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This essay explores the research practice of French geometer Michel Chasles (1793–1880), from his 1837 Aperçu historique up to the preparation of his courses on ‘higher geometry’ between 1846 and 1852. It argues that this scientific pursuit was jointly carried out on a historiographical and a mathematical terrain. Epistemic techniques such as the archival search for and comparison of manuscripts, the deconstructive historiography of past geometrical methods, and the epistemologically motivated periodization of the history of mathematics are shown to have played a crucial role in the shaping of Chasles's own theories. In particular, we present Chasles's approach to the ‘material history’ of algebraic symbolism and argue that it motivated and informed his subsequent invention of a novel notational technology for the writing of geometrical proofs and propositions. In return, this technology allowed Chasles to carry out a programme for the modernization of geometry in keeping with epistemic requirements he had also delineated via a form of historical writing.

A formula for the magnetostatic energy of a finite magnet is proven. In contrast to common approaches, the new energy identity does not rely on evaluation of a nonlocal boundary integral inside the magnet or the solution of an equivalent Dirichlet problem. The formula is therefore computationally efficient, which is also shown numerically. Algorithms for the simulation of magnetic materials could benefit from incorporating the presented representation of the energy. In addition, a natural analogue for the energy via the magnetic induction is given. Proofs are carried out within a setting which is suitable for common discretizations in computational micromagnetics.

Pictures and diagrams help high school geometry students develop reasoning and proof-writing skills.

The mathematical literature knows several quite different ways to determine the area of a parabolic segment. In this article they are brought together, compared and varied. Finally two new proofs are added to the collection. Each proof is displayed by an expressive figure. A colorful compilation of those figures accompanies the article as an appendix.

This is an expository article detailing results concerning large arcs in finite projective spaces. It is not strictly a survey but attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known old and more recent theorems. The article is mostly self-contained and includes a proof of the most general form of Segre’s lemma of tangents and a short proof of the MDS conjecture over prime fields based on this lemma.

Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs are given under the headings: standard proof, elegant proof, and the proof without words. The solutions were obtained through a combination of mathematical tools and by dynamic investigation of the geometrical properties.

Two results, proved synthetically in plane Euclidean geometry by Chisini in 1924—stating that: (i) if MAB is an isosceles triangle, with \({MA\equiv MB}\), inscribed in a circle \({\mathcal{C}}\), P1 and P2 are two points on \({{\mathcal C}}\) such that {B, Pi} separates {A, M} for \({i \in \{1,2}\}\), and {B, P2} separates {M, P1}, then \({AP_1+BP_1 < AP_2+BP_2}\), and (ii) of all triangles inscribed in a given circle the equilateral triangle has the greatest perimeter—are proved inside Hilbert’s absolute geometry.

This paper reports the results of an international comparative study on the nature of proof to be taught in geometry. Proofs in French and Japanese lower secondary schools were explored by analyzing curricular documents: mathematics textbooks and national curricula. Analyses on the three aspects of proof—statement, proof, and theory—suggested by the notion of Mathematical Theorem showed differences in these aspects and also differences in the three functions of proof—justification, systematization, and communication—that are seemingly commonly performed in these countries. The results of analyses imply two major elements that form the nature of proof: (a) the nature of the geometrical theory that is chosen to teach and (b) the principal function of proof related to that theory. This paper suggests alternative approaches to teach proof and proving and shows that these approaches are deeply related to the way geometry is taught.

In the chapter Geometry and trigonometry, the majority of problems can be treated either by classical proof techniques and properties provided in the regular school program or by using some additional proof techniques and theorems that are considered to be outside of the scope of the regular school program. The classical proof techniques, theorems, properties or mathematical laws used in the proofs included in this chapter are the properties of congruence of triangles, properties of similarities of triangles, the Pythagorean theorem, the properties of the right-angled triangle, law of sines and law of cosines, the properties of an area, the properties of inscribed and circumscribed quadrilaterals, trigonometric methods, the method of coordinates and applications of geometric inequalities (for more applications, see Sedrakyan and Sedrakyan, Geometric Inequalities, 2017, [1]). A large number of problems are related with the maxima and minima properties, such problems can be treated either by the different methods of estimations or by transformations and simplifications of the trigonometric expressions. In some cases, when we need to find the sum of the given trigonometric expressions, the main trick is to represent each of the summands as the difference of two expressions and then perform the corresponding simplifications in order to find the required sum. For the readers convenience, we list and provide the formulations of all used theorems and proof techniques that are outside of the scope of the regular school program.

