ARS MATHEMATICA CONTEMPORANEA
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 12 (2017) 315-327
A chiral 4-polytope in R3
Daniel Pellicer
Centro de Ciencias Matemáticas, UNAM, Campus Morelia C.P. 58190, Morelia, Michoacan, México
Received 8 March 2016, accepted 29 August 2016, published online 21 January 2017
Abstract
In this paper we describe an infinite chiral 4-polytope in the Euclidean 3-space. This builds on previous work of Bracho, Hubard and the author, where a finite chiral 4-polytope in the Euclidean 4-space is constructed. These two polytopes show that there are finite and infinite chiral polytopes of full rank as defined by McMullen.
Keywords: Chiral 4-polytope, full rank polytope. Math. Subj. Class.: 52B11
1 Introduction
In this paper we regard n-polytopes as combinatorial structures in Rd constructed from (n-1)-polytopes as building blocks, where 0- and 1-polytopes are points and line segments, respectively.
Regular polytopes are those admitting the highest degree of symmetry in the sense that any two flags are equivalent under the symmetry group. In some way they admit all possible abstract reflections as symmetries.
Nowadays we have plenty of examples of regular polytopes, the most obvious being the convex regular polytopes (see for example [2]) and the tessellations by n-dimensional cubes. Other examples of regular polytopes can be found in [4, 5, 6, 7].
Chiral polytopes have two orbits of flags under the symmetry group with the property that adjacent flags belong to distinct orbits. They admit all abstract rotations as symmetries, but no abstract reflection.
There is very little published work on chiral polytopes on Euclidean spaces. There are no convex chiral polytopes and no chiral tessellations of Euclidean spaces. This illustrates the difficulty to find 'natural' families of chiral polytopes.
In 2005 Schulte classified all chiral polyhedra in R3 (see [11] and [12]). They are all infinite; some have finite faces, and some have infinite faces.
E-mail address: pellicer@matmor.unam.mx (Daniel Pellicer)
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
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In [5, Theorem 11.2] it was claimed that for any positive integer d there are neither finite chiral d-polytopes in Rd, nor infinite chiral (d + 1)-polytopes in Rd.
The first known finite chiral 4-polytope in R4 was discovered in [1] in 2014 proving false one half of the claim in [5]. In this paper we describe the first known infinite chiral 4-polytope in R3, proving false the remaining half of the claim.
Definitions and basic results are given in Section 2. In Section 3 we describe the building blocks of the 4-polytope, which is constructed in Section 4. Finally, in Section 5 we discuss the combinatorial symmetry of the 4-polytope.
2 Preliminaries
In this section we recall some concepts and results of the Euclidean space and of polytopes embedded on it.
2.1 Symmetries of the Euclidean space
The rotation group of the octahedron, that we shall denote by [3,4]+, is one of the finite groups of isometries of R3. It contains 24 elements, out of which six are 4-fold rotations, eight are 3-fold rotations, nine are half-turns, and the remaining one is the identity. In particular, all of its elements preserve orientation (see for example [3] for a more detailed description of this group).
A lattice is the orbit of the origin o under a discrete translation group of R3 generated by translations with respect to three linearly independent vectors. Up to similarity, there are three lattices that are invariant under the action of the group [3,4]+ (see [8, Section 6D]).
The cubic lattice, denoted by A(1j0,o), consists of the points of R3 with integer coordinates. The translations by the vectors (1,0,0), (0,1,0) and (0,0,1) constitute a basis for the translation group of this lattice.
The face-centred cubic lattice, denoted by A(110), is generated by the translations by the vectors (1,1,0), (1,0,1) and (0,1,1). It contains the set of points of R3 with integer coordinates, whose sum is an even number. The cubic lattice is the union
A(i,i,o) u (A(i,i,o) + (1,0,0))
of two isometric copies of the face-centred cubic lattice.
Finally, the body centred cubic lattice is the set of points with integer coordinates such that either all of them are even, or all of them are odd. It is generated by the translations by the vectors (1,1,1), (-1,1,1) and (1, -1,1), and it is denoted by A(111). The cubic lattice is the union
A(i,i,o) u (A(i,i,o) + (1,0,0)) u (A(i,i,o) + (0,1,0)) u (A(i,i,o) + (0,0,1))
of four isometric copies of the body-centred cubic lattice.
