Hjelmslev's geometry of reality

During the first half of the 20th century the Danish geometer Johannes Hjelmslev developed what he called a geometry of reality. It was presented as an alternative to the idealized Euclidean paradigm that had recently been completed by Hilbert. Hjelmslev argued that his geometry of reality was superior to the Euclidean geometry both didactically, scientifically and in practice: Didactically, because it was closer to experience and intuition, in practice because it was in accordance with the real geometrical drawing practice of the engineer, and scientifically because it was based on a smaller axiomatic basis than Hilbertian Euclidean geometry but still included the important theorems of ordinary geometry. In this paper, I shall primarily analyze the scientific aspect of Hjelmslev's new approach to geometry that gave rise to the so-called Hjelmslev (incidence) geometry or ring geometry.     

A characterization of subshifts with bounded powers

We consider minimal, aperiodic symbolic subshifts and show how to characterize the combinatorial property of bounded powers by means of a metric property. For this purpose we construct a family of graphs which all approximate the subshift space, and define a metric on each graph, which extends to a metric on the subshift space. The characterization of bounded powers is then given by the Lipschitz equivalence of a suitably defined infimum metric with the corresponding supremum metric. We also introduce zeta-functions and relate their abscissa of convergence to various exponents of complexity of the subshift. Our results, following a previous work of two of the authors, are based on constructions in non commutative geometry.     

Geometries of the Group PSL(2,11)

We determine all residually weakly primitive flag-transitive geometries for the groups PSL(2,11) and PGL(2,11). For the first of these we prove the existence by simple constructions while uniqueness, namely the fact that the lists are complete, relies on MAGMA programs. A central role is played by the subgroups Alt(5) in PSL(2,11). The highest rank of a geometry in our lists is four. Our work is related to various atlases of coset geometries.     

Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem

It may seem odd that Abel, a protagonist of Cauchy's new rigor, spoke of “exceptions” when he criticized Cauchy's theorem on the continuity of sums of continuous functions. However, when interpreted contextually, exceptions appear as both valid and viable entities in the early 19th century. First, Abel's use of the term “exception” and the role of the exception in his binomial paper is documented and analyzed. Second, it is suggested how Abel may have acquainted himself with the exception and his use of it in a process denoted critical revision is discussed. Finally, an interpretation of Abel's exception is given that identifies it as a representative example of a more general transition in the understanding of mathematical objects that took place during the period. With this interpretation, exceptions find their place in a fundamental transition during the early 19th century from a formal approach to analysis toward a more conceptual one.     

Explication as a lens for the formalization of mathematical theory through guided reinvention

Realistic Mathematics Education supports students’ formalization of their mathematical activity through guided reinvention. To operationalize “formalization” in a proof-oriented instructional context, I adapt Sjogren's (2010) claim that formal proof explicates (Carnap, 1950) informal proof. Explication means replacing unscientific or informal concepts with scientific ones. I use Carnap's criteria for successful explication – similarity, exactness, and fruitfulness – to demonstrate how the elements of mathematical theory – definitions, axioms, theorems, proofs – can each explicate their less formal correlates. This lens supports an express goal of the instructional project, which is to help students coordinate semantic (informal) and syntactic (formal) mathematical activity. I demonstrate the analytical value of the explication lens by applying it to examples of students’ mathematical activity drawn from a design experiment in undergraduate, neutral axiomatic geometry. I analyze the chains of meanings (Thompson, 2013) that emerged when formal elements were presented readymade alongside those emerging from guided reinvention.     