Pursuing abstractness and generality, a number of books and articles in econometrics rely heavily on standard algebraic proofs. However, most of the theorems proved in such a way have a strong geometric appeal. This paper demonstrates how shorter, less technical and even more beautiful the proofs can be when they are based on geometric theorems. We show that this technique can be extended to explain concepts in probability and statistics. Although most of the theorems and ideas are not completely new and there are even a few works on the geometry in econometrics and statistics, this paper introduces alternative explanations and more general results in some cases. Further, there are algebraic proofs provided parallel to geometric ones and illustrations which are our own work.

This study inspected the relationships between self-reported implementation of instructional strategies using a dynamic geometry approach and the students’ engagement in making, testing, and proving conjectures Data collected includes a self-reported questionnaire given to all of the project’s participating high school geometry teachers, collecting both quantitative and qualitative data. The results of the linear model, with proving conjectures as a response variable, indicate that students spent less time proving or disproving their conjectures when working alone regardless of whether they were in a regular or advanced level geometry class. Time spent making conjectures and testing conjectures were positively and significantly correlated with the frequency of teachers’ implementation of class discussions. Furthermore, giving instruction that prompted group work had a significant and positive correlation with students proving conjectures in Regular geometry classes.

These notes are based on lectures delivered in the summer school ``Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory'', held in Bedlewo, Poland, in the summer of 2011. The course is an exposition of Furstenberg's conjectures on ``transversality'' of the maps $x\rightarrow ax $mod1 and $x\mapsto bx$mod1 for multiplicatively independent integers $a,b$, and of the associated problems on intersections and sums of invariant sets for these maps. The first part of the course is a short introduction to fractal geometry. The second part develops the theory of Furstenberg's CP-chains and local entropy averages, ending in proofs of the sumset problem and of the known case of the intersections conjecture.

Proof validation is important in school mathematics because it can provide a basis upon which to critique mathematical arguments. While there has been some previous research on proof validation, the need for studies with school students is pressing. For this paper, we focus on proof validation and modification during secondary school geometry. For that purpose, we employ Lakatos’ notion of local counterexample that rejects a specific step in a proof. By using Toulmin’s framework to analyze data from a task-based questionnaire completed by 32 ninth-grade students in a class in Japan, we identify what attempts the students made in producing local counterexamples to their proofs and modifying their proofs to deal with local counterexamples. We found that student difficulties related to producing diagrams that satisfied the condition of the set proof problem and to generating acceptable warrants for claims. The classroom use of tasks that entail student discovery of local counterexamples may help to improve students’ learning of proof and proving.

The study aimed to evaluate the potentialities of Dynamic Geometry Environments (DGE) in the teaching and learning of mathematical proof by analyzing students’ cognitive processes while solving an open geometry problem. The study was conducted in a grade-10 math class in a Lebanese school. Data were collected through whole-class observation with analysis of paper-based data and closer observation of 12 pairs of students. The analysis focused on the mistakes that occurred at the three stages of the proving process: the construction and manipulation of the figure; the formulation of the conjecture; and the proof itself. The results suggest the development of a “framework of failed proving processes” that classifies errors by type and by explanations for the failure.

We survey the applications of ruled surface theory in incidence geometry. We discuss some of the proofs and raise some open questions.

Summary This is the translation of the Mizar article containing readable Mizar proofs of some axiomatic geometry theorems formulated by the great Polish mathematician Alfred Tarski [8], and we hope to continue this work. The article is an extension and upgrading of the source code written by the first author with the help of miz3 tool; his primary goal was to use proof checkers to help teach rigorous axiomatic geometry in high school using Hilbert’s axioms. This is largely a Mizar port of Julien Narboux’s Coq pseudo-code [6]. We partially prove the theorem of [7] that Tarski’s (extremely weak!) plane geometry axioms imply Hilbert’s axioms. Specifically, we obtain Gupta’s amazing proof which implies Hilbert’s axiom I1 that two points determine a line. The primary Mizar coding was heavily influenced by [9] on axioms of incidence geometry. The original development was much improved using Mizar adjectives instead of predicates only, and to use this machinery in full extent, we have to construct some models of Tarski geometry. These are listed in the second section, together with appropriate registrations of clusters. Also models of Tarski’s geometry related to real planes were constructed.

