Sphere In En Research Articles

Dual characterizations of the sphere and the hyperbolic space in arbitrary dimension

There are several well-known characterizations of the sphere as a regular surface in the Euclidean space. On the contrary, there are not so many characterizations of the hyperbolic space, the spacelike sphere in the Minkowski space. By means of a purely synthetic technique, we get a rigidity result for the sphere in 𝔼n+1 without any curvature conditions, nor completeness or compactness, as well as a dual result for the n-dimensional hyperbolic space in 𝕃n+1.

A new class of infinite sphere packings

The oscillatory or Apollonian packing in two dimensions is well known and is described for example in [13]. Recently we investigated the three dimensional osculatory packing of a sphere [4] However, the results of that paper indicate that, for N > 3, ΛΓ-dimensional osculatory packings are irregular and not invariant under inversion as is the case for N = 2 and 3. In this paper we introduce a class of packings which we call discrete packings, and produce some examples. This class is analogous to the class of lattice packings which appear in the theory of packings of equal spheres. We shall use the systems of polyspherical coordinates developed in [4]. Section 2 contains a description of these as well as the proofs of some additional results needed here. The idea of the separation Δ{X, Y) between two spheres X and Y will again play an important role. In § 3 we consider inversively generated configurations obviously generalizing the construction used in [4]. That is, we begin with a 'cluster' of (N + 2) disjoint spheres and by successive inversions replace the spheres one at a time with new spheres in such a way that the separations between the spheres in the new cluster are the same as for the initial cluster. In terms of polyspherical coordinates the necessary inversions are represented by matrices which preserve a certain indefinite quadratic form. Repetition of the process leads to a configuration of spheres in EN which may or may not be a packing, depending on the initial cluster. In § 4 we give sufficient conditions under which an inversively generated configuration is a packing. The conditions force the separations between the spheres in the configuration to lie in a discrete subset of the rational numbers, hence the name 'discrete packing'. In addition to the two and three dimensional osculatory packings, we give examples of discrete packings for dimensions 2, 3, 4, 5, and 9. We do not know yet whether such packings exist in all dimensions. The examples we have found are given in § 6. The packings described in § 4 are not in general osculatory; that is, the largest possible sphere is not generated at each step. However,

Characterizations of Steiner points

To each convex compact A in Euclidian space En there corresponds a point S (A) from En such that 1) S(x) = x for x ∈ En, 2) S(A + B) = S(A) + S(B), 3) S (Ai) →θ, if Ai converges in the Hausdorff metric to the unit sphere in En, then S(A) is the Steiner point of the set A. The theorem improves certain earlier results on characterizations of the Steiner point.

Tame polyhedra in wild cells and spheres

It is shown that each arc on a disk D in E 4 {E^4} can be homeomorphically approximated by an arc in D which is tame in E 4 {E^4} . Some applications of this are given. Also, we construct an everywhere wild ( n − 1 ) (n - 1) -sphere in E n , n ≧ 3 {E^n},n \geqq 3 , each of whose arcs is tame in E n {E^n} .

Free curves inE3

A 2-sphere in Euclidean 3-dimensional space Ez is called free if it can be pushed into either complementary domain by a map moving no point more than e, for arbitrary e. Such 2-spheres have been the object of much recent attention, although the basic problem of whether they must be tame or not remains unsolved. The purpose of this paper is to take a different direction in this study. We introduce a natural generalization of the term free so that it can be used to describe a A -sphere in En, then direct our attention to free 1-spheres and 2-spheres in E3. Our main tool is Theorem 1, which, roughly speaking, should be viewed as follows: It is well known that if D and E are both polyhedral disks in Ez intersecting only in their interiors (in general position), then E may be altered via a disk replacement process to miss D. Theorem 1 states that even if D were a singular disk in E3, this process would remain valid to an extent.

A note on sets of constant width

1. Let En be the n-dimensional Euclidean space and let K be a bounded closed convex set in En. For any direction u there are exactly two supporting hyperplanes of K, orthogonal to u, and the distance D,(K) between them is called the width of K in the direction u. By a hyperplane we always mean one of dimension n -1. K is called a set of constant width 1, or briefly of width 1, if SD(K)=1 for all u. A sphere S,(x) denotes the closed solid sphere in En of radius r about the center x. S,(x) is the circum-sphere (the in-sphere) of K if KCS,(x)(Sr(x)CK) and the radius r is smallest (largest) possible. We shall prove the following