A multidimensional generalization of Lagrange’s theorem on continued fractions

A multidimensional geometric analog of Lagrange’s theorem on continued fractions is proposed. The multidimensional generalization of the geometric interpretation of a continued fraction uses the notion of a Klein polyhedron, that is, the convex hull of the set of nonzero points in the lattice ? ⁿ contained inside some n-dimensional simplicial cone with vertex at the origin. A criterion for the semiperiodicity of the boundary of a Klein polyhedron is obtained, and a statement about the nonempty intersection of the boundaries of the Klein polyhedra corresponding to a given simplicial cone and to a certain modification of this cone is proved.     

Sails and Hilbert bases

A Klein polyhedron is the convex hull of the nonzero integral points of a simplicial coneC⊂ ℝⁿ. There are relationships between these polyhedra and the Hilbert bases of monoids of integral points contained in a simplicial cone. In the two-dimensional case, the set of integral points lying on the boundary of a Klein polyhedron contains a Hilbert base of the corresponding monoid. In general, this is not the case if the dimension is greater than or equal to three (e.g., [2]). However, in the three-dimensional case, we give a characterization of the polyhedra that still have this property. We give an example of such a sail and show that our criterion does not hold if the dimension is four. CEREMADE, University Paris 9. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 43–49, April–June, 2000. Translated by J.-O. Moussafir     

Klein polyhedra for the fourth extremal cubic form

Davenport and Swinnerton-Dyer found the first 19 extremal ternary cubic formsg _i; they have the same meaning as the familiar Markov forms in the binary quadratic case. The Klein polyhedra for the formsg ₁,g ₂,g ₃ were recently computed by Bryuno and Parusnikov. The same authors computed the convergents for certain matrix generalizations of the continued fraction algorithm and studied their arrangement with respect to the Klein polyhedra. Here we consider similar problems for the fourth formg ₄. Namely, the Klein polyhedra forg ₄ and the conjugate formg ₄* are computed. They turn out to be essentially different. Their periods and fundamental domains are found. The matrix algorithm expansions of the vectors of these forms are calculated. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 110–128, January, 2000.     

Comparison of various generalizations of continued fractions

We use the Euler, Jacobi, Poincaré, and Brun matrix algorithms as well as two new algorithms to evaluate the continued fraction expansions of two vectorsL related to two Davenport cubic formsg ₁ andg ₂. The Klein polyhedra ofg ₁ andg ₂ were calculated in another paper. Here the integer convergentsP _k given by the cited algorithms are considered with respect to the Klein polyhedra. We also study the periods of these expansions. It turns out that only the Jacobi and Bryuno algorithms can be regarded as satisfactory. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 339–348, March, 1997. Translated by V. E. Nazaikinskii     

Chiral Polyhedra in Ordinary Space, I

Chiral polyhedra in ordinary euclidean space E³ are nearly regular polyhedra; their geometric symmetry groups have two orbits on the flags, such that adjacent flags are in distinct orbits. This paper completely enumerates the discrete infinite chiral polyhedra in E³ with finite skew faces and finite skew vertex-figures. There are several families of such polyhedra of types {4,6}, {6,4} and {6,6}. Their geometry and combinatorics are discussed in detail. It is also proved that a chiral polyhedron in E³ cannot be finite. Part II of the paper will complete the classification of all chiral polyhedra in E³. All chiral polyhedra not described in Part I have infinite, helical faces and again occur in families. So, in effect, Part I enumerates all chiral polyhedra in E³ with finite faces.     