We consider questions posed in a recent paper of Mandayam, Bandyopadhyay, Grassl and Wootters [10] on the nature of "unextendible mutually unbiased bases." We describe a conceptual framework to study these questions, using a connection proved by the author in [19] between the set of nonidentity generalized Pauli operators on the Hilbert space of $N$ $d$-level quantum systems, $d$ a prime, and the geometry of non-degenerate alternating bilinear forms of rank $N$ over finite fields $\mathbb{F}_d$. We then supply alternative and short proofs of results obtained in [10], as well as new general bounds for the problems considered in loc. cit. In this setting, we also solve Conjecture 1 of [10], and speculate on variations of this conjecture., 16 pages; submitted (July 2014)

Recommendations that teachers promote argument and discourse in their mathematics classrooms anticipate researchers' needs for methods for examining and analyzing such talk. One form of discourse is oral arguments, including proofs. We ask: How can we track the development of an oral argument by a teacher and her/his students? We illustrate a method that combines Systemic Functional Linguistics (SFL) and Toulmin's argumentation scheme to examine how speakers logically connect different parts of an argument. We suggest that conjunction analysis can aid a researcher to map the content of a proof that has been constructed in class discussion. Using data from a discussion of a geometry proof, we show that different types of conjunctions enabled the teacher and the students to connect various components of an argument and, also, different arguments. The article illustrates how conjunction analysis can support and deepen what Toulmin's scheme for arguments can reveal about oral discussions.

