- dent proof avoiding case-by-case checking. We count now many such matrices exist. Observe that there are 8 possibilities for the first row R1. The second and third rows R2,R3 are all possible choices of R2 = ±R1, R3 = ±R1, with four choices of the ± signs, so in total there are 8 × 4 = 32 such matrices. In other words, the equivalent conditions (1)+(2) or (1)+(3) characterise the elements of S that are not 0. 112 i. stewart Proposition 4.1 lets us give an alternative proof that S is a semigroup, without listing all products: Theorem 4.2. The collection of matrices satisfying (1)+(2), together with the zero matrix, is a semigroup. Proof. As observed, conditions (1) and (2) also imply (3). Redefine S to be the set of matrices satisfying these conditions, together with 0. Let A,B ∈ S. Clearly 0 = 0.0 = 0.A = A.0. It remains to show that AB ∈ S when A,B 6= 0. Permuting rows we can write A = P   XX εX   , ε = ±1 , where P is a permutation matrix, and X = [x,y,z] is a row vector. Dually, permuting columns we can write B = [ Y Y δY ] Q, δ = ±1 , where Q is a permutation matrix, and Y = [u,v,w]T is a column vector. Then AB = P   X ·Y X ·Y δX ·YX ·Y X ·Y δX ·Y εX ·Y εX ·Y εδX ·Y  Q. Either X ·Y = 0 and AB is the zero matrix, or X ·Y = ±1 and the matrix in the middle clearly also satisfies (1) and (2). Now P and Q permute its rows and columns, leaving properties (1) and (2) unchanged. Remark 4.3. As stated above, the four matrices Si are distinguished from the twelve matrices SiSj (i 6= j) by symmetry. For the symmetric matrices Si, we have X = Y and X.Y = X.X = (±1)2 + (±1)2 + (±1)2 ≡ 0 (mod 3). For the asymmetric matrices SiSj (i 6= j) this does not happen, and X.Y ≡ ±1 (mod 3). This is consistent with the relations S2i = 0 but (SiSj) 2 6= 0. The semigroup S exhibits a lot of symmetry. We find its automorphism group. Some automorphisms are inherited from the symmetry group Σ of the tetrahedron ∆, which has order 24 and is isomorphic to S4. Since Σ permutes the faces of ∆, it permutes the Si by conjugation: Si 7→ σSiσ−1. tetrahedral chains and a curious semigroup 113 This action extends to −Si since conjugation commutes with the negative of the identity, and hence the action extends uniquely to any element of S. Clearly this action defines automorphisms σ̂ of S, given by σ̂(Si) = Sσ(i) , σ̂(−Si) = −Sσ(i) , σ ∈ S4 , 0 ≤ i ≤ 3 . (4.1) Define Ŝ4 = { σ̂ : σ ∈ S4 } which is isomorphic to S4. We now prove that the σ̂ are the only automor- phisms. Proposition 4.4. The automorphism group of S is the group Ŝ4 with action (4.1). Proof. Suppose that α is an automorphism not in Ŝ4. The elements in the subset T = {±Si : 1 ≤ i ≤ 4} are the only nonzero elements with square 0, so any automorphism α permutes them. Moreover, within this set the annihilator of Si is{ U ∈T : USi = 0 = SiU } = { Si,−Si : 1 ≤ i ≤ 4 } . Therefore if α maps Si to ±Sj then it must map −Si to ∓Sj. Composing with a suitable permutation in Ŝ4 we can assume α(Si) = ±Si for all i. Since α 6∈ Ŝ4, we have α(Si) = −Si for some i. For all j 6= i the relation SiSjSi = Si in (3.1) implies that (−Si)(α(Sj))(−Si) = −Si, so α(Sj) = −Sj for all j. But this contradicts the second relation SiSjSk = −SiSk in (3.1) which applies when all i,j,k are different. Thus every automorphism lies in Ŝ4. Remark 4.5. Without using the above result, it is clear that the elements ±Si satisfy (±Si)2 = 0. The products SiSj for i 6= j are idempotent: (SiSj) 2 = SiSj. The elements −SiSj for i 6= j are not idempotent, but their squares are: (−SiSj)2 = SiSj. Thus these three classes are distinguished by simple automorphism-invariant properties. 