The tessellation {4, 3,4} by cubes of R3 is the only regular tessellation of the Euclidean space. A Petrie polygon of {4, 3,4} is a helix with vertex and edge sets contained in those of {4,3,4}, where every two consecutive edges belong to some square; every three consecutive edges belong to the same cube, but not to the same square; and no four consecutive edges belong to the same cube. The direction vectors of any three consecutive edges of a Petrie polygons of {4,3,4} are precisely (1,0,0),(0,1,0) and (0,0,1) in some order. These helices have axes with direction vectors (1,1,1), (1,1, -1), (1, -1,1) and
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(-1,1,1). Every edge of a Petrie polygon h is a translate of any edge that is 3k steps apart in h (that is, there are 3k - 1 edges between them).
Any two Petrie polygons of {4,3,4} are isometric. However, any given Petrie polygon is equivalent under orientation preserving isometries (translations, rotations and twists) to only half of the Petrie polygons. We say that a Petrie polygon is a right helix if it can be obtained from the helix
..., (1,0,0), (0, 0,0), (0,1,0), (0,1,1), (-1,1,1),... by an orientation preserving isometry. The remaining helices are called left helices. 2.2 Polyhedra and 4-polytopes
When studying highly symmetric polytopes we need to move away from convexity to get a richer theory. The definitions below follow the spirit of [4] and subsequent papers.
For us a polygon (or 2-polytope) in R3 is a discrete set of points called vertices or 0-faces together with a set of line segments called edges or 1-faces between pairs of vertices, such that the resulting graph is connected and 2-regular. The edges are allowed to intersect in interior points, but there are no vertices in the interior of edges.
A polyhedron (or 3-polytope) in R3 is a set of polygons, called 2-faces, with the extra properties that every edge belongs to exactly two polygons, the set of vertices is discrete, the graph determined by the vertices and edges is connected, and the vertex-figures at all vertices are connected. Here the vertex-figure at a vertex v is the polygon (or in principle polygons) whose vertices are the neighbours of v, two of which are adjacent if and only if they are the two neighbours of v in a 2-face.
A 4-polytope in R3 is a set of polyhedra, called 3-faces with the extra properties that every 2-face belongs to exactly two polyhedra, the set of vertices is discrete, the graph determined by the vertices and edges is connected, and the vertex-figures at all vertices are polyhedra. The vertex-figure at a vertex v in this case consists of the polygons that are the vertex-figures at v in the polyhedra containing it.
Defined as above, polyhedra and 4-polytopes in R3 are precisely Euclidean realisations of abstract polyhedra and 4-polytopes as defined in [8, Section 5]. Due to this relationship, we say that two elements of an n-polytope are incident if one is contained in the other as geometric objects. Vertices, edges, polygons and polyhedra are then regarded as objects of rank 0,1, 2 and 3, respectively.
The facets of an n-polytope are its (n - 1)-faces (n g {3,4}). The 1-skeleton of a polyhedron or of a 4-poltyope is the graph determined by its sets of vertices and edges. The 2-skeleton of a 4-polytope consists of the sets of vertices, edges and 2-faces.
A flag of an n-polytope p is a set of n mutually incident elements of p, one of each rank. That is, a flag of a polygon is a pair of incident vertex and edge, a flag of a polyhedron is a triple of mutually incident vertex, edge and 2-face, and a flag of a 4-polytope contains a vertex, an edge, a polygon and a polyhedron, all incident to the other three.
Given any flag $ of an n-polytope and given i g{0,...,n - 1} there exists a unique flag that differs from $ only on the face of rank i. The flag is known as the i-adjacent flag of $. We extend recursively this notion and for any word w on the elements in {0,..., n - 1} we define $wi := ($w)\
By a symmetry of an n-polytope p we mean an isometry of R3 that preserves p. An n-polytope is said to be regular whenever its symmetry group acts transitively on the flags
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of p. Clearly, the facets of a regular 4-polytope are regular polyhedra. There are 48 regular polyhedra and 8 regular 4-polytopes in R3; they were thoroughly studied in [7].
An n-polytope is said to be chiral whenever its symmetry group induces two orbits on the flags, with adjacent flags in distinct orbits. The term 'chiral' often carries the meaning of being handed, that is, not admitting a mirror symmetry. In our context, where only highly symmetric objects are of interest, chiral polytopes denote the most symmetric polytopes that do not admit a symmetry mapping a flag to an adjacent flag, which is the combinatorial equivalent to mirror symmetry.