Problem Durumu: Matematiğin önemli bir parçası olan geometri, okullarda ve günlük yaşamda önemli bir yer edinmiştir. Geometri eğitimine verilen öneme rağmen, birçok kişi geometri karşısında çaresiz kalmakta, buna bağlı olarak da başarısızlık göstermektedir. Matematiği ezberlenmesi gereken kurallar bütünü olarak gören öğrenciler, matematikten kaçmaktadırlar. Buda onların matematik derslerinde başarısı gösterememesinin nedenlerinden birini oluşturmaktadır. Hâlbuki matematik belli bir mantık içinde gelişmiş, kavramların bir biri ile uyum içinde olduğu bir disiplindir. Matematiğin bu uyumlu mantıksal ilişkisini anlamanın yolu, kuralları ezberlemeden ziyade, kavramların nasıl geliştiğini görmekten geçmektedir. Bunun anlamanın yolu ise onları elde ederken yapılan ispatları incelemekten geçmektedir. Nasıl kalp insana hayat verirse, ispatlarda matematiğe hayat vermektedirler. Fakat ispatlar karmaşık gözüktüğünden, birçok öğretmen tarafından derslerde pek kullanılmamaktadır. Oysaki eğitimin ilk yıllarından itibaren, öğrencilerde bu becerinin gelişmesi için öğretmenlerin derslerde ispata yer vermeleri gerekmektedir. Bu açıdan bakıldığında öğretmenlerin derslerde ispat kullanımına yönelik görüşleri ve ispat yaparken ki süreçlerinin incelenmesi önem arz etmektedir. Araştırmanın Amacı: Bu çalışmanın amacı öğretmen adaylarının geometrik bir teoremi ispatlarken, kullandıkları ispat şemalarını, ispat yaparken karşılaştıkları zorlukları ve derslerde ispat kullanımına yönelik görüşlerini belirlemektir. Araştırmanın Yöntemi: Araştırmada nitel araştırma desenlerinden durum çalışması kullanılmıştır. Çünkü bir konuda derinlemesine bilgi edinmek istenildiğinde durum çalışmasının kullanılması uygundur. Çalışmanın örneklemini, seçkisiz desenlerden amaçlı örnekleme yöntemine göre belirlenen ikisi kız, diğeri erkek üç öğretmen adayı oluşturmaktadır. Çalışmada ilk önce öğretmen adaylardan bir üçgenin iç açıları ölçüleri toplamının 1800 olduğunu göstermeleri istenmiştir. Daha sonra yaptıkları ispata ve derslerde ispat kullanımına yönelik yaklaşık otuz dakika süren yarı yapılandırılmış görüşmeler yapılmıştır. Yapılan görüşmeler analiz edilmek üzere kayıt altına alınmıştır. Öğretmen adaylarının ispat şemalarına belirlemek için, elde edilen veriler her bir araştırmacı tarafından ayrı ayrı incelenmiştir. Sonuç olarak araştırmacıların uyum birliğine vardıkları görülmüştür Araştırmanın Bulguları: Araştırmadan elde edilen verilere göre, öğretmen adayları derslerdeki başarılarını ispat yaparken gösterememişlerdir ve temel geometrik kavramlara yönelik bilgilerinde eksiklikler olduğu görülmüştür. Örneğin Ayşe derste başarılı olmasına rağmen kendisinden istenen ispatı uygun şekilde göstermekte zorlanmıştır. İspatı yaparken ilköğretim yıllarındaki öğretmeninin verdiği örneği temel alarak ispatı yapmaya çalışmıştır. İlk denemesinde başarısız olan Ayşe, ikinci denemesinde ise ispatı özel bir durum üzerinde (ikizkenar dik üçgen için) göstermeye çalışmıştır. Yani genellemeye gidememiştir. Elif orta düzey başarı gösteren bir öğrencidir. İspat yaparken rastgele çizimler yapmaya çalışmıştır. Bu durum Elif'in tam olarak ne yapacağını bilmediğini göstermektedir. Ali derslerde başarılı olmamasına rağmen ispat yaparken iyi bir performans göstermiştir. İşlemleri düzenli bir şekilde yapmıştır. Fakat temel geometrik kavramlara yönelik bilgilerinde eksiklik olduğu görülmüştür. Bu ise Ali'nin tanımları tam olarak öğrenemediğinin göstergesidir. Ayrıca Ali'nin üçgeni oluştururken ışınları kullanması farklı bir bulgudur. Ali bunu daha önceki öğrenmelerine bağlamıştır. Araştırmanın Sonuçları ve Önerileri: Sonuç olarak Ayşe'nin ispatı yaparken bir örnek ile başlaması ve öğretmenin verdiği örneği temel alarak ispatı yapmaya çalışması, onun tümevarımsal ispat şeması yaklaşımında olduğunu göstermektedir. Elif'in rastgele şekiller çizmesi, ne yapacağını tam olarak bilememesi onun sembolik ispat şeması yaklaşımında olduğunu göstermektedir. Ali'nin ispatı daha önceki öğrenmelerine göre yapması onun ritüel ispat şeması yaklaşımda olduğunu göstermektedir. Genel olarak bakıldığında Ayşe deneysel, Elif ve Ali dışsal bir yaklaşım göstermişlerdir. Bununla birlikte Coe ve Ruthven (1994) çalışmalarında birçok öğrencinin deneysel ispat şeması yaklaşımına sahip olduklarını görmüştür. Ayrıca öğretmen adayları derslerde ispat kullanımının yararlı olduğunu düşünmektedirler. Böylece kuralların ezberlenmeyip, mantıksal olarak öğrenileceğini belirtmektedirler. Fakat öğrencilik yıllarında derslerde ispat kullanılmışsa da ilgili olmadıkları görülmektedir. Benzer şekilde Ceo ve Ruthen (1994) çalışmalarında da çok az sayıda öğrencinin kuralların nasıl oluştuğunu öğrenmeye meraklı olduklarını belirtmişlerdir. Matematiğin kalbi olan