5. Density We now use the classification of closed subgroups of O(3) to prove a den- sity theorem for R. This result is too weak to imply the result of [9] that almost closed non-embedded chains with arbitrarily small gaps exist, because 114 i. stewart it factors out the translations in the affine reflections, but it leads to the equidistribution theorem of Section 6 and is of independent interest. Here the notations SO(2),SO(3),O(3) refer to Lie groups defined over the real num- bers. The full classification is not required explicitly; we just need a simple consequence: Lemma 5.1. Let G be a closed subgroup of O(3). Then one of the follow- ing conditions is valid: (a) G is finite. (b) The subgroup of G consisting of elements with determinant +1 is con- jugate to SO(2). (c) G = O(3) or G = SO(3). Proof. This can be read off from the classification of closed subgroups of O(3); see for example [14, Theorem XIII.9.2]. Theorem 5.2. The group R is dense in O(3). Proof. Let R be the closure of R. This is a closed subgroup of O(3), so Lemma 5.1 applies. We consider the three cases in turn. (a) This case does not apply: the group R is infinite since it is a free product. (b) This case also does not apply. Since SO(2) is abelian, elements of R with determinant 1 commute. In particular R0R1 and R2R3 commute. So R0R1R2R3 = R2R3R0R1 contrary to Theorem 3.1. (c) This is the only remaining case. The generators Ri do not belong to SO(3), so R = O(3). Therefore R is dense in O(3). Corollary 5.3. The subgroup R2 ⊆ R generated by all products RiRj (0 ≤ i 6= j ≤ 3) is dense in SO(3). Proof. Since det Ri = −1, the subgroup R2 consists precisely of the ele- ments of R that have determinant 1. Therefore R2 = R∩ SO(3), which is dense in SO(3). tetrahedral chains and a curious semigroup 115 6. Equidistribution In this section we prove that the orbit of any point of the unit 2-sphere S2 for the action of the group R is equidistributed in S2, in a sense made precise in Definition 6.1 below. This is a natural analogue of the theorem of Arnold and Krylov [2], and we prove it using similar methods, including simplifications suggested by one reviewer that eliminate the use of spherical harmonics. Figure 3: The Cayley graph of R (schematic). Dots (nodes) indicate group elements, with the identity being at the centre. Each edge represents left multiplication by a reflection Ri. Since R 2 i = I these edges are bidirectional. There are four types of edge, for 0 ≤ i ≤ 3, and each node lies on one edge of each type. The tree structure continues recursively to infinity. It is convenient to motivate the method in terms of the Cayley graph C(R) of R, for the generating set {R0,R1,R2,R3}, see [5, 18]. The nodes of C(R) correspond to elements of R. Edges (of type i) join node γ to Riγ. The graph C(R) is an infinite tree, every node of which has valence 4, indicated schematically in Figure 3. Right multiplication by an element γ ∈R induces an automorphism of C(R) that preserves edge types, because Ri(δγ) = (Riδ)γ for all δ ∈ R, so an edge of type i from δ to Riδ maps to an edge of type i from δγ to Ri(δγ). In particular, C(R) is homogeneous in the sense that 116 i. stewart its automorphism group acts transitively. This is a hint that orbits might be equidistributed. If x ∈ S2, the orbit Rx wraps the nodes of C(R) around S2, sending γ ∈R to γx. We can therefore use the structure of C(R) to represent the orbit. The length n of a product of reflections RinRin−1 · · ·Ri1 = γ ∈ R corresponds to the length of a path in C(R) from the identity to that element. Such paths may intersect themselves, or repeat edges. Consider a random walk on the Cayley graph of R, where each edge of type i occurs with equal probability 1 4 for 0 ≤ i ≤ 3. After n steps the random walk reaches the group element RinRin−1 · · ·Ri1 (6.1) where successive Rik are chosen randomly from {0, 1, 