OOn Roli’s Cube
Barry Monson ∗ University of New BrunswickFredericton, New Brunswick, Canada E3B 5A3February 18, 2021
Abstract
First described in 2014,
Roli’s cube R is a chiral 4-polytope, faithfully realized inEuclidean 4-space (a situation earlier thought to be impossible). Here we describe R in a new way, determine its minimal regular cover, and reveal connections to theM¨obius-Kantor configuration.Key Words: regular and chiral polytopes; realizations of polytopesAMS Subject Classification (2000): Primary: 51M20. Secondary: 52B15. Actually
Roli’s cube R isn’t a cube, although it does share the 1-skeleton of a 4-cube.First described by Javier (Roli) Bracho, Isabel Hubard and Daniel Pellicer in [3], R isa chiral 4-polytope of type { , , } , faithfully realized in E (a situation earlier thoughtimpossible). Of course, Roli didn’t himself name R ; but the eponym is pleasing to hiscolleagues and has taken hold.Chiral polytopes with realizations of ‘full rank’ had (incorrectly) been shown not toexist by Peter McMullen in [11, Theorem 11.2]. Mind you, these objects do seem to beelusive. Pellicer has proved in [15] that chiral polytopes of full rank can exist only inranks 4 or 5.Roli’s cube R was constructed in [3] as a colourful polytope , starting from a hemi-4-cube in projective 3-space. (For more on this, see Section 3.) The construction given here ∗ This work was generously supported at the Universidad Nacional Aut´onoma de M´exico (Morelia) byPAPIIT-UNAM grant a r X i v : . [ m a t h . M G ] F e b n Section 6 is a bit different, though certainly closely related. In Section 7 we can theneasily manufacture the minimal regular cover T for R , and give both a presentation andfaithful representation for its automorphism group. Along the way, we encounter boththe M¨obius-Kantor Configuration 8 and the regular complex polygon 3 { } -cube: convex, abstract and colourful The most familiar of the regular convex polytopes in Euclidean space E is surely the4-cube P = { , , } . A familiar projection of P into E is displayed in Figure 1. v w−v Figure 1: A 2-dimensional look at a 3-dimensional projection of the 4-cube.Let us equip E with its usual basis b , . . . , b and inner product. Then we may takethe vertices of P to be the 16 sign change vectors e = ( ε , . . . , ε ) ∈ {± } . (1)At any such vertex there is an edge (of length 2) running in each of the 4 coordinatedirections, so that P has 32 = · edges. Similarly we count the 24 squares { } as facesof dimension 2. Finally, P has 8 facets; these faces of dimension 3 are ordinary cubes { , } . They lie in four pairs of supporting hyperplanes orthogonal to the coordinate axes.It is enjoyable to hunt for these faces in Figure 2, where the 8 parallel edges in each ofthe coordinate directions have colours black, red, blue and green, respectively.Let us turn to the symmetry group G for P . Each symmetry γ is determined by itsaction on the vertices, which clearly can be permuted with sign changes in all possibleways. Thus G has order 2 ·
4! = 384, and we may think of it as being comprised of all4 × x x x −v v = (1,1,1,1)w Figure 2: The most symmetric 2-dimensional projection of the 4-cube.In fact, G can be generated by reflections ρ , ρ , ρ , ρ in hyperplanes. Here ρ negatesthe first coordinate x (reflection in the coordinate hyperplane orthogonal to b ); and, for1 (cid:54) j (cid:54) ρ j transposes coordinates x j , x j +1 (reflection in the hyperlane orthogonal to b j − b j +1 ).Note that the reflection in the j -th coordinate hyperplane is ρ ρ ··· ρ j − for 1 (cid:54) j (cid:54) γ η := η − γη .) The product of these 4 special reflections, in anyorder, is the central element ζ : t (cid:55)→ − t . It is easy to check as well that ζ = ( ρ ρ ρ ρ ) . (2)The Petrie symmetry π = ρ ρ ρ ρ therefore has period 8.For purposes of calculation, we note that G (cid:39) C (cid:111) S is a semidirect product. Underthis isomorphism, each γ ∈ G factors uniquely as γ = eµ , where µ ∈ S n is a permutationof { , . . . , } (labelling the coordinates); and e is a sign change vector, as in (1). Notethat e µ = ( ε , ε , ε , ε ) µ = ( ε (1) µ − , ε (2) µ − , ε (3) µ − , ε (4) µ − ) . Now really γ is a signed permutation matrix. But it is convenient to abuse notation,keeping in mind that each e corresponds to a diagonal matrix of signs and each µ to apermutation matrix. Thus we might write π = ρ ρ ρ ρ = ( − , , , · (4 , , ,
