Extensions of distance preserving mappings in euclidean and hyperbolic geometry

Suppose that X is a real inner product space of (finite or infinite) dimension at least 2. A distance preserving mapping , where is a (finite or infinite) subset of a finite-dimensional subspace of X, can be extended to an isometry of X. This holds true for euclidean as well as for hyperbolic geometry. To both geometries there exist examples of non-extentable distance preserving , where S is not contained in a finite-dimensional subspace of X.     

Manifold structures on abstract regular polytopes

Summary Abstract regular polytopes are complexes which generalize the classical regular polytopes. This paper discusses the topology of abstract regular polytopes whose vertex-figures are spherical and whose facets are topologically distinct from balls. The case of toroidal facets is particularly interesting and was studied earlier by Coxeter, Shephard and Grünbaum. Ann-dimensional manifold is associated with many abstract (n + 1)-polytopes. This is decomposed inton-dimensional manifolds-with-boundary (such as solid tori). For some polytopes with few faces the topological type or certain topological invariants of these manifolds are determined. For 4-polytopes with toroidal facets the manifolds include the 3-sphereS ³, connected sums of handlesS ¹ × S ² , euclidean and spherical space forms, and other examples with non-trivial fundamental group.     

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.     

On the addition of convex sets in the hyperbolic plane

Analogue to the definition $K + L := \bigcup_{x\in K}(x + L)$ of the Minkowski addition in the euclidean geometry it is proposed to define the (noncommutative) addition $K \vdash L := \bigcup_{0\, \leqsl\, \rho\,\leqsl\, a(\varphi),0\,\leqsl\,\varphi\,<\, 2\pi}T_{\rho}^{(\varphi)}(L)$ for compact, convex and smoothly bounded sets K and L in the hyperbolic plane $\Omega$ (Kleins model). Here $\rho = a(\varphi)$ is the representation of the boundary $\partial$ K in geodesic polar coordinates and $T_{\rho}^{(\varphi)}$ is the hyperbolic translation of $\Omega$ of length $\rho$ along the line through the origin o of direction $\varphi$. In general this addition does not preserve convexity but nevertheless we may prove as main results: (1) $o \in$ int $K, o \in$ int L and K,L horocyclic convex imply the strict convexity of $K \vdash L$, and (2) in this case there exists a hyperbolic mixed volume $V_h(K,L)$ of K and L which has a representation by a suitable integral over the unit circle.     

A 10-point circle is associated with any general point of the ellipse. New properties of Fagnano’s point

Let H be an ellipse with semiaxes a and b (a > b). Two circles concentric with H, and with radii a − b and a + b, are described, each of them being the locus of the intersections between couples of noteworthy H-related lines (Theorems 1 and 2). Tight, as well as unexpected links among such circles and Monge’s circle are shown (Theorems 4, 5, and 6). A surprising pythagorean relationship involving segments related to the ellipse is shown (Theorem 3). A set of 10 concyclic points is associated with any general point of H (Theorem 9). New properties of Fagnano’s point are described (Theorems 10 through 13). Only elementary facts from trigonometry and analytic geometry are used.      

Sur le tétraèdre régulier dans un espace projectif complexe

Résumé On montre l'existence, dans un espace projectif complexe, d'un tétraèdre régulier ayant un groupe de symétrie isomorphe à celui du tétraèdre régulier euclidien. On précise la classe d'homologie de dimension 2, qui contient le complexe simplicial qui lui est associé.

---

We show the existence, in a complex projective space, of a regular tetrahedron which has its symmetry group isomorphic to that of the euclidean regular tetrahedron. We give precisely the two-dimensional homology class, containing the simplicial complexe associated to it.

     

The dimension theorem in axiomatic geometry

Given a matroid and an integer n 0, eleven conditions are shown to be equivalent to the validity of the rank formula r(E F) + r(E F = r(E) + r(F) for subspaces satisfying r(EF)n. For n=0 one finds the projective geometries. The case n=1 also includes the affine and the hyperbolic geometries, the case n=2 the Möbius geometries. The general case covers the incidence geometries of grade n of Wille.     

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.     

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.     

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.     

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.     

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).     

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.     

Realizations of regular apeirotopes

Summary In an earlier paper, a theory of realizations of (finite) regular polytopes in euclidean spaces was developed. Here, the analogous problem of realizing regular apeirotopes (infinite polytopes) is investigated. While no complete theory is expounded, several basic results are established. Among these is the curious fact that, if a regular apeirotope has a discrete realization, then it has one with no translations in its symmetry group.     

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.

---

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.

     

Constructions of rank five geometries for the Mathieu group M 22

We construct nine rank five incidence geometries that are firm and residually connected and on which the Mathieu group M²² acts flag-transitively. The constructions use mainly objects arising from the Steiner systemS(3, 6, 22). One of these geometries was constructed by Meixner and Pasini in [10]. Three of them are obtained from the geometry of Meixner and Pasini using doubling (see [8] or [12]) or similar constructions. The remaining five are new and four of them have a star diagram. These latter four geometries are constructed using special partitions of the 22 points of the Steiner system S(3, 6, 22).     

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.     

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.