2, 3}, each with proba- bility 1 4 . Let Jn be the set of all index sequences j = j1, . . . ,jn, where 0 ≤ jk ≤ 3 for k = 1, . . . ,n. For j ∈ Jn define Rj = RjnRjn−1 · · ·Rj1 and let the length of the sequence be λ(j ) = n. Assume that the random walk starts at the identity of R. Let U be an open (or, more generally, measurable) subset of S2. For fixed but arbitrary x ∈ S2, let Pn(U) be the probability that after n steps of the random walk the point RinRin−1 · · ·Ri1x belongs to U. Let µ be normalised surface Lebesgue measure on S2, that is, Lebesgue measure divided by 4π, so that µ(S2) = 1. Heuristically, the orbit of x is equidistributed provided that lim n→∞ Pn(U) = µ(U) (6.2) for all U. This motivates the following discussion, leading to the definition of equidistribution that we employ in this paper, Definition 6.1. Let C(S2) be the space of all continuous maps f : S2 → R with inner product 〈f,g〉 = ∫ S2 fg dµ. tetrahedral chains and a curious semigroup 117 The integral is finite because S2 is compact, and the inner product gives C(S2) a Hilbert space structure, inducing the norm ‖f‖ = √ 〈f,f〉 . (6.3) The group O(3) of rotations in R3 has a natural action as isometries (norm- preserving maps) of C(S2), defined as follows. Suppose that γ ∈ O(3). The natural action of O(3) on R3 leaves S2 invariant, so for each γ ∈ O(3) there is an operator Gγ on C(S2) defined by (Gγf)(x) = f ( γ−1x ) . (6.4) The inverse ensures that this is a left action: Gγδf = GγGδf. Since µ is O(3)-invariant, ‖Gγf‖ = ‖f‖ (6.5) for all γ ∈ O(3). Therefore every Gγ is an isometry of C(S2). In particular, there are isometries Ri (0 ≤ i ≤ 3) of C(S2) such that (Rif)(x) = f(R −1 i x) = f(Rix) where the latter equality follows from R2i = I. If we define R = R0 + R1 + R2 + R3 4 then the powers R n correspond bijectively to paths of length n through the Cayley graph, weighted by the probability 4−n of each such path. This moti- vates the following definition: Definition 6.1. Let x ∈ S2, and let fn(x) = (R n f)(x) = 1 4n ∑ j∈Jn f(Rj x) be the average value of f evaluated at images of x under elements of R having length n (corresponding to paths of length n in the Cayley graph). Then the R-orbit of x ∈ S2 is equidistributed if and only if lim n→∞ fn(x) = ∫ S2 f dµ (6.6) for any f ∈C(S2). 118 i. stewart Recall that the norm of a linear operator A on a Banach space B is defined by ‖A‖ = sup ‖x‖=1 ‖Ax‖ (x ∈B) . Immediate consequences are: ‖Ax‖≤‖A‖‖x‖ , ‖AB‖≤‖A‖‖B‖ . (6.7) As in Arnold and Krylov [2], we also need the following lemma: Lemma 6.2. Let v1, . . . ,vs ∈C(S2). Suppose that ‖vi‖ 6= 0 for 1 ≤ i ≤ s, and ‖v1 + · · · + vs‖ = ‖v1‖ + · · · + ‖vs‖ . (6.8) Then there exists v ∈ C(S2) and positive real numbers ri (1 ≤ i ≤ s) such that vi = riv. Proof. The triangle inequality implies that ‖v1 + · · · + vs‖ ≤ ‖v1‖ + · · · + ‖vs‖ for any vi ∈ C(S2). We claim that the conditions of the lemma imply that this is an equality with the stated properties of the vi. To prove the claim, recall that a normed vector space is strictly convex if x,y 6= 0 and ‖x + y‖ = ‖x‖+‖y‖ imply that x = cy for some real constant c > 0. The space C(S2) is a Hilbert space, hence strictly convex [16, 6]. For such spaces, equality occurs in the triangle inequality if and only if all vi are multiples of each other by nonnegative real numbers. A simple induction completes the proof. Corollary 6.3. If ‖vi‖ = 1 for 1 ≤ i ≤ s, and (6.8) holds, then v1 = · · · = vs . (6.9) We can now state and prove the main theorem of this section: Theorem 6.4. If x ∈ S2 then the orbit Rx is equidistributed in the sense of (6.6). Proof. Define a polynomial function in C(S2) to be the restriction to S2 of a polynomial function R3 → R in Cartesian coordinates (x,y,z). By the