Rigid Ball-Polyhedra in Euclidean 3-Space

A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean $3$ -space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a standard ball-polyhedron if its vertex–edge–face structure is a lattice (with respect to containment). To each edge of a ball-polyhedron, one can assign an inner dihedral angle and say that the given ball-polyhedron is locally rigid with respect to its inner dihedral angles if the vertex–edge–face structure of the ball-polyhedron and its inner dihedral angles determine the ball-polyhedron up to congruence locally. The main result of this paper is a Cauchy-type rigidity theorem for ball-polyhedra stating that any simple and standard ball-polyhedron is locally rigid with respect to its inner dihedral angles.     

Extended Rectifying Curves in Minkowski 3-Space

In this paper, we consider nonhelical rectifying curves using the any orthonormal moving frame in Minkowski 3?space. Then, we give some relations between nonhelical rectifying curves and their Darboux vectors. Also, we demonstrate that the modified Darboux vectors of the curves are rectifying curves. So, this study is a general expression for the known rectifying curves. In the last section, we give some examples and draw their pictures.     

求过n个可重复定点的参数方程

本文给出了过n个定点的平面曲线参数方程与空间曲线参数方程的一般性方法.由于这n个定点可重复,从而所得的参数曲线可以是首尾相连的,也可以是局部有环的.并用Matlab软件给出了空间曲线图.     

A Note on Minimal Surfaces in Euclidean 3-Space

In this note, a construction of minimal surfaces in Euclidean 3-space is given. By using the product of Weierstrass data of two known minimal surfaces, one gets a new Weierstrass data and a corresponding minimal surface from the Weierstrass representation.     

Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space

The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. In particular, we prove that if C(n) denotes the largest number of touching pairs in a packing of n>1 congruent balls in Euclidean 3-space, then $0.695<\frac{6n-C(n)}{n^{\frac{2}{3}}}< \sqrt[3]{486}=7.862\dots$ for all $n=\frac{k(2k^{2}+1)}{3}$ with k??2.     

Ruled surfaces generated by some special curves in Euclidean 3-Space

In this paper, a family of ruled surfaces generated by some special curves using a Frenet frame of that curves in Euclidean 3-space is investigated. Some important results are obtained in the case of general helices as well as slant helices. Moreover, as an application, circular general helices, spherical general helices, Salkowski curves and circular slant helices, which illustrate the results, are provided and graphed.     

Fundamentals of Quaternionic Kinematics in Euclidean 4-Space

Based on a kinematic mapping for the group SE(4) of displacements of Euclidean 4-space, we show that the mapping of basic elements (points, oriented lines, oriented planes, oriented hyperplanes, instantaneous screws) can be written compactly in terms of 2 × 2 quaternionic matrices. Moreover we discuss the kinematics on the velocity level by investigating instantaneous screws and their geometric parameters.     

A Polyhedral Model in Euclidean 3-Space of the Six-Pentagon Map of the Projective Plane

In a private communication, Branko Grünbaum asked: “I wonder whether you know anything about the possibility of realizing as a polyhedron in Euclidean 3-space the family of six pentagons, that is a model of the projective plane arising by identifying antipodal points of the regular dodecahedron. Naturally, any realization must have some self-intersections—but is there any realization that is not completely contained in a plane?”We show that it is possible to realize this polyhedron; in our realization five of the six faces are simple polygons. In this model there are sets of three faces, which form a realization of the Möbius strip without self-intersections. There are four variants of the model. We conjecture that in any model of this polyhedron there must be at least one self-intersecting face.     

Spinor Frenet Equations in Three Dimensional Lie Groups

In this paper, we study spinor Frenet equations in three dimensional Lie groups with a bi-invariant metric. Also, we obtain spinor Frenet equations for some special cases of three dimensional Lie groups.     

Almost Disjoint Triangles in 3-Space

Abstract. Two triangles are called almost disjoint if they are either disjoint or their intersection consists of one common vertex. Let f(n) denote the maximum number of pairwise almost disjoint triangles that can be found on some vertex set of n points in 3-space. Here we prove that f(n)=Ω(n ^3/2 ) .     