There are no finite chiral polyhedra in R3 (see for example [11, Theorem 3.1]). Infinite chiral polyhedra were classified in [11] and [12] in six families. One of this polyhedra is described in detail in Section 3.
Regular and chiral n-polytopes admit a set of distinguished abstract rotations as symmetries. These are isometries Si that map a given base flag $ to the flag	for i e {1,... ,n - 1}. Such an isometry needs not be a rotation around an axis in R3. However, the term 'rotation' is in no way inadequate, since their combinatorial impact is similar to that of a rotation on a polygon. S1 cyclically permutes the vertices and edges of the base 2-face, S2 cyclically permutes the edges and polygons around the base vertex contained in the base polyhedron, and S3 cyclically permutes the polygons and polyhedra around the base edge.
The symmetry group of a chiral n-polytope is generated by its distinguished abstract rotations. The group generated by all distinguished abstract rotations of a regular n-polytope has index at most 2 in the full symmetry group. The tetrahedron {3,3} and its Petrial {4, 3}3 are examples of polyhedra where the subgroups generated by the distinguished abstract rotations have index 2 and 1, respectively.
Conversely, an n-polytope whose symmetry group contains all possible distinguished abstract rotations is either regular or chiral, and it is regular if and only if the symmetry group contains an element moving the base vertex but fixing all other elements of the base flag (see [13] for the combinatorial analogue to these claims).
3 The polyhedron P1 (1, 0)
It is time now to describe the chiral polyhedron P1 (1,0) as a particular case of the general description of the polyhedra P1(a, b) in [9]. Other description, using the technique known as Wythoff's construction, can be found in [12].
Throughout, t will denote the cubical tessellation {4, 3,4} of R3 with vertices on Z3, and n : R3 ^ n the orthogonal projection into the plane n through the origin o with normal vector (1,1,1). It is well known that the image under n of the 1-skeleton of t is the 1-skeleton of a tessellation t' by equilateral triangles, and that A(111)n is the vertex set of a tessellation by equilateral trangles whose edge length is twice as that of Z3n. The preimage under n of any edge of t' intersects t in a collection of parallel edges.
Figure 1 shows the tessellation t' of n on pale gray and black lines. Assume that the origin o projects to the fat vertex and that the coordinate axes project as indicated. That is, assume that one endpoint of the edge to the left of the fat vertex is the projection of (1,0,0), that the black edge at the fat vertex that does not belong to the dotted path ends at the projection of (0,0,1), and that one endpoint of the remaining black edge incident to the fat vertex is the projection of (0,1,0).
Under these assumptions the polyhedron P1(1,0) can be described as follows. Its ver-
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y
Figure 1: Projection of the 1-skeleton of Pi(1,0)
tex set is Z3. The edge set of P1 (a, b) consists of all edges e of t such that en is a black edge in Figure 1, that is,
•	the edge between (x, y, z) and (x + 1, y, z) for every (x, y, z) g A(111) and (x,y,z) g A(i,i,i) + (0,0,1),
•	the edge between (x, y, z) and (x, y + 1, z) for every (x, y, z) g A(1,1,1) and (x, y, z) g A(1,1,1) + (1,0,0),
•	the edge between (x, y, z) and (x, y, z + 1) for every (x, y, z) g A(1,1,1) and (x, y, z) g A(1,1,1) + (0,1,0).
Finally, the 2-faces are all Petrie polygons of t living in this 1-skeleton.
The six edges incident to any given vertex of t project by n to three gray edges and three black edges. This can be used to show that all vertices of P^a, b) have degree 3. Since no two black edges at the same vertex in Figure 1 are collinear, the set of three edges incident to a vertex of P1(a, b) are translates of the three edges incident to some vertex of the cube c with vertex set {(x, y, z) : x, y, z, g {0,1}}. The precise arrangement of edges at each vertex is explained by the following straightforward lemma.
Lemma 3.1. The three edges incident to any vertex of P1(1,0) in A(111) are translates of the three edges incident to (0,0,0) in the cube C defined as above. Similarly, the three edges incident to any vertex of P1(1,0) in A(111) + (1,0,0) (resp. in A(111) + (0,1,0) and A(111) + (0,0,1)) are translates of the three edges of C incident to (1,0,1) (resp. to (1,1,0) and to (0,1,1)).