Klein polyhedra for three extremal cubic forms

Davenport and Swinnerton-Dyer found the first 19 extremal ternary cubic forms g _i, which have the same meaning as the well-known Markov forms in the binary quadratic case. Bryuno and Parusnikov recently computed the Klein polyhedra for the forms g ₁ – g ₄. They also computed the convergents for various matrix generalizations of the continued fractions algorithm for multiple root vectors and studied their position with respect to the Klein polyhedra. In the present paper, we compute the Klein polyhedra for the forms g ₅, – g ₇ and the adjoint form g ₇ ^* . Their periods and fundamental domains are found and the expansions of the multiple root vectors of these forms by means of the matrix algorithms due to Euler, Jacobi, Poincaré, Brun, Parusnikov, and Bryuno, are computed. The position of the convergents of the continued fractions with respect to the Klein polyhedron is used as a measure of quality of the algorithms. Eulers and Poincarés algorithms proved to be the worst ones from this point of view, and the Bryuno one is the best. However, none of the algorithms generalizes all the properties of continued fractions.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 566–583.Original Russian Text Copyright © 2005 by V. I. Parusnikov.This revised version was published online in April 2005 with a corrected issue number.     

Klein polyhedra and relative minima of lattices

We prove that in ?³, the relative minima of almost any lattice belong to the surface of the corresponding Klein polyhedron. We also prove, for almost any lattice in ?³, that the set of relative minima with nonnegative coordinates coincides with the union of the set of extremal points of the Klein polyhedron and a set of special points belonging to the triangular faces of the Klein polyhedron.     

The boundary characteristic and the volume of lattice polyhedra

The boundary characteristic — introduced by Ding and Reay — is a functional defined for a given planar tiling which associates with a given lattice figure, some integer. It appeared to be a very useful parameter to determine the area of lattice figures in the planar tilings with congruent regular polygons. The purpose of this paper is to extend the notion of the boundary characteristic to lattice polyhedra inR³. Studying some of its properties we show, in particular, that it can be applied to determine the volume of lattice polyhedra.     

The Klein correspondence of line congruences immersed in an elliptic spaceS 3 onto the hyperquadricP 4

This article is devoted to the investigation and the construction of the Klein correspondence of line congruences referred to a specialized moving frame in a 3-dimensional elliptic spaceS ₃ to the hyperquadricP ⁴ of the Klein 5-dimensional elliptic spaceS ₅. The Klein correspondence is given and characterized by Theorems 1, 2. The methods adapted here are based on Cartan's differential calculus [1], [6].     

On the order of an automorphism of a smooth hypersurface

In this paper we give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d ≥ 3, n ≥ 2, (n, d) ≠ (2, 4), and gcd(q, d) = gcd(q, d ? 1) = 1. This allows us to give a complete criterion in the case where q = p is a prime number. In particular, we show the following result: If X is a smooth hypersurface of dimension n and degree d admitting an automorphism of prime order p then p < (d ? 1) ⁿ⁺¹; and if p > (d ? 1) ⁿ then X is isomorphic to the Klein hypersurface, n = 2 or n + 2 is prime, and p = Φ _n+2(1 ? d) where Φ _n+2 is the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.     

A sign test for detecting the equivalence of Sylvester Hadamard matrices

In this paper we show that every matrix in the class of Sylvester Hadamard matrices of order 2 ^k under H-equivalence can have full row and column sign spectrum, meaning that tabulating the numbers of sign interchanges along any row (or column) gives all integers 0,1,...,2 ^k  − 1 in some order. The construction and properties of Yates Hadamard matrices are presented and is established their equivalence with the Sylvester Hadamard matrices of the same order. Finally, is proved that every normalized Hadamard matrix has full column or row sign spectrum if and only if is H-equivalent to a Sylvester Hadamard matrix. This provides us with an efficient criterion identifying the equivalence of Sylvester Hadamard matrices.     

On the Description of the Content-Equality by a Relation

In the geometry of polyhedra we understand by an elementary content-functional a real valued, non-negative, finite additive measure on the set of polyhedra which is invariant under isometries. There are close relations between the content-measurement and the relation of equidecomposability. Two polyhedra are called equidecomposable if they are decomposed into pairwise congruent pieces. For an example we consider the set of all polygons in the euclidean plane. It is well known that planar polygons have the same area if and only if they are equidecomposable. In the three-dimensional euclidean space one also can describe the content-equality of polyhedra by a relation. Two polyhedra have the same volume if they are equidecomposable with respect to equiaffine mappings (see [3]). In [4] the concept of an invariant content of polyhedra in a topological Klein space is introduced. Each regular closed quasicompact set ot the space is called polyhedron. Under this supposition two polyhedra have equal contents if they are equivalent by decomposition. The relation “equivalent by decomposition” is closely related to the relation “equidecomposable”.     