ispatlar, bir teoremin doğru olup olmadığını anlamaya yarar ve matematiğin gelişmesine öncülük eder. Rav (1999) yaptığı bir analojide ispatları yollar ağına, teoremleri de otobüs durağına benzetmiştir. İspatlar matematiğin temelleri olmasına rağmen, birçok araştırmada öğrencilerin başarısız oldukları görülmüştür. Bunun birçok nedeni vardır. Öğrencilerin ispatı hiç kullanmaması, ne anlama geldiğini bilmemesi bunlardan bir kaçını oluşturmaktadır. Bundan dolayı öğrenciler teoremleri anlamada ve uygulamada yetersiz kalmaktadır. Tall (2004) ise üniversite öğrenimine kadar ispatlar ile ilgilenmemiş öğrencilerin, ispat yapmada zorlandıklarını belirtmiştir. Öğrencilerin başarısızlığın bir diğer nedeni öğretmenlerin ispata yönelik tutumundan olabilir. Çünkü programlar incelendiğinde ispat yapma becerisini geliştirecek şekilde düzenlenmediği görülmektedir (Tall, 1995). Programda ispatlara yeterince yer verilmediğinden dolayı öğrencilerin ispat becerilerinin geliştirilmesinde temel kaynak sınıf öğretmenleridir. Benzer şekilde Martin ve Harel (1989) öğretmen adaylarının ispatlara bakış açısının önemli olduğunu belirtmektedirler. Hersh (1993) ise derslerde sınırlı zamandan dolayı ispat süreçlerinin göz ardı edilmesine neden olduğunu belirtmektedir. Bu çalışmanın sonuçları üç öğretmen adayının ispat şemaları ile sınırlıdır. Bundan dolayı genelleme yapmak mümkün değildir. Bununla birlikte bu çalışmanın sonuçlarına göre, sınıf öğretmeni adaylarının basit bir geometrik teoremi ispatlama da bile zorlandıkları görülmüştür. Öğrencilerin geometrik kavramlardaki eksik bilgileri ve kavram yanılgıları ise göze çarpmaktadır. Öğrencilerin önceki eğitimlerinde ispatlar ile ilgili yeterince tecrübe kazanmamaları ve kavramsal bilgilerin göz ardı edilmesi buna neden olmuş olabilir. Oysaki İspatlar öğrencilerin işlemsel bakış açısından kavramsal bakış açısına geçmelerine yardım eder (Weber, 2001; Jones, 2002). Öğretmen adayları da ispatlamanın bilgiyi elde etmede önemli olduğunu belirtmişlerdir. Sonuç olarak ispat yapabilme matematik eğitiminde geliştirilmesi gereken bir beceridir. Bundan dolayı matematik eğitimi ispatları içermelidir., Problem Statement: Recent research and evaluation reports show that students are not learning geometry efficiently. One identifier of student understanding related to geometry is teachers' knowledge structures. Understanding what a proof is and writing proofs are essential for success in mathematics. Thus, school mathematics should include proving activities. Proofs are at the heart of mathematics, and proving is complex; teachers should help their students develop these processes in the early grades. The success of this process depends on teachers' views about the essence and forms of proofs. Hence, it is necessary to investigate the classroom teachers' perceptions related to proofs. Purpose of the Study: The purpose of this study is to determine the proof scheme of pre-service teachers when proving a geometry theorem. In this sense, the study is oriented by the research question: which proof schemes do pre-service teachers use when making proofs in geometry? Method: The current case study is a detailed examination of a particular subject. Firstly, an open ended question was asked, and then semistructured interviews were conducted. The three students investigated in this study were selected by considering their Basic Mathematics scores. Two girls having maximum and average scores and a boy having a minimum score voluntarily participated. The students were asked to proof "the sum of the interior angle measurements of a triangle is 180º". After proving this, each student was interviewed about what they think about proofs and proving. Findings: The findings of the study reveal that pre-service classroom teachers have difficulties related to proving. Also, the participants' attitudes are not parallel to their achievements in the lesson. Another result of this study concerns using proofs in teaching and learning processes. When students are asked about their opinions regarding proofs, it is understood that they have the common idea that the lessons should be made with proofs. Conclusion and Recommendations: The results of this study and other studies in the field reveal that pre-service teachers are not able to prove even a simple geometry theorem. What underlies this is that pre-service teachers are thought to have insufficient knowledge about the definitions of geometric concepts as well as the misconceptions concerning the topic. Another reason can be that participants do not experience any proving processes in previous education. Hence, students should realize how valuable proving and acquiring knowledge is through the counsel of a teacher

