The calculus of reflections and the order relation in hyperbolic geometry

It is well known that the calculus of reflections (developed by Hjelmslev, Bachmann et?al.) enables the derivation of a large part of Euclidean and non-Euclidean geometry without using assumptions about order and continuity. We show in this article that the calculus of reflections can conversely be used to introduce a relation of order in hyperbolic geometry. Our investigations are based on the famous ??Endenrechnung?? of Hilbert which was formulated purely in terms of the calculus of reflections by F. Bachmann. We then discuss some implications of these results and show that the calculus of reflections enables (1) the introduction of an order relation in a Pappian projective line and (2) to define an axiom system for hyperbolic planes which seems to be as simple as the famous axiom system of Menger who only used the notion of point-line incidence to axiomatize plane hyperbolic geometry.     

Ferdinand Douwe Veldkamp: 5 September 1931–3 August 1999

On 3 August 1999, Ferdinand Douwe Veldkamp, one of the managing editors of Geometriae Dedicata since 1981 passed away. Professor Veldkamp was a passionate mathematician. His area of research comprised large parts of algebra and geometry; it extended from algebraic groups and Lie algebras to ring geometries and their axiomatizations. Veldkamp's pioneering work on polar spaces has become a lasting part of the theory of buildings. From octonian geometries, especially Moufang Hjelmslev planes, as a starting point he opened completely new vistas into the geometry of rings. This becomes evident if one observes that the most general ring geometries which can be reasonably treated are today called Veldkamp planes.     

几何代数基础新视角下的初步探讨

提出有关几何代数基础的一个问题:在给定了变换群的几何上,可能建立哪些代数结构?首先证明,不可能在欧氏平面上的点之间定义一种在保距变换下不变的运算,使之在此运算下形成阿贝尔群.进一步的讨论证明,只有将欧氏几何扩大为质点几何,才能在其上建立在保距变换下不变的可交换可结合的运算,而且这种运算只能是质点几何中的加法.如果希望在此运算下构成阿贝尔群,就必须引入向量.最后讨论了所获结果的意义,并提出若干问题.     

Validation of Solutions of Construction Problems in Dynamic Geometry Environments

This paper discusses issues concerning the validation of solutions of construction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and compass in classic Euclidean geometry. We then demonstrate that the drag test criterion may permit constructions created using measurement tools to be considered valid; however, these constructions prove inconsistent with classical geometry. This inconsistency raises the question of whether dragging is an adequate test of validity, and the issue of measurement versus straightedge-and-compass. Without claiming that the inconsistency between what counts as valid solution of a construction problem in the two geometry worlds is necessarily problematic, we examine what would constitute the analogue of the straightedge-and-compass criterion in the domain of DGEs. Discovery of this analogue would enrich our understanding of DGEs with a mathematical idea that has been the distinguishing feature of Euclidean geometry since its genesis. To advance our goal, we introduce the compatibility criterion, a new but not necessarily superior criterion to the drag test criterion of validation of solutions of construction problems in DGEs. The discussion of the two criteria anatomizes the complexity characteristic of the relationship between DGEs and the paper-and-pencil Euclidean geometry environment, advances our understanding of the notion of geometrical constructions in DGEs, and raises the issue of validation practice maintaining the pace of ever-changing software.     

Möbius hypersurfaces in S S+1with three distinct principal curvatures

On a hypersurface of a unit sphere without umbilical points, we know that three Möbius invariants can be defined and analogous to Euclidean case, we have the concepts of Möbius isoparametric and isotropic hypersurfaces. In this paper, we study the relationship between Euclidean geometry and Möbius geometry, and prove that a hypersurface in a sphere with constant length of the second fundamental form is Euclidean isoparametric if and only if it is Möbius isoparametric. When restricting to the case of three distinct principal curvatures, we show that such a hypersurface is either Möbius isoparametric or isotropic if the Blaschke tensor has constant eigenvalues.     