1) = −
11 0 0 00 1 0 00 0 1 0 . (3)3ext we use the group G = (cid:104) ρ , ρ , ρ , ρ (cid:105) to remanufacture the cube. In this (ge-ometric) version of Wythoff ’s construction [7, § base vertex v fixed bythe subgroup G := (cid:104) ρ , ρ , ρ (cid:105) (which permutes the coordinates in all ways). Thus, v = c (1 , , ,
1) for some c ∈ R . To avoid a trivial construction we take c (cid:54) = 0, so, up tosimilarity, we may use c = 1. Then the orbit of v under G is just the set of 16 points in(1); and their convex hull returns P to us. Since G is the full stabilizer of v in G , thevertices correspond to right cosets G γ .The beauty of Wythoff’s construction is that all faces of P can be constructed in asimilar way by induction on dimension ([13, Section 1B], [4] and [12]). For example, thevertices v = (1 , , ,
1) and vρ = ( − , , ,
1) of the base edge of P are just the orbit of v under the subgroup G := (cid:104) ρ , ρ , ρ (cid:105) ; and edges of P correspond to right cosets of thenew subgroup G . Furthermore, a more careful look reveals that a vertex is incident withan edge just when the corresponding cosets have non-trivial intersection.Pursuing this, we see that the face lattice of P can be recontructed as a coset geometrybased on subgroups G , G , G , G , where G j := (cid:104) ρ , . . . , (cid:98) ρ j , . . . , ρ (cid:105) . (4)From this point of view, P becomes an abstract regular -polytope , a partially ordered setwhose automorphism group is G . Notice that the distinguished subgroups in (4) providethe proper faces in a flag in P , namely a mutually incident vertex, edge, square and3-cube.The crucial structural property of G is that it should be a string C-group with respectto the generators ρ j . A string C-group is a quotient of a Coxeter group with linear diagramunder which an ‘intersection condition’ on subgroups generated by subsets of generators,such as those in (4), is preserved [13, Sections 2E].For the 4-cube P , G is actually isomorphic to the Coxeter group B with diagram • • • • (5)Comparing the geometric and abstract points of view, we say that the convex 4-cube is a realization of its face lattice (the abstract 4-cube).When we think of a polytope from the abstract point of view, we often use the term rank instead of ‘dimension’. An abstract polytope Q is said to be regular if its automor-phism group is transitive on flags (maximal chains in Q ). Intuitively, regular polytopeshave maximal symmetry (by reflections). Next up are chiral polytopes, with exactlytwo flag orbits and such that adjacent flags are always in different orbits (so maximalsymmetry by rotations, but without reflections).We will soon encounter less familiar abstract regular or chiral polytopes, with theirrealizations. For a first example, suppose that we map (by central projection) the facesof P onto the 3-sphere S centred at the origin. We can then reinterpret P as a regular spherical polytope (or tessellation), with the same symmetry group G . Now the centre of4 is the subgroup (cid:104) ζ (cid:105) of order 2. The quotient group G/ (cid:104) ζ (cid:105) has order 192 and is still astring C-group. The corresponding regular polytope is the hemi- -cube H = { , , } ,now realized in projective space P [13, Section 6C]; see Figure 3. By (2), the product ofthe four generators of G/ (cid:104) ζ (cid:105) has order 4; this is recorded as the subscript in the Schl¨aflisymbol for H .Now we can outline the construction of Roli’s cube given in [3]. The image in Figure 1 or on the left in Figure 2 can just as well be understood as agraph G , namely the 1 -skeleton of the 4-cube P . In fact, we can recreate the abstract (orcombinatorial) structure of P from just the edge colouring of G : for 0 (cid:54) j (cid:54)