The 2-faces of P1(1,0) are helices over triangles and belong to four parallel classes h1, h2, h3, h4. Helices in h1 project to the triangles with black edges in Figure 1. Every helix in h2 projects to n either in the path with dashed lines or in one of its translates. Helices in H3 and h4 project to images of helices in h2 under rotations by 2n/3 and by 4n/3, respectively.
The axis of every helix in h1 has direction vector (1,1,1). There is precisely one helix in h1 projecting to each triangle in Figure 1. For example, the helix
..., (-1, 0,-1), (-1,0, 0), (-1,1, 0), (0,1, 0), (0,1,1), (0, 2,1),... (3.1)
is the only helix that projects to the triangle with gray interior. All other helices in h1 are obtained by translating this helix by integer combinations of (1,1, -1) and (1, -1,1).
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In contrast, infinitely many helices in h2 project to the dotted path. They are the helix
..., (1,-1, -1), (1,0, -1), (1, 0, 0), (0, 0, 0), (0,1, 0), (0,1,1), (-1,1,1),... (3.2)
and its translates by m(1,1,1) for m g Z. The remaining helices in h2 are obtained by translating these helices by m(1,1, -1) for m g Z. They all have direction vector
(-1,1,1).
The parallel classes h3 and h4 are respectively represented by the helices
..., (-1,1, -1), (-1,1,0), (0,1, 0), (0, 0, 0), (0, 0,1), (1,0,1), (1, -1,1),..., (3.3) ..., (-1, -1,1), (0, -1,1), (0, 0,1), (0, 0, 0), (1, 0, 0), (1,1, 0), (1,1, -1),..., (3.4)
which have the same projection to n as their translates by m(1,1,1) with m g Z. All other helices in each of these classes are obtained by translating these helices by m(-1,1,1) for m g Z. The axis of every helix in h3 (resp. h4) has direction vector (1, -1,1) (resp. (1,1, -1)).
It should be clear now that every edge in a black-edged triangle in Figure 1 is in the projection of a helix in hi and of a helix in some other parallel class. The horizontal edges in black-edged triangles are the projection of helices in h1 and h3; those edges in black-edged triangles that are translates of the edge in the gray triangle that belongs also to the dotted path are projections of helices in h1 and h2; and the remaining edges in black-edged triangles are projections of helices in h1 and h4. Similarly, every black edge in Figure 1 that is not in a triangle belongs to the projection of helices in precisely two of the parallel classes h2, h3 and h4. From this it is easy to see that every edge of P1(1,0) belongs precisely to two helices. Furthermore, the parallel classes of the helices containing an edge e are completely determined by whether or not en belongs to a black-edged triangle, together with its direction vector in Figure 1.
In order to discuss the symmetries of P1(1,0) we take as base flag $ the one containing the origin o, the edge between o and (0,1,0), and the helix in (3.2). Let S1 be the screw motion
(x,y, z) ^ (-y + 1,z, -x)	(3.5)
with axis {1 (1,1,0) + k(-1,1,1) : k g M}, translation vector 1 (1, -1, -1) and rotation component of 2n/3. Let S2 be the rotation
(x,y, z) ^ (z,x,y)	(3.6)
about the axis {k(1,1,1) : k g M} by an angle of 2n/3. By applying these isometries to the edges of -F1(1,0) we can see that S1 and S2 preserve the 1-skeleton of P1(1,0). Hence, S1 and S2 also preserve the set of 2-faces of P1(a, b). Furthermore, S1 cyclically permutes the vertices of the base 2-face and S2 cyclically permutes the three helices around o. This implies that P1 (1,0) admits symmetries acting like the distinguished abstract rotations and therefore it is either regular or chiral.
The polyhedron P1 (1,0) turns out to be chiral. Indeed, the only isometry T preserving the base edge and base helix, but interchanging the base vertex o with (0,1,0) is the halfturn
(x,y, z) ^ (z, -y + 1, x)
with axis {(0,1/2,0) + k(1,0,1) : k g M}. However, such a T maps the edge between o and (0,0,1) to the edge between (0,1,0) and (1,1,0), which is not an edge of P1(1,0), and hence does not preserve the 1-skeleton of P1 (1,0).
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The base helix of Pi (1,0) is a right helix as explained in Section 2. The symmetries Si and S2 defined above are both orientation preserving. It follows that all helices in P1 (1,0) are right helices.