Conformal infinitesimal variations of submanifolds

This paper belongs to the realm of conformal geometry and deals with Euclidean submanifolds that admit smooth variations that are infinitesimally conformal. Conformal variations of Euclidean submanifolds are a classical subject in differential geometry. In fact, already in 1917 Cartan classified parametrically the Euclidean hypersurfaces that admit nontrivial conformal variations. Our first main result is a Fundamental theorem for conformal infinitesimal variations. The second is a rigidity theorem for Euclidean submanifolds that lie in low codimension.     

Regular Hjelmslev planes

A projective Hjelmslev plane is called regular iff it admits an Abelian collineation group that is regular on both the points and lines of the plane and that splits into a summand regular on the elements of any given neighborhood and another summand permuting the points and lines of the projective image plane regularly. Regular Hjelmslev planes are shown to correspond to so-called special difference sets. We construct regular Hjelmslev planes with parameters (qⁿ, q) for any prime power q and any natural number n as well as for infinitely many series of parameters (t, q), where t is not a power of q. Our construction also yields series of parameters for which the existence of a Hjelmslev plane was not known up to now as well as the first information on the existence of nontrivial collineations in the case of parameters (t, q) with t not a power of q.     

Order and topology in projective Hjelmslev planes

The concept of an ordered projective Hjelmslev plane was intuitively introduced by Hjelmslev in Einleitung in die allgemeine Kongruenglehre ([9], [10]).This paper is concerned with formalizing and examing preorderings and orderings for projective Hjelmslev planes. In addition we show that orderings generated topologies of the point and line sets which render the plane a topological Hjelmslev plane ([19], [13]). These planes — unlike the ordinary ordered planes ([18]) — are, due to the existence of infinitesimals, non-archimedian, non-compact and disconnected with the neighbour classes as certain quasi-components.The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.     

Constant mean curvature tori with spherical curvature lines in noneuclidean geometry

A previous result in Euclidean geometry [7] on H-tori with plane and spherical curvature lines is extended here to the two noneuclidean geometries. There result families of H-tori with only spherical curvature lines, which are explicitly representable by elliptic and theta functions (or ordinary integrals of elementary functions). Among the geometric properties, it is shown that the midpoints of the generating spheres vary on geodesics. The hyperbolic case is more similar to the Euclidean situation than the elliptic one. In elliptic geometry the constructed surfaces depend on one additional rational parameter and, as a limiting case, there are even countably many minimal tori of this type.     

Torsten Brodén's work on the foundations of Euclidean geometry

The Swedish mathematician Torsten Brodén (1857–1931) wrote two articles on the foundations of Euclidean geometry. The first was published in 1890, almost a decade before Hilbert's first attempt, and the second was published in 1912. Brodén's philosophical view of the nature of geometry is discussed and his thoughts on axiomatic systems are described. His axiomatic system for Euclidean geometry from 1890 is considered in detail and compared with his later work on the foundations of geometry. The two continuity axioms given are compared to and proved to imply Hilbert's two continuity axioms of 1903.     

An Introduction to Constant Curvature Spaces in the Commutative (Segre) Quaternion Geometry

It is known that complex numbers can be associated with plane Euclidean geometry and their functions are successfully used for studying extensions of Euclidean geometry, i.e., non-Euclidean geometries and surfaces differential geometry. In this paper we begin to study the constant curvature spaces associated with the geometry generated by commutative elliptic-quaternions and we show how the “mathematics” they generate allows us to introduce these spaces and obtain the geodesic equations without developing a complete mathematical apparatus as the one developed for Riemannian geometry.     

