A geometric inequality for a finite set of points on a sphere and its applications

In this paper we present a geometric inequality for a finite number of points on an (n–1)-dimensional sphere S _n–1(R). As an application, we obtain a geometric inequality for finitely many points in the n-dimensional Euclidean space E ⁿ.     

Three geometric inequalities for a simplex

In this paper, we obtain three new geometric inequalities for ann-dimensional simplex in then-dimensional Euclidean spaceE ⁿ . As special cases we find two known inequalities from L. Fejes Tóth and M. S. Klamkin, respectively.     

Some inequalities on areas of bisection planes of dihedral angles of a simplex

In this paper we prove some inequalities on areas of bisection planes of dihedral angles of a simplex in E ⁿ.     

Analogues of the nine-point circle for orthocentric n-simplexes

This article presents a generalization of some theorems connected with the nine-point circle for n-dimensional Euclidean space.     

Angles between Euclidean subspaces

The angle between two subspaces of dimensions p and q in a Euclidean space is considered by using exterior algebra. Some properties of angles are obtained. The relation between such a higher dimensional angle and the usual principal angles is also given. And finally, an application to Grassmann manifolds is briefly discussed.Supported by the National Natural Science Fund of P.R. China.     

The isoperimetric point and the point(s) of equal detour in a triangle

A point P in the plane of triangle ABC is said to be an isoperimetric point if PA + PB + AB = PB + PC + BC = PC + PA + CA, and is said to be a point of equal detour if PA + PB − AB = PB + PC − BC = PC + PA − CA. Incorrect conditions for the existence and uniqueness of such points were given by G. R. Veldkamp in Amer. Math. Monthly 92 (1985) 546-558. In this paper, we use a much simpler approach that yields correct versions of these conditions and that exhibits the relations of these points to the centers of the Soddy circles. Mowaffaq Hajja: This work is supported by a research grant from Yarmouk University.     

Classification of disordered tilings

A tilingT is a disordered realization of a periodic tilingP with symmetry group Γ if we can map the complement of a compact set ofT onto the quotientP/Γ in such a way that this map respects the features of the tilingT andP. We show that the global type of a 2-dimensional tilingT is determined by the periodic tilingP it is a disordered realization of, a conjugacy class of Γ which can be associated toT and a winding number. In some cases, we need in addition some kind of orientation. For higher-dimensional tilings of spaces which are simply connected at infinity, e.g. ℝ ⁿ withn≥3, the associated periodic tiling alone is sufficient.     

The perpendicular bisector construction

LetQ ₁,Q ₂,Q ₃ be plane quadrilaterals such that the sides ofQ _i are the perpendicular bisectors of the sides ofQ _i–1 (i=2, 3). This note gives a simple trigonometrical solution of the long outstanding problem of showing thatQ ₁ andQ ₃ are similar.     

An extension of the Erdős-Szekeres theorem on large angles

The existence of a functionn(ε) (ε>0) is established such that given a finite setV in the plane there exists a subsetW⊆V, |W|<n(ε) with the property that for anyv εV\ W there are two pointsw ₁,w ₂ εW such that the angle ∢(w ₁ vw ₂)>π-ε.     

Feuerbach's relation and Ptolemy's theorem in R n

Ptolemy's equality for four points on a circle is related to a Feuerbach-type area relation. This suggested an extension of Ptolemy's inequality to a Feuerbach type volume relation between simplexes formed from n+2 points in R ⁿ (n2). Extensions of the Möbius-Neuberg and Pompeiu Theorems in R ² are given for R ⁿ . Ptolemy's inequality is also extended to convex n-gons in R ² yielding an extension of Fuhrmann's hexagon theorem to 2n-gons in R ² (n3).     

On hyperbolic right-angled polygons

In this paper we find conditions in order to construct hyperbolic right-angledN-gons with the lengths ofN-3 sides given.Explicit formulae for the length of a side in terms of the lengths ofN-3 non-adjacent sides are obtained.Partially supported by CICYT.     