4, the j -facesof P can be identified with the components of those subgraphs obtained by keeping justedges with some selection of the j colours (over all such choices). We therefore say that P is a colourful polytope .Such polytopes were introduced in [1]. In general, one begins with a finite, connected d -valent graph G admitting a (proper) edge colouring, say by the symbols 1 , . . . , d . Thuseach of the colours provides a 1-factor for G . The graph G determines an (abstract)colourful polytope P G as follows. For 0 (cid:54) j (cid:54) d , a typical j -face ( C, v ) is identified withthe set all vertices of G connected to a given vertex v by a path using only colours fromsome subset C of size j taken from { , . . . , d } . The j -face ( C, v ) is incident with the k -face( D, w ) just when C ⊆ D and w can be reached from v by a D -coloured path. (This meansthat j (cid:54) k ; and we can just as well take w = v . The minimal face of rank − P G isformal.) Notice that P G is a simple d -polytope whose 1-skeleton is just G itself. From[1, Theorem 4.1], the automorphism group of P G is isomorphic to the group of colour-preserving graph automorphisms of G . (Such automorphisms are allowed to permute the1-factors.)It is easy to see that the hemi-4-cube H is also colourful. Its 1-skeleton is the completebipartite graph K , found in Figure 3. We obtain this graph from Figure 1 or Figure 2by identifying antipodal pairs of points, like v and − v .If we lift H , as it is now, to S , we regain the coloured 4-cube P . Now keep K , embedded in P , as in Figure 3. But, following [3], observe that K , admits the automor-phism α which cyclically permutes, say, the first three vertices y, w, x in the top block,leaving the rest fixed. Clearly, α is a non-colour-preserving automorphism of K , , so itseffect is to recolour 12 of the edges in the embedded graph. On the abstract level nothinghas changed for the resulting colourful polytope; it is still the hemi-4-cube H . But facesof ranks 2 and 3 are now differently embedded in P . For example, the red-blue 2-face on v , which is planar in Figure 3(b), becomes a helical quadrangle Figure 3(c) and therebyacquires an orientation. According to Definition 4.1, these helical polygons are Petriepolygons for the standard realization of H in Figure 3(b).5 y w x(a) (b) (c) α vvw x y wx y Figure 3: The graph K , in (a) is the 1-skeleton of the hemi-4-cube { , , } (b),(c).The newly coloured geometric object, which we might label H R , is a chiral realization of the abstract regular polytope H . Comforted by the fact that P is orientable, we couldjust as well apply α − to obtain the left-handed version H L . These two enantiomorphs are oppositely embedded in P , though both remain isomorphic to H as partially orderedsets. If we lift either enantiomorph to S , we obtain a chiral 4-polytope faithfully realizedin E [3, Theorem 2]. This is Roli’s cube R .Next we set the stage for a slightly different construction of R , without the use of P . -cube Let us consider the progress of the base vertex v = (1 , , ,
1) as we apply successivepowers of π in (3). We get a centrally symmetric 8-cycle of vertices v → (1 , , , − → (1 , , − , − → (1 , − , − , − → − v = ( − , − , − , − → . . . . Starting from v in Figure 2 we therefore proceed in coordinate directions 4, 3, 2, 1 (indi-cated by different colours), then repeat again. This traces out the peripheral octagon C ,which in fact is a Petrie polygon for P . Definition 4.1.
A Petrie polygon of a -polytope is an edge-path such that any consec-utive edges, but no , belong to a -face. We then say that a Petrie polygon of a -polytope Q is an edge-path such that any consecutive edges, but no , belong to ( a Petrie polygonof ) a facet of Q . For the cube P , the parenthetical condition is actually superfluous; compare [9].6learly, we can begin a Petrie polygon at any vertex, taking any of the 4! orderingsof the colours. But this counts each octagon in 16 ways. We conclude that P has 24Petrie polygons. What we really use here is the fact that G is transitive on vertices, andthat at any fixed vertex, G permutes the edges in all possible ways. We see that G actstransitively on Petrie polygons.But the (global) stabilizer of C (constructed above with the help of π and v ) is thedihedral group K of order 16 generated by µ = ρ ρ ρ ρ = ( − , , , · (2 , , and µ = µ π = ρ ρ ρ ρ ρ ρ = (1 , , , · (1 , , C (cid:111) S . Note that any 4consecutive vertices of C form a basis of E .) We confirm that P has 24 = 384 /