Historical origins of the nine-point conic. The contribution of Eugenio Beltrami

In this paper, we examine the evolution of a specific mathematical problem, i.e. the nine-point conic, a generalisation of the nine-point circle due to Steiner. We will follow this evolution from Steiner to the Neapolitan school (Trudi and Battaglini) and finally to the contribution of Beltrami that closed this journey, at least from a mathematical point of view (scholars of elementary geometry, in fact, will continue to resume the problem from the second half of the 19th to the beginning of the 20th century). We believe that such evolution may indicate the steady development of the mathematical methods from Euclidean metric to projective, and finally, with Beltrami, with the use of quadratic transformations. In this sense, the work of Beltrami appears similar to the recent (after the anticipations of Magnus and Steiner) results of Schiaparelli and Cremona. Moreover, Beltrami's methods are closely related to the study of birational transformations, which in the same period were becoming one of the main topics of algebraic geometry. Finally, our work emphasises the role played by the nine-point conic problem in the studies of young Beltrami who, under Cremona's guidance, was then developing his mathematical skills. To this end, we make considerable use of the unedited correspondence Beltrami – Cremona, preserved in the Istituto Mazziniano, Genoa.     

Analysis on the minimal representation of O(p,q) I. Realization via conformal geometry

This is the first in a series of papers devoted to an analogue of the metaplectic representation, namely the minimal unitary representation of an indefinite orthogonal group; this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of Kirillov–Kostant. We begin by applying methods from conformal geometry of pseudo-Riemannian manifolds to a general construction of an infinite-dimensional representation of the conformal group on the solution space of the Yamabe equation. By functoriality of the constructions, we obtain different models of the unitary representation, as well as giving new proofs of unitarity and irreducibility. The results in this paper play a basic role in the subsequent papers, where we give explicit branching formulae, and prove unitarization in the various models.     

Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany

Remainder problems have a long tradition and were widely disseminated in books on calculation, algebra, and recreational mathematics from the 13th century until the 18th century. Many singular solution methods for particular cases were known, but Bachet de Méziriac was the first to see how these methods connected with the Euclidean algorithm and with Diophantine analysis (1624). His general solution method contributed to the theory of equations in France, but went largely unnoticed elsewhere. Later Euler independently rediscovered similar methods, while von Clausberg generalized and systematized methods that used the greatest common divisor procedure. These were followed by Euler's and Lagrange's continued fraction solution methods and Hindenburg's combinatorial solution. Shortly afterwards, Gauss, in the Disquisitiones Arithmeticae, proposed a new formalism based on his method of congruences and created the modular arithmetic framework in which these problems are posed today.     

A proportional view: The mathematics of James Glenie (1750–1817)

The mathematical work of James Glenie (1750–1817) was published at irregular intervals during a turbulent life. His ideas, mostly deriving from his time as an Assistant in Mathematics at St Andrews University in Scotland, were developed intermittently over a period of thirty-seven years. His mathematical achievements, underestimated by previous historians, were deeply rooted in Euclidean geometry and his own generalized theory of proportion. Among them are many new geometrical constructions and proofs, a novel demonstration of the binomial theorem, and an alternative approach to the differential calculus.     

Uniform K-theory,and Poincaré duality for uniform K-homology

We revisit ?pakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology.We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with uniform K-homology and prove Poincaré duality between uniform K-theory and uniform K-homology on spin^c manifolds of bounded geometry.     

Another Constructive Axiomatization of Euclidean Planes

H. Tietze has proved algebraically that the geometry of uniquely determined ruler and compass constructions coincides with the geometry of ruler and set square constructions. We provide a new proof of this result via new universal axiom systems for Euclidean planes of characteristic ≠ 2 in languages containing only operation symbols.     

Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets

In this paper, we give a new lifting construction of “hyperbolic” type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to m-ovoids and i-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.     

Automatic deduction in (dynamic) geometry: Loci computation

A symbolic tool based on open source software that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction is presented. The prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction, namely with the open source dynamic geometry system GeoGebra or using the common file format for dynamic geometry developed by the Intergeo project. Locus computation algorithms based on Automatic Deduction techniques are recalled and presented as basic for an efficient treatment of advanced methods in dynamic geometry. Moreover, an algorithm to eliminate extraneous parts in symbolically computed loci is discussed. The algorithm, based on a recent work on the Gröbner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Several examples are shown in detail.     