A combinatorial theory of Grünbaum's new regular polyhedra,Part II: Complete enumeration

The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE ³ and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE ³, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE ⁿ is worked out in §9.     

Vertex-minimal simplicial immersions of the Klein bottle in three space

Although the Klein bottle cannot be embedded inR ³, it can be immersed there, and in more than one way. Smooth examples of these immersions have been studied extensively, but little is known about their simplicial versions. The vertices of a triangulation play a crucial role in understanding immersions, so it is reasonable to ask: How few vertices are required to immerse the Klein bottle inR ³? Several examples that use only nine vertices are given in Section 3, and since any triangulation of the Klein bottle must have at least eight vertices, the question becomes: Can the Klein bottle be immersed inR ³ using only eight vertices? In this paper, we show that, in fact, eight isnot enough, nine are required. The proof consists of three parts: first exhibiting examples of 9-vertex immersions; second determining all possible 8-vertex triangulations ofK ²; and third showing that none of these can be immersed inR ³.     

Chiral Polyhedra in Ordinary Space, II

A chiral polyhedron has a geometric symmetry group with two orbits on the flags, such that adjacent flags are in distinct orbits. Part I of the paper described the discrete chiral polyhedra in ordinary euclidean space E³ with finite skew faces and finite skew vertex-figures; they occur in infinite families and are of types {4,6}, {6,4} and {6,6}. Part II completes the enumeration of all discrete chiral polyhedra in E³. There exist several families of chiral polyhedra of types {∞,3} and {∞,4} with infinite, helical faces. In particular, there are no discrete chiral polyhedra with finite faces in addition to those described in Part I.     

Sphere Packings, II

An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition of R ³ into polyhedra. The polyhedra are divided into two classes. The first class of polyhedra, called quasi-regular tetrahedra, have density at most that of a regular tetrahedron. The polyhedra in the remaining class have density at most that of a regular octahedron (about 0.7209). Received April 24, 1995, and in revised form April 11, 1996.     

Completely classifying all vertex-transitive and edge-transitive polyhedra

Recently A. Dress completed the classification of the regular polyhedra in E ³ by adding one class to the enumeration given by Grünbaum on this subject. This classification is the only systematic study of a collection of polyhedra possessing special symmetries which uses the generalized definition of a polygon allowing for skew polygons as well as planar polygons in E ³. This study gives necessary conditions for polyhedra to be vertex-transitive and edge-transitive. These conditions are restrictive enough to make the task of completely enumerating such polyhedra realizable and efficient. Examples of this process are given, and an explanation of the basic process is discussed. These new polyhedra are appearing more frequently in applications of geometry, and this examination is a beginning of the classifications of polyhedra having special symmetries even though there are many other such classes which lack this scrutiny.     

Classifying foliations of 3-manifolds via branched surfaces

We use branched surfaces to define an equivalence relation on C¹ codimension one foliations of any closed orientable 3-manifold that are transverse to some fixed nonsingular flow. There is a discrete metric on the set of equivalence classes with the property that foliations that are sufficiently close (up to equivalence) share important topological properties.     

Dihedral Angles of Convex Polyhedra

Convex polyhedra in H ³ are not determined by (their combinatorics and) their edge lengths. Convex space-like polyhedra in the de Sitter space S ³ ₁ are determined neither by their dihedral angles nor by their edge lengths. The same holds of convex polyhedra in S ³ . Received November 16, 1998, and in revised form March 8, 1999.     

Caps Embedded in Grassmannians

This paper is concerned with constructing caps embedded in line Grassmannians. In particular, we construct a cap of size q³ +2q²+1 embedded in the Klein quadric of PG(5,q) for even q, and show that any cap maximally embedded in the Klein quadric which is larger than this one must have size equal to the theoretical upper bound, namely q³+2q²+q+2. It is not known if caps achieving this upper bound exist for even q > 2.