The symmetry S3 of P1(1,0) is the translation by the vector (1, -1, -1). The conjugates of this translation by S2 and S2 are the translations by the vectors (-1,1, -1) and (-1, -1,1), respectively. The next proposition follows.
Proposition 3.2. The symmetry group of P1(1,0) contains the translations by all vectors with endpoints in A(1j11).
Since P1 (1,0) is chiral, it is also helix-transitive, implying the next remark.
Remark 3.3. The orthogonal projections of P1(1,0) in the directions (1,1, -1), (1, -1,1) and (-1, 1, 1) of the axes of the helices are all isometric to the projection in the direction (1,1,1) in Figure 1.
It is interesting to note that the set of gray edges in Figure 1 is isometric to the set of black edges, and one can be obtained from the other by a half-turn around the fat point. The polyhedron constructed from the preimages in t of the gray edges under the projection n is clearly isometric to P1 (1,0) but it contains only left helices. They are precisely the images of the helices of P1 (1,0) under the isometry mapping x to -x for every x g R3.
4 The chiral 4-polytope V{<^,3,4}
The polyhedron P1(1,0) just described is the building block of the chiral 4-polytope p{œ,3,4}. The vertex and edge sets of p{to,3,4} are the vertex and edge sets of the cubic tessellation t. The 2-faces of p{to,3,4} are all right Petrie polygons of t. This set constitutes the regular polygonal complex k7(1,1) in [10].
The facets of p{to,3,4} are P1(1,0) and its images under the group [3,4]+ of rotations of the octahedron. Recall that [3,4]+ has 24 elements. Since P1(1,0) is invariant under three-fold rotations around the line through o with direction vector (1,1,1), there are at most 8 images of P1(1,0) under [3,4] + .
In fact, P{TOj3j4} has precisely 8 facets. They are described next. Recall from Section 3 that the three neighbours of o in P1(1,0) are e1 = (1,0,0), e2 = (0,1,0) and e3 = (0,0,1). This motivates to denote this polyhedron as a facet of p{to,3,4} by F(+,+,+). The group [3,4]+ acts transitively on the set of octants of R3 and hence P{TOj3j4} has precisely 8 facets. They are denoted F(ai,a2,a3), where a takes the value '+' whenever in that facet ej is a neighbour of o, and the value '-' otherwise. For example, the orbit of F(+,+,+)
under the 4-fold rotation around the z axis mapping (x, y, z) to (y, -x, z) is
(F(+,+,+), F(+,-,+), F(-,-,+), F(-,+,+)).
In order to better understand the combinatorics of p{to,3,4} it is convenient to compute the image of all its facets under the projection n as defined in Section 3. This can be done by directly applying the orientation preserving isometries in [3,4]+ and then n to the edges of Pl(1, 0).
Alternatively, we can use the fact that the helices of P1(1,0) split in four classes h1, h2, H3 and h4, consisting of helices with axes having direction vector (1,1,1), (-1,1,1), (1, -1,1) and (1,1, -1), respectively. Every isometry T g [3,4]+ permutes the four directions (1,1,1), (1,1, -1), (1, -1,1) and (-1,1,1) of the axes of helices of P1(1,0) and
so ,+,+)T must have a parallel class of helices with axes in the direction of (1,1,1). These helices project orthogonally into triangles on the plane n; furthermore, this triangles must be pointing up, since they are precisely the images of right helices, whereas the left helices project into triangles pointing down. Similarly, the helices in the three remaining parallel classes must project into isometric copies of the dotted path in Figure 1. This information, together with the three neighbours of o on each facet and Remark 3.3, determines the projections of the eight facets of P{TOj3j4} as in Figure 2, where the fat dot represents the origin o.
We can see that F(+ + +) and	are the only facets where o does not belong to a
helix with axis in the direction of (1,1,1). In both instances o belongs to helices with axes in the directions of (-1,1,1), (1, -1,1) and (1,1, -1).
We choose the following seven isometries T(ai,a2,a3) € [3,4]+ mapping F(+ + +) to
F(ai,a2,a3)-
(x, -y, -z), (-x, y, -z),
( X ^ z)
(-z, y, x), ^ z)
(x, z, -y), ( x, -y, -z).
We also denote by Hi, H2, H3 and H4 the helices in (3.1), (3.2), (3.3) and (3.4), respectively. Recall that the helices in F(+ + +) with direction vector (1,1,1) are Hi and its translates by vectors with endpoints in A(1j11).