Stone-Weierstrass Theorem [20, 22, 23, 29], polynomial functions are dense tetrahedral chains and a curious semigroup 119 in C(S2) with the topology of uniform convergence. Therefore it is enough to prove (6.6) when f is polynomial. This equality is obvious when f is constant, so it suffices to prove that lim(R n f)(x) = 0 for any polynomial function whose integral over S2 is zero. Let Pl be the vector space of polynomial functions p : S2 → R of degree ≤ l such that ∫ S2 p dµ = 0. Since this space is finite-dimensional, any two Hausdorff vector topologies coincide, so the topology of pointwise convergence is the same as as that given by the norm (6.3). It is therefore sufficient to show that lim‖Rnf‖ = 0 for all f ∈ Pl. Consider the linear operators R n : Pl → Pl. We have ‖Rif‖ = ‖f‖, since reflections preserve µ. Hence ‖R‖≤ 1, so ‖Rn‖≤ 1 for any n ∈ N. Suppose that some ‖Rm‖ = k < 1 where m ≥ 1. Then (6.7) implies that ‖Rnf‖≤ kbn/mc. Therefore limn→∞fn(x) = 0, proving (6.6). Otherwise we must have ‖Rn‖ = 1 for all n ∈ N. We will show that this cannot occur. For a contradiction, suppose it does. The unit ball of Pl is compact, so there exists fn ∈ Pl with ‖R n fn‖ = ‖fn‖ = 1; hence also ‖Rifn‖ = 1 for all i ≤ n. A subsequence of the fn converges to a polynomial f ∈ Pl with ‖R n f‖ = ‖f‖ = 1 for all n. Now Lemma 6.2, with the vi being all Rj f for j ∈ Jn, implies that R0f = R1f = R2f = R3f , R0R1f = R0R2f = · · · = R2R3f , . . . and more generally, Rj f = Rkf for all j ,k ∈ Jn . (6.10) Let Km = { Rj : λ(j ) = m } be the set of all words in the Ri of length m, allowing consecutive repetitions. Since R2i = I, we have K0 ⊆ K2 ⊆ K4 ⊆ ···⊆ K2n ⊆ ··· . Applying these group elements to x ∈ S2, K0x ⊆ K2x ⊆ K4x ⊆ ···⊆ K2nx ⊆ ··· . 120 i. stewart Therefore the orbit of x under R2 ⊆ SO(3) is the union R2x = ⋃ n∈N K2nx. Now (6.10) implies that f ( R−1j x ) = f ( R−1k x ) for all Rj , Rk ∈ K2n and all n ∈ N. Thus f is constant on the R2-orbit of x. By Corollary 5.3, R2 is dense in SO(3), so by continuity f is constant on the SO(3)-orbit of x. By definition, elements of Pl have zero integral over S2, so this contradicts ‖f‖ = 1. Remark 6.5. (a) It is also plausible that R is equidistributed in O(3) with respect to Haar measure [15]. Equivalently, R2 is equidistributed in SO(3). However, we have not sought a proof. (b) A similar argument shows that if G is a subgroup of O(n) generated by finitely many reflections, and the orbit Gx of some point x in the unit sphere Sn−1 is dense, then all orbits are equidistributed in Sn−1 in the sense of the obvious generalisation of Definition 6.1. Acknowledgements I am grateful to Stan Wagon and Staszek Janeczko for discussions about the fascinating mathematics associated with chains of tetrahe- dra, and for helpful comments on a draft of this paper. I thank the two anonymous reviewers, who suggested numerous improvements, especially to the proof of Theorem 6.4. The presentation also benefited from a lec- ture given to the Warwick University Mathematics Society in November 2018, which motivated a closer examination of some subtle features of the proofs and removed a number of typographical errors. References [1] T.M. Apostol, Kronecker’s approximation theorem: the one-dimensional case, in “ Modular Functions and Dirichlet Series in Number Theory ”, Sec- ond edition, Graduate Texts in Mathematics, 41, Springer-Verlag, New York, 1990, 148 – 155. [2] V.I. Arnold, A.L. Krylov, Uniform distribution of points on a sphere and certain ergodic properties of solutions of linear ordinary differential equations in a complex domain, Dokl. Akad. Nauk SSSR 148 (1963), 9 – 12 (Russian). English translation: Soviet Math. Dokl. 4 (1963), 1 – 5. tetrahedral chains and a curious semigroup 121 [3] H. Babiker, S. Janeczko, Combinatorial representation of tetrahedral chains, Commun. Inf. Syst. 15 (2015), 331 – 359. [4] A.H. Boerdijk, Some remarks concerning close-packing of equal spheres, Philips Research Rep. 7 (1952), 303 – 313. [5] A. Cayley, Desiderata and suggestions: No. 2. The theory of groups: graphi- cal representation, Amer. J. Math. 1 (1878), 174- 176. Collected Mathemati- cal Papers 10 (reprint), Scholarly Publishing Office, University of Michigan Library, Michigan, 2005, 403 – 405. [6] J.A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396 – 414. [7] H.S.M. Coxeter, “ Regular Complex Polytopes ”, Cambridge University Press, London-New York, 1974. [8] T.J. Dekker, On reflections in Euclidean spaces generating free products, Nieuw Arch. Wisk. (3) 7 (1959), 57 – 60. [9] M. Elgersma, S. Wagon, Closing a Platonic gap, Math. Intelligencer 37 (1) (2015), 54 – 61. [10] M. Elgersma, S. Wagon, The quadrahelix: a nearly perfect loop of tetra- hedra, arXiv:1610.00280 [math.MG]. [11] M. Elgersma, S. Wagon, An asymptotically closed loop of tetrahedra, Math. Intelligencer 39 (3) (2017), 40 – 45. [12] R.B. Fuller, “ Synergetics ”, Macmillan, New York, 1975. [13] A. Gorodnik, A. Nevo, On Arnold’s and Kazhdan’s equidistribution prob- lems, Ergodic Theory Dynam. Systems 32 (2012), 1972 – 1990. [14] M. Golubitsky, I.N. Stewart, D.G. Schaeffer, “ Singularities and Groups in Bifurcation Theory ”, Vol. II, Applied Mathematical Sciences 69, Springer-Verlag, New York, 1988. [15] A. Haar, Der Massbegriff in der Theorie der kontinuierlichen Gruppen (Ger- man), Ann. of Math. 34 (1933), 147 – 169. [16] O. Hanner, On the uniform convexity of Lp and `p, Ark. Mat. 3 (1956), 239 – 244. [17] L. Kronecker, Näherungsweise ganzzahlige Auflösung linearer Gleichungen, Berl. Ber. (1884), 1179 – 1193, 1271 – 1299. [18] W. Magnus, A. Karrass, D. Solitar, “ Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations ”, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966; (2nd ed., Dover, New York, 2005). [19] J.H. Mason, Can regular tetrahedra be glued together face to face to form a ring? Math. Gaz. 56 (1972), 194 – 197. [20] W. Rudin, “ Functional Analysis ”, McGraw-Hill, New York, 1973. [21] H. Steinhaus, Problème 175, Colloq. Math. 4 (1957), 243. [22] M.H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375 – 481. 122 i. stewart [23] M.H. Stone, The generalized Weierstrass approximation theorem, Math. Mag. 21 (1948), 167 – 184, 237 – 254. [24] S. Świerczkowski, On a free group of rotations of the Euclidean space, Indag. Math. 20 (1958), 376 – 378. [25] S. Świerczkowski, On chains of regular tetrahedra, Colloq. Math. 7 (1959), 9 – 10. [26] S. Świerczkowski, “ Looking Astern ”, unpublished memoir. [27] G. Tomkowicz, S. Wagon, “ The Banach–Tarski Paradox ” (2nd. ed.), En- cyclopedia of Mathematics and its Applications 163, Cambridge University Press, New York, 2016. [28] S. Ulam, The Scottish Book (typed English translation sent by Stan Ulam to Professor Copson in Edinburgh on January 28, 1958), http://kielich.amu.edu.pl/Stefan Banach/pdf/ks-szkocka/ks- szkocka3ang.pdf. [29] K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Sitzungsber. Königlich Preussischen Akad. Wiss. Berlin 2 (1885), 633 –639, 789 – 805. [30] The Lviv Scottish Book, http://www.math.lviv.ua/szkocka/view.php. http://kielich.amu.edu.pl/Stefan_Banach/pdf/ks-szkocka/ks-szkocka3ang.pdf http://kielich.amu.edu.pl/Stefan_Banach/pdf/ks-szkocka/ks-szkocka3ang.pdf http://www.math.lviv.ua/szkocka/view.php Introduction Reflections in faces of the tetrahedron Symmetries of the tetrahedron. The 3 3 matrices. Factoring out translations. Free product structure Non-existence of a closed chain. Properties of the semigroup Density Equidistribution