16 Petriepolygons.Now we move to the rotation subgroup G + = (cid:104) ρ ρ , ρ ρ , ρ ρ (cid:105) . It has order 192 and consists of the signed permutation matrices of determinant +1. Notethat
K < G + . Thus, under the action of G + , there are two orbits of Petrie polygons of12 each. Let’s label these two chiral classes R and L for right- and left-handed, taking C in class R .The two chiral classes must be swapped by any non-rotation, such as any ρ j . Todistinguish them, we could take the determinant of the matrix whose rows are any 4consecutive vertices on a Petrie polygon. The two chiral classes R and L then havedeterminants +8, respectively, −
8. Or starting from a common vertex, the edge-coloursequence along a polygon in one class is an odd permutation of the colour sequence for apolygon in the other class.The inner octagram C ∗ in Figure 2 is another Petrie polygon. Start at the vertex w = ( v ) ρ ρ ρ = (1 , − , ,
1) which is adjacent to v along a red edge; then proceed indirections 4, 1, 2, 3 and repeat. However, the remaining Petrie polygons appear in lesssymmetrical fashion in Figure 2.Note that µ actually acts on the diagram in Figure 2 as a reflection in a vertical line,whereas π rotates the octagon C and octagram C ∗ in opposite senses. On the other hand, µ = ρ ρ ρ ρ = (1 , − , , · (1 ,
3) is an element of G + which swaps C and C ∗ . Thusthere are 6 such unordered pairs like C , C ∗ in class R and another 6 pairs in class L . Remark . It can be shown that Figure 2 is the most symmetric orthogonal projectionof P to a plane [6, § isometric .The Petrie symmetry π is one instance of a Coxeter element in the group G = B ,namely a product of the four generators in some order. All such Coxeter elements areconjugate. Each of them has invariant planes which give rise to the sort of orthogonalprojection displayed in Figure 2. A procedure for finding these planes is detailed in [10,7.17]. For π , the two planes are spanned by the rows of (cid:34) √ −
12 1 √ (cid:35) and (cid:34) √ − − √ (cid:35) . These planes are orthogonal complements; and π acts on them by rotations through 45 ◦ and 135 ◦ , respectively. Figure 2 results from projecting P onto the first plane. (cid:3) M and the M¨obius-Kantor Configuration C , C ∗ in Figure 2. Now working around therim clockwise from v delete the edges coloured blue, red, black, green, and repeat. Weare left with the trivalent graph L displayed in Figure 4.
13 5 720 46 v
Figure 4: The Levi graph L = { } + { } for the configuration 8 .In fact, L is the generalized Petersen graph { } + { } , studied in detail by Coxeterin [5, Section 5]. The graph is 2-arc transitive, so that its automorphism group has order96 = 16 · · − [2, Chapter 18]. We return to this group later.We have labelled alternate vertices of L by the residues 0 , , , , , , , . The remaining(unlabelled) vertices of L represent the 8 lines in the configuration. Thus we have lines813 (represented by the ‘north-west’ vertex ( − , − , , v .Notice that we can interpret the configuration as being comprised of two quadrangleswith vertices 0 , , , , , , each inscribed in the other : vertex 0 lies on edge 13,vertex 1 lies on edge 24, and so on.So far this configuration 8 is purely abstract. In fact, it can be realized as a point-lineconfiguration in a projective (or affine) plane over any field in which z − z + 1 = 0has a root, certainly over C . However, 8 cannot be realized in the real plane.Coxeter made other observations in [5], including the fact that the graph L is a sub-1-skeleton of the 4-cube. Altogether L contains 6 Petrie polygons, which we can brieflydescribe by their alternate vertices:0246(= C ∗ ) 0541 12561357(= C ) 2367 0743Hence, the configuration can be regarded as a pair of mutually inscribed quadrangles inthree ways.Observe that each edge of L lies on exactly two of the 6 octagons. For example, thetop edge with vertices labelled 1 and v lies on octagons 1357 and 1256. (It does not matterthat two such octagons then share a second edge opposite the first.) Furthermore, eachvertex