The right Petrie polygons of t with axes in the direction of (1,1,1) are the ones in F(+,+,+) together with their translates by (1,0,0), (0,1,0) and (0,0,1). It can be seen from Figure 2 that the helices in F(+ + +) with axis in the direction of (1,1,1) are also
T(+,_	_) :	(x, y, z)	
T(-,+	_) :	(x, y, z)	
T(_,_	,+) :	(x, y, z)	
T(-,+	,+) :	(x,y,z)	
T(+,_	,+) :	(x,y,z)	
T(+,+	,_) :	(x, y, z)	
T(_,_	,_) :	(x, y, z)	
H3 and H4 the helices			
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helices of F(_,-,-). Furthermore, the helices with axes in the direction of (1,1,1) of F(+i_,_) and of F(+ - +) are the translates of those in F(+ + +) by (0,0,1), the helices with axes in the direction of (1,1,1) of
in F

_) and of F(+j+j_
are the translates of those

by (1,0,0), and the helices with axes in the direction of (1,1,1) of

and of F(_j+ +) are the translates of those in F(+ + +) by (0,1,0). This can also be verified by noting that
Hi
Hi + (1,-1,1) Hi + (1,-1, -1)
HiT(-,-,_ H2T(+,-,-) + (0,0,1)
) + (-1,1,1),
H2T(+,-,+) + (0,0,1),
HsT(-,+,-) + (1,0,0) = HsT(+i+,-) + (1,0,0), Hi = H4T(-,-,+) + (0,1,0) = H4T(_i+i+) + (0,1,0),
together with the fact that [3,4]+ permutes the four directions of the axes of the helices of F(+,+,+) and that the set of helices of F(+ + +) in any of the four directions is invariant by translations by vectors with endpoints in A(1,1,1). This shows that all right Petrie polygons of t with axes in the direction of (1,1,1) belong to at least two facets of P{TOj3j4} . Recall that the set of helices on P1(1,0) with axis in the direction of (1,1,1) can be obtained by translating H1 by vectors with endpoints in A(111), and similarly the set of helices of P1(1,0) with axis in the direction of (-1,1,1), (1, -1,1) or (1,1, -1) can be obtained by translating H2, H3 or H4, respectively, by vectors with endpoints in A(111). This shows that each right Petrie polygon of t with axis in the direction of (1,1,1) belongs to precesely two facets of P{TOj3j4} . The fact that t and P{TOj3j4} are symmetric under [3,4]+ implies the following lemma.
Lemma 4.1. Every helix of P{TOj3j4} belongs to precisely two facets.
Table 1 indicates the direction vector of the helices that two facets have in common (if any). In the table, (a1, a2, a3) indicates the facet F(ai a2 a3). In particular one can conclude that the facets F(ai a2 a3) and F(bljb2jb3) have a helix in common if (a1, a2, a3) and (b1, b2, b3) coincide either in two coordinates or in none. The entries on the table can be obtained by applying the isometries T(ai,a2,a3) to the helices of F(+ + +), or by a careful inspection of Figure 2.
Facets		(+, +,-)	(+,-, +)	(-, +, +)	(+,-, -)	(-, +, -)	(-,-, +)	(-,-, -)
(+, +, +)	—	(-1, 1,1)	(1,1,-1)	(1,-1,1)	none	none	none	(1,1, 1)
(+, +,-)	(-1, 1, 1)	—	none	none	(1,-1,1)	(1,1, 1)	(1, 1,-1)	none
(+,-, +)	(1,1,-1)	none	—	none	(1,1, 1)	(1,-1,1)	(-1, 1,1)	none
(- +, +)	(1,-1, 1)	none	none	—	(-1, 1,1)	(1, 1,-1)	(1,1, 1)	none
(+,-,-)	none	(1,-1,1)	(1,1, 1)	(-1, 1,1)	—	none	none	(1, 1,-1)
(-, +,-)	none	(1, 1,1)	(1,-1,1)	(1, 1,-1)	none	—	none	(-1, 1,1)
(-,-, +)	none	(1,1,-1)	(-1, 1,1)	(1,1, 1)	none	none	—	(1,-1,1)
(-,-,-)	(1, 1,1)	none	none	none	(1, 1,-1)	(-1, 1,1)	(1,-1,1)	—
Table 1: Helices shared by two facets of P{TOj3j4}.