Analoga der steinerschen konstruktion in einer absoluten geometrie

We consider an absolute geometry with the following base of axioms: Hilbert's plane axioms of incidence, order and congruence and a circle axiom. Thus no parallelism and not much continuity is involved. In this geometry the metric cannot be determined by Steiner's basic structure “fixed circle with centre”. In this work it will be proved that the following basic figures are suitable for such an absolute geometry in the sense that, after tracing any one of them, all constructions of second order can be done only with a ruler:

1. Two non-concentric circles, one of them with centre.
2. A unit-turner and a non-concentric circle without centre.
3. A circle with centreO and a line segmentA B with midpointM, the linesA B andO M being not orthogonal.
4. A circle with centre and two orthogonal lines, none of them passing through the centre.
5. A circle with centre and a distance-line (with their two branches).

In the basic structures 1, 3, 4, 5, instead of a circle with centre, a finite arc of a circle with centre or two concentric circles without centre may be taken.     

Noncommutative geometry and conformal geometry. III. Vafa–Witten inequality and Poincaré duality

This paper is the third part of a series of papers whose aim is to use the framework of twisted spectral triples to study conformal geometry from a noncommutative geometric viewpoint. In this paper we reformulate the inequality of Vafa–Witten [42] in the setting of twisted spectral triples. This involves a notion of Poincaré duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa–Witten, in the sense of an explicit control of the Vafa–Witten bound under conformal changes of metrics. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated with conformal weights on noncommutative tori, and spectral triples associated with duals of torsion-free discrete cocompact subgroups satisfying the Baum–Connes conjecture.     

Asymmetry in the Converse of Schur's Lemma

We show that the converse of Schur's Lemma can hold in the category of right modules, but not the category of left modules, over an appropriate ring. We exhibit classes of rings over which this left-right asymmetry does not occur, and provide new constructions of rings over whose module categories the converse of Schur's Lemma holds. We propose various open problems and avenues for further research concomitant to our work.     

Unpacking learner’s growth in geometric understanding when solving problems in a dynamic geometry environment: Coordinating two frames

The emergence of dynamic geometry environments challenges researchers in mathematics education to develop theories that capture learner’s growth in geometric understanding in this particular environment. This study coordinated the Pirie-Kieren theory and instrumental genesis to examine learner’s growth in geometric understanding when solving problems in a dynamic geometry environment. Data analysis suggested that coordinating the two theoretical approaches provided a productive means to capture the dynamic interaction between the growth in mathematical understanding and the formation/application of utilization scheme during a learner’s mathematical exploration with dynamic geometry software. The analysis of one episode on inscribing a square in a triangle was shared to illustrate this approach. This study contributes to the continuing conversation of “networking theories” in the mathematics education research community. By networking the two theoretical approaches, this paper presents a model for studying learner's growth in mathematical understanding in a dynamic learning environment while accounting for interaction with digital tools.     

Johann Heinrich Lambert,mathematician and scientist, 1728 – 1777

1977 is the two hundredth anniversary of the death of Johann Heinrich Lambert, a little known but nonetheless intriguing figure in 18th century science. In the general histories of science and mathematics Lambert's contributions are often described piecemeal, with each discovery and invention usually divorced both from the method by which he arrived at it and from the totality of his intellectual endeavour. To the student of optics he is remembered for his cosine law in photometry, to the astronomer for his work on comets, to the meteorologist for his design of a gut hygrometer, and to the mathematician for his work on non-Euclidean geometry and his demonstration of the irrationality of π and e. There is no doubt that each of these contributions had a definite importance of its own; but it is not the aim of the present article to enumerate in this way the high points of Lambert's scientific and mathematical work, rather to describe it for once as a unified whole, and to relate it to the contemporary intellectual outlook.     

On the acceptance of trigonometry in wasan: Evidence from a text of Aida Yasuaki

The Dutch introduced trigonometry to Japan in the middle of the 17th century, but the use of trigonometry was rarely seen until the 18th century, and its use was limited to practical purposes such as surveying and astronomy. It was rarely used to solve geometry problems currently called wasan, the reason for which is not yet fully explained. In this paper, I summarize a book by Aida Yasuaki (1747–1817) in which he strongly criticizes the use of trigonometry for wasan, and try to find the origins of his antipathy.