lies on the three octagons determined by choices of two edges. We can therebyconstruct a 3-polytope M of type { , } , with 6 octagonal faces, whose 1-skeleton is L .In short, M is realized by substructures of the 4-cube P .Moving sideways, we can reinterpret M in a more familiar topological way as a mapon a compact orientable surface of genus 2. Recall that M is covered by the tessellation { , } of the hyperbolic plane, as indicated in Figure 5.Now return to E where the combinatorial structure of M is handed to us as faithfullyrealized. Drawing on [8, Section 8.1], we have that the rotation group Γ ( M ) + for M isgenerated by two special Euclidean symmetries: σ = π = ρ ρ ρ ρ = ( − , , , · (4 , , ,
1) (preserving the base octagon C ); and σ = ρ ρ ρ ρ = (1 , , , · (1 , ,
4) (preserving the base vertex v on C ).The order of Γ ( M ) + must then be twice the number of edges in M , namely 48. Letus assemble these and further observations in Proposition 5.1. (a)
The -polytope M is abstractly regular of type { , } , here realizedin E in a geometrically chiral way. (b) The rotation subgroup Γ ( M ) + = (cid:104) σ , σ (cid:105) has order and presentation (cid:104) σ , σ | σ = σ = ( σ σ ) = ( σ − σ ) = 1 (cid:105) (6)9 Figure 5: Part of the tessellation { , } of the hyperbolic plane.(c) The full automorphism group Γ ( M ) has order and presentation (cid:104) τ , τ , τ | τ = τ = τ = ( τ τ ) = ( τ τ ) = ( τ τ ) = (( τ τ ) τ τ ) = 1 (cid:105) (7) Proof . We begin with (b), where it is easy to check that the relations in (6) do hold for thematrix group (cid:104) σ , σ (cid:105) . By a straightforward coset enumeration [8, Chapter 2], we concludefrom the presentation in (6) that the subgroup (cid:104) σ (cid:105) has the 6 coset representatives1 , σ , σ , σ σ − , σ σ , σ σ − σ . (We abuse notation by passing freely between the matrix group and abstract group.) Thisfinishes (b).We next note that (cid:104) σ (cid:105) ∩ (cid:104) σ (cid:105) = { } , since σ j fixes v only for j ≡ M is regular (ratherthan just chiral) if and only if the mapping σ (cid:55)→ σ − , σ (cid:55)→ σ σ induces an involutoryautomorphism τ of Γ ( M ) + . But the new relations induced by applying the mapping to(6) are easily verified formally, or even by matrices. For instance, since σ σ = σ − σ − ,we have ( σ σ ) = ( σ σ − σ − ) = σ σ − σ − = 1 . M is abstractly regular and Γ ( M ) has order 96. The presentation in (7) follows atonce by extending Γ ( M ) + by (cid:104) τ (cid:105) , then letting τ := τ, τ := τ σ , τ := τ σ σ .It remains to check that our realization is geometrically chiral. This means that τ isnot represented by a symmetry of M as realized in E . From the combinatorial structure, τ would have to swap vertices 1 and v while preserving the two Petrie polygons on thatedge. This means that τ would have to act just like µ , that is, just like reflection in avertical line in Figure 2. But µ does not preserve the set of 8 edges deleted to give L inFigure 4. (cid:3) Remark . It is helpful to note that the centre of Γ ( M ) + is generated by σ . Referringto [8, Section 6.6], we find that Γ ( M ) + is isomorphic to the group (cid:104)− , | (cid:105) , which in turnis an extension by C of the binary tetrahedral group (cid:104) , , (cid:105) . Indeed, a = σ − σ σ − , b = σ σ satisfy a = b = ( ab ) (= ζ ). Thus, (cid:104) , , (cid:105) (cid:47) Γ ( M ) + . R of type { , , } Under the action of G + we expect to find 4 = 192 /
48 copies of M . To understand thisbetter, recall that there are 12 Petrie polygons in one chiral class, say R . As with C and C ∗ , each polygon D is paired with a unique polygon D ∗ (with the disjoint set of 8 vertices).For each D there are then two ways to remove 8 edges so as to get a copy of L and hencea copy of M . Since M has six 2-faces like C , we once more find 12 · / M .Each Petrie polygon lies on 2 copies of M , again from the two ways to remove 8 edges.For example, C lies on both M and ( M ) µ . (The same is true for C ∗ .)The pointwise stabilizer in G + of the base edge joining v = (1 , , ,
1) and ( v ) µ =( − , , ,