Sur les m-uples F-réguliers dans les espaces projectifs complexes

Résumé On étudie dans P ⁿ les m-uples de points, appelés F-réguliers, dont les sous-triplets ordonnés sont deux à deux isométriques. On montre qu'il existe au plus deux classes d'isométrie de quintuplets F-réguliers contenant un triangle équilatère T donné. On étudie aussi les m-uples F-réguliers, dont les sous k-uples (k<m) non ordonnés sont deux à deux isométriques. Ces m-uples sont appelés k-réguliers. On montre que la 4-régularité implique la k-régularité pour tous les k5.

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We investigate in P ⁿ m-tuples of points in which all ordered triples are pairwise isometric. Such m-tuples are called F-regular. We show that for a given triangle T there exist at most two isometry classes of F-regular quintuples containing T. We also investigate F-regular m-tuples in which all (unordered) k-tuples (k<m) are pairwise isometric. Such m-tuples are called k-regular. We show that 4-regularity implies k-regularity for all k5.

     

A characterization of the Corrado Segre variety

In this paper we give a combinatorial characterization of the Corrado Segre variety of type {n,m} in terms of its incidence structure of points and lines.     

Metric and Periodic Lines in Real Inner Product Space Geometries

The lines of euclidean and hyperbolic geometries are characterized as metric lines in the sense of Blumenthal–Menger, and the lines of spherical and elliptic geometry as periodic ones. In euclidean geometries there do not exist periodic lines.     

The generalized tilt formula

A convex hull construction in Minkowski space defines a canonical cell decomposition for a cusped hyperbolicn-manifold. An algorithm to compute the canonical cell decomposition uses the concept of the tilt of ann-simplex relative to each of its (n–1)-dimensional faces. An essential tool for computing tilts is the tilt theorem. The tilt theorem was previously known only in dimensionsn3, and the proof was needlessly complicated. Here we offer a new, simplified proof which applies in all dimensions. We also offer a second geometric interpretation of the tilt.     

Geraden-Abbildungen,die den Flächeninhalt 1 von Dreiecken erhalten

Zusammenfassung und Vorwort Zu jeder Geometrie, in der Geraden definiert sind, gehört eine Geradengeometrie, in der interessante Fragestellungen auftauchen. Die klassische Frage in der Liniengeometrie Plückers lautet: Gegeben sei ein dreidimensionaler projektiver Raum. Wie lauten diejenigen bijektiven Geradenabbildungen, für die sich zwei Geraden genau dann schneiden, wenn sich deren Bilder schneiden? Eine solche Abbildung ist entweder induziert durch eine Kollineation des projektiven Raumes oder eine Dualität. W. L. Chow und H. Brauner haben in [5] und [4] diese Fragestellung verallgemeinert. W. Benz hat in [1] die klassische Frage Plückers im ⁿ fürn 3 beantwortet. J. Lester sucht in [6] diejenigen Geradenabbildungen des dreidimensionalen euklidischen Raumes, die den Abstand 1 von Geraden in beiden Richtungen erhalten.In meiner Arbeit bestimme ich diejenigen Geradenabbildungen desn-dimensionalen reellen euklidischen Raumes, die den Inhalt 1 von Dreiecken unverändert lassen (n 2). Dieser Frage hat sich auch schon Frau Lester gewidmet ([8]), aber unter der zusätzlichen, starken Voraussetzung, daß bijektiv sein soll. Ich werde zeigen, daß diese Forderung unnötig ist; der Nachweis der Surjektivität beinhaltet die eigentliche Schwierigkeit.

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On the products of cross-ratios on diagonals of polygons

The purpose of this paper is to establish a conjecture of B. Grünbaum, which states that in every n-polygon P in the plane, n 5, some diagonals intersect in a pattern that defines a new n-polygon (P), such that the product of the cross-rations on the diagonals of P is equal to the product of the corresponding cross-ratios on the diagonals of (P).     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

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.     

Lattice-point examples for a question of Erdős

Here we establish a set of eight points in general position in the plane, i.e. no three on a line, no four on a circle, and they determine 7 distinct distances, so that, thei-th distance occursi times,i = 1, 2, , 7. The points are embedded in a triangular net, and the distances are not ordered by size or in any other way. We shall show that some known and unknown examples forn < 8 with the above properties may also be lattice points of a similar net.Research (partially) supported by the Hungarian National Foundation for Scientific Research (OTKA) grant, no. 1808.