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Recall that the vertex-figure at o in p{to,3,4} consists of the eight triangles that are the vertex-figures at o of the eight facets of p{to,3,4}. From the construction it is immediate that the vertex-figure at o in p{to,3,4} is an octahedron.
As a consequence of Proposition 3.2, the three neighbours in F(+,+,+) of any vertex v in A(i,i,i) are v+(1,0,0), v+(0,1,0) and v+(0,0,1). A similar statement can be made for the remaining seven facets of p{to,3,4}. Indeed, since F(ai,a2,a3) is the image of F(+,+,+) under some isometry in [3,4]+ and A(111) is invariant under the entire group [3,4]+, the translations by vectors with endpoints in A(111) are symmetries of F(a1, a2, a3). This implies that the vertex-figure at any vertex of p{to,3,4} in A(111) is an octahedron. This statement is in fact true for any vertex of p{to,3,4}.
Lemma 4.2. The vertex-figure of any vertex of p{to,3,4} is an octahedron.
Proof. Since every facet of p{to,3,4} is invariant under translations by vectors with endpoints in A(111), we only need to show that the result holds for a representative of each translation class. Since A(10,o) is the disjoint union of four translates of A(111), there are only four orbits of vertices of p{to,3,4} under the action of the translations by vectors with endpoints in A(111). We take o, (1,0,0), (0,1,0) and (0,0,1) as representatives of these orbits.
The previous discussion shows that the result holds for o (and hence for vertices in A(111)). A close inspection to Figure 2 (or direct verification) shows that the neighbours of the remaining three representatives in facet F(ai,a2,a3) are as in Table 2, where an entry
(b1, b2, b3) indicates that v has as neighbour v + ej when b is '+', and v - ej when b is
	F(+,+,+)	F(+,+,-)	F(+,-,+)		F(+,-,-)	F(-,+,-)	F(-,-,+)	
(1,0,0)	(-, +, -)	(-,-,-)	(-, +, +)	(+,-, +)	(-,-, +)	(+, +, +)	(+,-,-)	(+, +,-)
(0,1,0)	(-,-, +)	(+,-, +)	(+, +,-)	(-,-,-)	(-, +,-)	(+,-,-)	(+, +, +)	(-, +, +)
(0,0, 1)	(+,-, -)	(-, +, +)	(-,-,-)	(+, +,-)	(+, +, +)	(-,-, +)	(-, +,-)	(+,-, +)
Table 2: Neighbours of (1,0,0), (0,1,0) and (0,0,1) on the facets of P{to,3,4}.
The entry of Table 2 corresponding to vertex v and facet F indicates the octant determined by the three neighbours of v in F. The vertex-figure of v at F is then a triangle in that octant (determined by the three neighbours of v). All octants appear exactly once on each row, implying that the vertex-figures are all octahedra.	□
We are now ready for our main result.
Theorem 4.3. The structure p{to,3,4} is a chiral 4-polytope in R3. Proof. We know that p{to,3,4} is the set of polyhedra
{P1(1, 0)a : a € [3, 4] +},
and that the set of vertices is descrete. We also know that its 1-skeleton coincides with that of the tessellation by cubes, and hence it is connected. Every 2-face belongs to precisely two facets by Lemma 4.1 and all vertex-figures are polyhedra by Lemma 4.2. Hence p{^,3,4} is a 4-polytope.
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By construction, p{to,3,4} is invariant under the rotation S2 defined in (3.6) and the rotation S3 given by
(x,y, z) ^ (z, y, -x),
since they just permute the images of P1 (1,0) under [3,4]+. Furthermore, the screw motion Si defined in (3.5) also preserves p{to,3,4}. In fact, by applying S1 to the edge set of F(ai,a2,a3) for each (a1,a2,a3) we can see that S1 fixes F(+,+,+) and F(+,+,_), and
induces the permutation
(F(+,_,+),F(_,_,_),F(_,+,+)) • (F(+,_,_,),F(_,_,+),F(_,+,_))
in the remaining 6 facets of p{to,3,4}.
These three isometries are the distinguished abstract rotations with respect to the flag ^ containing o, the edge between o and (0,1,0), the helix in (3.2) and the facet F(+,+,+). Indeed, it is not hard to verify that = ^10, = ^21 and = ^32. Hence
P{to,3,4} is either regular or chiral, and since its facets are chiral, p{to,3,4} itself is chiral.