1) must consist of pure, unsigned even permutations of { , , } . Therefore it isgenerated by σ := ρ ρ = (1 , , , · (2 , , . It is easy to check that G + = (cid:104) σ , σ , σ (cid:105) .Since three consecutive edges of a Petrie polygon lie on two adjacent square faces in acubical facet of of P , it must be that every vertex of R has the same vertex-figure as P ,thus of tetrahedral type { , } .We have enumerated and (implicitly) assembled the faces of a 4-polytope R , faithfullyrealized in E and symmetric under the action of G + . Let’s take stock of its proper faces:rank stabilizer in G + order number of faces type0 (cid:104) σ , σ (cid:105)
12 16 vertex of cube P (cid:104) σ σ , σ (cid:105) P (cid:104) σ , σ σ (cid:105)
16 12 Petrie polygons of P in one class R (cid:104) σ , σ (cid:105)
48 4 copy of M It is not hard to see that our 4-polytope R is isomorphic to Roli’s cube, as constructedin [3] and as described in Section 3. 11 heorem 6.1. (a) The -polytope R is abstractly chiral of type { , , } . Its symmetrygroup Γ ( R ) (cid:39) G + has order and the presentation (cid:104) σ , σ , σ | σ = σ = σ = ( σ σ ) = ( σ σ ) = ( σ σ σ ) = 1 (8)( σ − σ ) = 1 (9)( σ − σ ) = 1 (cid:105) (10)(b) R is faithfully realized as a geometrically chiral polytope in E . Proof . The relations in (8) are standard for chiral 4-polytopes [16, Theorem 1]; and wehave seen that the relation in (9) is a special feature of the facet M . Enumerating cosetsof the subgroup (cid:104) σ , σ (cid:105) , which still has order 48, we find at most the 8 cosets representedby 1 , σ , σ , σ σ , σ σ , σ σ σ , σ σ σ , σ σ σ . Thus the group defined by (8) and (9) has order at most 384. But G + , where theserelations do hold, has order 192. We require an independent relation. In Section 7, wewill see why (10) is just what we need.To show that R is abstractly chiral we must demonstrate that the mapping σ (cid:55)→ σ − , σ (cid:55)→ σ σ , σ (cid:55)→ σ does not extend to an automorphism of G + . This is easy, since( σ σ ) = ζ whereas ( σ − σ ) = 1 . (11)Clearly, R is realized in a geometrically chiral way in E ; we have already seen thisfor its facet M .Our concrete geometrical arguments should suffice to convince the reader that we reallyhave described here a chiral 4-polytope identical to the original Roli’s cube. A skeptic cannail home the proof by applying [16, Theorem 1] to the group G + , as generated above. (cid:3) R The rotation group G + for the cube has order 192 and ‘standard’ generators ρ ρ , ρ ρ , ρ ρ .But for our purposes we use either of two alternate sets of generators. We already have σ = ( ρ ρ )( ρ ρ ) , σ = ( ρ ρ )( ρ ρ )( ρ ρ ) , σ = ρ ρ . (12)Now we also want σ = σ − , σ = σ σ , σ = σ . (13)Recalling our the shorthand for such matrices, we have σ = ( − , , , · (4 , , , σ = (1 , , , · (1 , , σ = (1 , , , · (2 , , , σ = (1 , , , − · (1 , , , σ = ( − , − , , , , σ = (1 , , , · (2 , , . We have seen that the group G + = (cid:104) σ , σ , σ (cid:105) (with these specified generators) isthe rotation (and full automorphism) group of the chiral polytope R of type { , , } .From [17, Section 3] we have that the (differently generated) group G + = (cid:104) σ , σ , σ (cid:105) isthe automorphism group for the enantiomorphic chiral polytope R . By generating thecommon group in these two ways we effectively exhibit right- and left-handed versions ofthe same polytope.Our geometrical realization of R began with the base vertex v = (1 , , ,
1) (which alsoserved as base vertex for the 4-cube P ). It is crucial here that v does span the subspacefixed by σ and σ . By instead taking G + with base vertex v = ( − , , ,
1) fixed by σ , and σ (cid:105) , we have a faithful geometric realization of R , still in E , of course.We will soon have good reason to mix G + and G + in a geometric way. Each groupacts irreducibly on E . Construct the block matrices, κ j = ( σ j , σ j ), j = 1 , ,