Coincidence Site Modules in 3-Space

The coincidence site lattice (CSL) problem and its generalization to ℤ-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class number 1 over real algebraic number fields.     

Almost Disjoint Triangles in 3-Space

   Abstract. Two triangles are called almost disjoint if they are either disjoint or their intersection consists of one common vertex. Let f(n) denote the maximum number of pairwise almost disjoint triangles that can be found on some vertex set of n points in 3-space. Here we prove that f(n)=Ω(n ^3/2 ) .     

Symmetries of Real Hypersurfaces in Complex 3-Space

The main result of the paper consists in the proof of the fact that for any germ of a real analytic hypersurface in complex 3-space the following alternative (dimension conjecture) takes place: either the dimension of the group of holomorphic symmetries of the germ is at most the dimension of that of a nondegenerate hyperquadric (the latter equals 15), or the group is infinite-dimensional. We also discuss mistakes found in A. Ershova’s paper.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 171–179.Original Russian Text Copyright © 2005 by V. K. Beloshapka.     

Linkless and Flat Embeddings in 3-Space

We consider piecewise linear embeddings of graphs in 3-space ℝ³. Such an embedding is linkless if every pair of disjoint cycles forms a trivial link (in the sense of knot theory). Robertson, Seymour and Thomas (J. Comb. Theory, Ser. B 64:185–227, 1995) showed that a graph has a linkless embedding in ℝ³ if and only if it does not contain as a minor any of seven graphs in Petersen’s family (graphs obtained from K ₆ by a series of YΔ and ΔY operations). They also showed that a graph is linklessly embeddable in ℝ³ if and only if it admits a flat embedding into ℝ³, i.e. an embedding such that for every cycle C of G there exists a closed 2-disk D⊆ℝ³ with D∩G=∂D=C. Clearly, every flat embedding is linkless, but the converse is not true. We consider the following algorithmic problem associated with embeddings in ℝ³:     

Boundaries of Flat Compact Surfaces in 3-Space

This paper deals with the problem `which knots or links in3-space bound flat (immersed) compact surfaces?' In aforthcoming paper by the author, it is proven that any simple closedspace curve can be deformed until it bounds a flat orientable compact(Seifert) surface. The main results of this paper are that there existknots that do not bound any flat compact surfaces. The lower bound oftotal curvature of a knot bounding an orientable nonnegatively curvedcompact surface can, for varying knot types, be arbitrarily much greaterthan the infimum of curvature needed for the knot to have its knot type.The number of 3-singular points (points of zero curvatureor if not then of zero torsion) on the boundary of a flat immersedcompact surface is greater than or equal to twice the absolute value ofthe Euler characteristic of the surface. A set of necessary and, in aweakened sense, sufficient conditions for a knot or link to be what wecall a generic boundary of a flat immersed compact surface withoutplanar regions is given.     

The Sub-Parabolic Lines in the Minkowski 3-Space

Results in Mathematics - In this paper, we investigate the focal surfaces obtained by the normal rectilinear congruence in the Minkowski 3-space. The sub-parabolic points and the ridge points are...     

A New Packing Density Bound in 3-Space

It is shown that every compact convex set K which is centrally symmetric and has a non-empty interior admits a lattice packing of Euclidean 3-space with density greater than or equal to 0.53835.... This is an improvement of the result in [8], which achieved a bound of 0.46421.... Minkowski combinations and the Brunn-Minkowski inequality are used in conjunction with the construction in [8] to achieve a better result.     

On Helicoidal Surfaces in a Conformally Flat 3-Space

In this paper, we discuss the problem of finding explicit parametrizations for the helicoidal surfaces in a conformally flat 3-space \(\mathbb {E}^3_F\) with prescribed extrinsic curvature or mean curvature given by smooth functions. Also, we give examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in \(\mathbb {E}^3_F\).     

Effects of Quadratic Singular Curves in Integrable Equations

In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo‐cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first‐order derivatives of the new singular solitary wave and periodic waves exist, their second‐order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo‐cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo‐cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.