Dynamic hyperbolic geometry: building intuition and understanding mediated by a Euclidean model

This paper explores a deep transformation in mathematical epistemology and its consequences for teaching and learning. With the advent of non-Euclidean geometries, direct, iconic correspondences between physical space and the deductive structures of mathematical inquiry were broken. For non-Euclidean ideas even to become thinkable the mathematical community needed to accumulate over twenty centuries of reflection and effort: a precious instance of distributed intelligence at the cultural level. In geometry education after this crisis, relations between intuitions and geometrical reasoning must be established philosophically, rather than taken for granted. One approach seeks intuitive supports only for Euclidean explorations, viewing non-Euclidean inquiry as fundamentally non-intuitive in nature. We argue for moving beyond such an impoverished approach, using dynamic geometry environments to develop new intuitions even in the extremely challenging setting of hyperbolic geometry. Our efforts reverse the typical direction, using formal structures as a source for a new family of intuitions that emerge from exploring a digital model of hyperbolic geometry. This digital model is elaborated within a Euclidean dynamic geometry environment, enabling a conceptual dance that re-configures Euclidean knowledge as a support for building intuitions in hyperbolic space—intuitions based not directly on physical experience but on analogies extending Euclidean concepts.     

Some new invariant pairs (t,3) for projective hjelmslev planes

Associated with every finite projective Hjelmslev plane is an invariant pair (t,r): t is the number of neighbours of a given point on a given line passing through it and r is the order of the underlying projective plane. The Drake-Lenz method [2],[3] of using auxiliary matrices for the constructions of projective Hjelmslev planes has become standard by now. This paper is intended to give some new constructions of projective Hjelmslev planes with invariant pairs (t,3) by making use of the generalization and improvement of the Drake-Lenz theorem [3] obtained by the author in [6] and [7]. The results of this paper add 8 new values to the list ([5], example 3.7(ii)) of invariant pairs (t,3) with t 1,000 for projective Hjelmslev planes.     

Characterizing student mathematics teachers’ levels of understanding in spherical geometry

This article presents an exploratory study aimed at the identification of students’ levels of understanding in spherical geometry as van Hiele did for Euclidean geometry. To do this, we developed and implemented a spherical geometry course for student mathematics teachers. Six structured, task-based interviews were held with eight student mathematics teachers at particular times through the course to determine the spherical geometry learning levels. After identifying the properties of spherical geometry levels, we developed Understandings in Spherical Geometry Test to test whether or not the levels form hierarchy, and 58 student mathematics teachers took the test. The outcomes seemed to support our theoretical perspective that there are some understanding levels in spherical geometry that progress through a hierarchical order as van Hiele levels in Euclidean geometry.     

基于IOWA算子的欧式范数综合评价方法

针对综合评价信息不完整、分布不均匀以及现实中人们总是主观性地经常“向后看”这一问题,提出了基于区间数有序加权平均算子(IOWA算子)的欧式范数综合评价方法。本文首先介绍了IOWA算子的相关知识;然后依据IOWA算子的特点,运用正态分布确定其位置加权向量,并与欧式范数结合形成加权欧式范数;最后运用一个算例验证了方法的有效性,既能充分考虑评价信息的分布情况,又使得评价更加客观准确。     

Guided discovery of the nine-point circle theorem and its proof

The nine-point circle theorem is one of the most beautiful and surprising theorems in Euclidean geometry. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. This paper describes a set of instructional activities that can help students discover the nine-point circle theorem through investigation in a dynamic geometry environment, and consequently prove it using a method of guided discovery. The paper concludes with a variety of suggestions for the ways in which the whole set of activities can be implemented in geometry classrooms.     

From the Kneser–Poulsen conjecture to ball-polyhedra

A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser [M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass, Arch. Math. 6 (1955) 382–390] and Poulsen [E.T. Poulsen, Problem 10, Math. Scand. 2 (1954) 346]. According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that the distance between the centers of every pair of balls is decreased, then the volume of the union (resp., intersection) of the balls is decreased (resp., increased). In the first half of this paper we survey the state of the art of the Kneser–Poulsen conjecture in Euclidean, spherical as well as hyperbolic spaces with the emphases being on the Euclidean case. Based on that it seems very natural and important to study the geometry of intersections of finitely many congruent balls from the viewpoint of discrete geometry in Euclidean space. We call these sets ball-polyhedra. In the second half of this paper we survey a selection of fundamental results known on ball-polyhedra. Besides the obvious survey character of this paper we want to emphasize our definite intention to raise quite a number of open problems to motivate further research.