3, now actingon E and preserving two orthogonal subspaces of dimension 4. Obviously we may extendour notation for signed permutation matrices to the cubical group B acting on E . Thus,taking the second copy of E to have basis b , b , b , b , we may combine our descriptionsof σ j , σ j to get κ = ( − , , , , , , , − · (4 , , , , , , ,κ = (1 , , , , − , − , , · (1 , , , , ,κ = (1 , , , , , , , · (2 , , , , . Now let T + = (cid:104) κ , κ , κ (cid:105) . In slot-wise fashion, κ , κ , κ satisfy relations like thosein (8) and (9). From the proof of Theorem 6.1, we conclude that T + has order 384. Weeven get a presentation for it.Recall that the centre of G + is generated by ζ = σ = σ . Thus the centre of T + hasorder 4, with non-trivial elements( ζ,
1) = ( κ κ ) , (1 , ζ ) = ( κ − κ ) , and ( ζ, ζ ) = κ . (14)(This is at the heart of the proof that R is abstractly chiral.) Looking at (11), we seethat T + / (cid:104) (1 , ζ ) (cid:105) (cid:39) Γ ( R ) , and thus see the reason for the special relation in (10). Similarly, T + / (cid:104) ( ζ, (cid:105) (cid:39) Γ ( R ).Finally, we have T + / (cid:104) ( ζ, ζ ) (cid:105) (cid:39) Γ ( P ) + , (15)the rotation group of the 4-cube (isomorphic to G + generated in the customary way).13ow T + is clearly isomorphic to the mix G + ♦ G + described in [14, Theorem 7.2].Guided by that result, we seek an isometry τ of E which swaps the two orthogonalsubspaces, while conjugating each σ j to σ j . It is easy to check that τ = (1 , , , , , , , · (1 , , , , T + is the rotation subgroup of a string C-group T = (cid:104) τ , τ , τ , τ (cid:105) , where τ = τ κ , τ = τ κ κ , τ = τ κ κ κ . The corresponding directlyregular 4-polytope has type { , , } and must be the minimal regular cover of each of thechiral polytopes R and R . We consolidate all this in Theorem 7.1. (a)
The group T = (cid:104) τ , τ , τ , τ (cid:105) is a string C-group of order and withthe presentation (cid:104) τ , τ , τ , τ | τ j = ( τ τ ) = ( τ τ ) = ( τ τ ) = 1 , (cid:54) j (cid:54) , ( τ τ ) = ( τ τ ) = ( τ τ ) = (( τ τ ) τ τ ) = 1 (cid:105) (b) The corresponding regular -polytope T has type { , , } and is faithfully realizedin E , with base vertex ( v, v ) = (1 , , , , − , , , , . The polytope T is the minimalregular cover for Roli’s cube R and its enantiomorph R . It is also a double cover of the -cube P . (c) T (cid:39) {M , { , }} is the universal regular polytope with facets M and tetrahedalvertex-figures. Proof . The centre of T is generated by ( ζ, ζ ). It is easy to check that T / (cid:104) ( ζ, ζ ) (cid:105) (cid:39) G ,the full symmetry group of the cube; compare (15). In other words, the mapping τ j (cid:55)→ ρ j , (cid:54) j (cid:54)
3, induces an epimorphism ϕ : T → G . Since τ , τ , τ and ρ , ρ , ρ bothsatisfy the defining relations for Γ ( { , } ) (cid:39) S , ϕ is one-to-one on (cid:104) τ , τ , τ (cid:105) . By thequotient criterion in [13, 2E17], T really is a string C-group. The remaining details areroutine. For background on (c) we refer to [13, 4A]. (cid:3) Much as in the proof, the assignment κ j (cid:55)→ σ j , ( j = 1 , , ϕ R : T + → G + . On the abstract level, this in turn induces a covering (cid:101) ϕ R : T + → R , inother words, a rank- and adjacency-preserving surjection of polytopes as partially orderedsets. The corresponding covering of geometric polytopes is induced by the projection E → E ( x, y ) (cid:55)→ x The projection ( x, y ) (cid:55)→ y likewise induces the geometrical covering (cid:101) ϕ L : T + → R . Both (cid:101) ϕ R and (cid:101) ϕ L are 3-coverings, meaning here that each acts isomorphically on facets M and14ertex-figures { , } [13, page 43]. Notice that each face of R and R has two preimagesin T .The polytope T is also a double cover of the 4-cube P . But there is no natural wayto embed P in E to illustrate the geometric covering, since κ = − E , whereas ( ρ ρ ) = 1 for P . We noted earlier that 8 can be ‘realized’ as a point-line configuration in C . We willshow this here by first endowing E with a complex structure . Thus, we want a suitableorthogonal transformation J on E such that J = ζ . Keeping the addition, we thendefine ( a + ıb ) u = au + b ( uJ ) , for a, b ∈ R , u ∈ E . Thus ıu = uJ . Over C , E has dimension 2. Our choice for the matrix J is motivated byan orthogonal projection different from that in Figures 2 and 4.The vectors representing the vertices labelled 0 , . . . , O = { , , } ,one of two inscribed in P . In [7, Figure 4.2A], Coxeter gives a projection of O whichnicely displays certain 2-faces of O .