' ' ' ' □
5 Combinatorial symmetry
In the previous section we constructed a 4-polytope that is chiral as a geometric object. In this section we discuss its combinatorial nature. That is, we study p{to,3,4} as a partially ordered set with a rank function ranging in {0,1,2, 3}, whose elements are the vertices, edges, polygons and polyhedra of p{to,3,4} where two of them are incident if and only if one is contained in the other (see [8] for proper definitions of abstract polytopes).
An automorphism of an n-polytope is a bijection of its vertices, edges, etc. that preserves the incidence. A polytope is combinatorially regular (resp. combinatorially chiral) if its automorphism group acts transitively on its flags (resp. if its automorphism group induces two orbits on flags with adjacent flags in distinct orbits). The distinguished abstract rotations of p{to,3,4} as a geometric object induce automorphisms as an abstract object.
Due to the connectivity of p{to,3,4} and to the uniqueness of ¿-adjacent flags for i e {0,1,2,3}, any automorphism of p{to,3,4} is completely determined by the image on any flag. As a consequence of this, the group generated by the automorphisms given by the abstract distinguished rotations of p{to,3,4} has index at most 2 on the full automorphism group of P{to,3,4}. Furthermore, p{to,3,4} is combinatorially regular if and only if there is an automorphism mapping a flag $ to its 1-adjacent flag $1. We next show that this is not the case.
Theorem 5.1. The 4-polytope p{to,3,4} is combinatorially chiral.
Proof. We take as base flag ^ := {F0, F1, F2, F3} where F0 = o, F1 is the edge between o and (0,1,0), F2 is the helix in (3.2), and F3 = F(+,+,+). We will show that p{to,3,4} is abstractly chiral by assuming that there exists an automorphism R1 mapping ^ to and showing that the image of the vertex (1, -1,1) under such R1 is not well defined. In doing so we will abuse notation and use the geometric names and descriptions of the vertices, edges, 2-faces and facets, but the arguments to deduce the action of R1 will be purely combinatorial (not geometric).
Since R1 fixes F2 and o while moving F1, it must interchange the neighbours of o in F2, namely (1,0,0) and (0,1,0). The facet F3 also is fixed by R1, and o belongs to three
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edges in F3. Since R1 interchages the edge between o and (1,0,0) with Fi, it must fix the remaining edge, that is, the edge between o and (0,0,1); in particular (0,0,1)R1 = (0,0,1). This implies that the helices H3 in (3.3) and H4 in (3.4) are also interchanged by Ri, and so Ri interchanges (1,0,1) with (0, -1,1) and (1, -1,1) with (-1, -1,1). Now, since R1 fixes F3 but interchanges H3 and H4, it must also interchange the facets and F(+,_j+) since they are the only facets containing H3 and H4, respectively, other than F(+ + +). The edge between o and (0,0,1) is contained in precisely the four facets F(+ + +), F(_,+j+), F(+,_j+) and	The first of these facets is fixed by
Ri while the second and third are interchanged. Since Ri fixes the edge between o and (0,0,1), it must also fix the facet F(_ _ +).
Thus R1 fixes F(_ _ +) and the edge between o and (0,0,1). The remaining edges of F(_ _ +) containing o have their other endpoints in (-1,0,0) and (0, -1,0). The edge between o and (-1,0,0) is also an edge of F(_ + +) but not of F(+ _ +), whereas the edge between o and (0, -1,0) is also an edge of F(+ _ +) but not of F(_ + +). Therefore R1 must interchange the edge between o and (-1, 0,0) with the edge between o and (0, -1, 0). In F(_ _ +) there is only one helix containing these two edges, and so it must be preserved by R1. This helix is H2!"(_,_,+) with vertices
..., (-1,1, -1), (-1, 0, -1), (-1,0, 0), (0,0,0), (0, -1, 0), (0, -1,1), (1, -1,1),... ,
and so R1 must interchange (0, -1,1) with (-1,0, -1), and (1, -1,1) with (-1,1, -1). But we showed before that (1, -1,1)R1 = (-1, -1,1). This yields the desired contradiction.	□
6 Acknowledgements
This work was supported by PAPIIT-UNAM under grant IN101615. The author thanks the anonymous referee for helpful suggestions.
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