04 2613 57
Figure 6: Another projection of the cross-polytope O .In Figure 6, each vertex of either of the two concentric squares forms an equilateraltriangle with one edge of the other square. These 8 triangles correspond to the unlabellednodes in Figure 4, and also to the lines of the configuration 8 . (Any real triangle lies ona unique complex line in C .) We may take the vertices in Figure 6 to be ( ± , ±
1) and( ± r, , (0 , ± r ), where r = √ −
1. 15ut what plane Λ in E actually gives such a projection? Starting with an unknownbasis a , b for Λ, we can force a lot. For example, edge [2 ,
0] is the projection of (0 , , − , , , , , ◦ . Fromsuch details in the geometry, we soon find that Λ is uniquely determined and get a basissatisfying a · a = b · b and a · b = 0. But any such basis can still be rescaled or rotatedwithin Λ. Tweaking these finer details, we find it convenient to take a , b to be the firsttwo rows of the matrix L = 12 √ √ − − (2 + √ − √ √ − − −√ √ √ √ . (16)The last two rows a , b of L give a basis for the orthogonal complement Λ ⊥ .Since we want J to induce 90 ◦ rotations in both Λ and Λ ⊥ , we have J = 1 √ − − − − − − . (17)Notice that a J = b and a J = b , so , { a , a } is a C -basis for E ; and the planeΛ in Figure 6 is just z = 0 in the resulting complex coordinates. The points in theconfiguration 8 now have these complex coordinates:Label ( z , z )0 ( r, − ı )1 ( − ı, r )2 ( rı, − − ı )3 ( − − ı, − rı )4 ( − r, − ı )5 (1 − ı, − r )6 ( − rı, ı )0 (1 + ı, rı )(Recall that r = √ − ⊥ ( z = 0). There labels on theinner and outer squares are suitably swapped.A typical line in the configuration 8 , like that containing points 1 , ,
7, has equation r (1 − ı ) z + 2 z = 2 r (1 + ı ) . { }
3. Its symmetry group (of unitarytransformations on C ) is the group 3[3]3 with the presentation (cid:104) γ , γ | γ = 1 , γ γ γ = γ γ γ (cid:105) . (18)In fact, this group of order 24 is isomorphic to the binary tetrahedral group (cid:104) , , (cid:105) . Butin our context, we may identify it with the centralizer in G of the structure matrix J . Abit of computation shows that this subgroup of G is generated by γ = ρ ρ ρ ρ = (1 , ,
2) and γ = ρ ρ ρ ρ = ( − , , − , · (1 , , , which do satisfy the relations in (18). Acknowledgements . I want to thank Daniel Pellicer, both for his many geometricalideas and also for generously welcoming me to the Centro de Ciencias Matem´aticas atUNAM (Morelia).
References [1]
G. Araujo-Pardo, I. Hubard, D. Oliveros, and E. Schulte , Colorful Poly-topes and Graphs , Israel Journal of Mathematics, 195 (2013), pp. 647–675.[2]
N. Biggs , Algebraic Graph Theory , Cambridge University Press, Cambridge, UK,2nd ed., 1993.[3]
J. Bracho, I. Hubard, and D. Pellicer , A finite chiral 4-polytope in R , Dis-crete Comput. Geom., 52 (2014), pp. 799–805.[4] H. S. M. Coxeter , Wythoff ’s construction for uniform polytopes , Proc. LondonMath. Soc., 38 (1935), pp. 327–339. (Reprinted in
The Beauty of Geometry: TwelveEssays , Dover, NY, 1999).[5] ,
Self-dual configurations and regular graphs , Bull. Amer. Math. Soc., 56 (1950),pp. 413–455.[6] ,
Regular Polytopes , Dover, New York, 3rd ed., 1973.[7] ,
Regular Complex Polytopes , Cambridge University Press, Cambridge, UK,2nd ed., 1991.[8]
H. S. M. Coxeter and W. O. J. Moser , Generators and Relations for DiscreteGroups , Springer, New York, 3rd ed., 1972.179]
H. S. M. Coxeter and A. I. Weiss , Twisted Honeycombs { , , } t and TheirGroups , Geom. Dedicata, 17 (1984), pp. 169–179.[10] J. E. Humphreys , Reflection Groups and Coxeter Groups , Cambridge UniversityPress, Cambridge, UK, 1990.[11]
P. McMullen , Regular polytopes of full rank , Discrete Comput. Geom., 32 (2004),pp. 1–35.[12]
P. McMullen , Geometric Regular Polytopes , vol. 172 of Encyclopedia of Mathe-matics and its Applications, Cambridge University Press, Cambridge, UK, 2020.[13]
P. McMullen and E. Schulte , Abstract Regular Polytopes , vol. 92 of Encyclo-pedia of Mathematics and its Applications, Cambridge University Press, Cambridge,UK, 2002.[14]
B. Monson, D. Pellicer, and G. Williams , Mixing and Monodromy of AbstractPolytopes , Trans. Amer. Math. Soc, 366 (2014), pp. 2651–2681.[15]
D. Pellicer , Chiral polytopes of full rank exist only in ranks 4 and 5 , Beitr. AlgebraGeom., (2020).[16]
E. Schulte and A. I. Weiss , Chiral polytopes , in Applied Geometry and DiscreteMathematics: The Victor Klee Festschrift, P. Gritzmann and B. Sturmfels, eds.,vol. 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Amer. Math. Soc.,Assoc. Comput. Mach., 1991, pp. 493–516.[17]