In this chapter we give a survey of rigid geometry over non-Archimedean fields. The foundation of the theory was laid by Tate in his private Harvard notes dating back to 1961, which were later published in Inventiones mathematicae (Tate in Invent. Math. 12:257–289, 1971). Here we explain the main results from the classical point of view as studied in the late sixties; for proofs we refer to Bosch (Lectures on Formal and Rigid Geometry, vol. 2105, 2014). At that time rigid geometry was mainly inspired by complex analysis. Fundamental results were achieved by Kiehl, who introduced the Grothendieck topology and proved the basic facts concerning coherent sheaves. Moreover, Kiehl makes Serre’s theory (Serre in Ann. Inst. Fourier 6:1–42, 1956) of Geometrie Algebrique et Geometrie Analytique available for rigid analytic geometry, often referred to as GAGA; cf. (Kopf in Uber eigentliche Familien algebraischer Varietaten uber affinoiden Raumen, vol. 7, 1974).

Symbolic geometry software, such as Geometry Expressions, can guide students as they develop strategies for proofs.

This contribution is sort of an addendum to a recently published paper on circle-geometries in Cayley–Klein planes, see Martini and Spirova (Publicationes Mathematicae Debrecen 72:371–383, 2008b), as it deals with further generalisations and extensions of the author’s results to circle-geometries in all Cayley–Klein planes. The main methods in this paper are the interpretation of planar figures in space and the dualizing according to the duality principle of projective spaces. There are, in principle, only three types of \({\mathbb{R}^2}\)-ring structures and, thus, only three types of corresponding circle-geometries, see Benz (Geometrie der Algebren, 1973a). Therefore, each generalisation to non-Euclidean planes must turn out to be just another representation of the classical Euclidean cases. This point of view gives more insight into why some elementary geometric theorems remain valid when changing the place of action from the Euclidean plane to non-Euclidean circle planes and makes explicit proofs of such elementary geometric theorems in non-Euclidean circle planes superfluous. We believe that even the Euclidean cases of circle-geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle-geometries might also be of interest for their own. For example, among the planar Cayley–Klein geometries the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated, similar to the isotropic Mobius geometry, by suitable projections of the Blaschke cylinder.

Within the domain of geometry, proof and proof development continues to be a problematic area for students. Battista (2007) suggested that the investigation of knowledge components that students bring to understanding and constructing geometry proofs could provide important insights into the above issue. This issue also features prominently in the deliberations of the 2009 International Commission on Mathematics Instruction Study on the learning and teaching of proofs in mathematics, in general, and geometry, in particular. In the study reported here, we consider knowledge use by a cohort of 166 Sri Lankan students during the construction of geometry proofs. Three knowledge components were hypothesised to influence the students’ attempts at proof development: geometry content knowledge, general problem-solving skills and geometry reasoning skills. Regression analyses supported our conjecture that all 3 knowledge components played important functions in developing proofs. We suggest that whilst students have to acquire a robust body of geometric content knowledge, the activation and the utilisation of this knowledge during the construction of proof need to be guided by general problem-solving and reasoning skills. Refereed/Peer-reviewed

The article analyzes in detail the assumptions and the proofs typical of the research field of the geometry of burning mirrors. It emphasizes the role of two propositions of the Archimedean Quadratura parabolae, never brought to bear on this subject, and of a complex system of projections reducing a sumptōma of a parabola to some specific linear lemmas. On the grounds of this case-study, the much-debated problem of the heuristic role of analysis is also discussed.

The theorems that we will discuss are well-known in mathematics. They are related to the foundations of geometry, to geometry “in the large” and to the history of geometry. Namely, we are dealing with three beautiful ancient theorems whose authors are Archimedes (the theorem on the dropping of a stone), Euclid and an Egyptian writer, Ahmes (problems from Egyptian papyruses). It seems astonishing that the aforementioned theorem of Euclid went unnoticed as a generalization of the fundamental uniqueness theorems of A. Cauchy and H. Minkowski concerning convex closed polyhedra. The three theorems discussed are absolutely flawless, but their theoretical and historical interpretations are still rather inadequate. In our opinion, these theorems belong to the ancient civilizations of Babylon, Egypt and Sumer which were superior to our modern civilization in numerous aspects. This opinion will be supported by generalizations, proofs and a precise reconstruction of ancient theorems.

The purpose of this article is to provide examples of 'non-traditional' theorems that can be explored in a dynamic geometry environment by university and high school students. These theorems were encountered in the dynamic geometry environment. The author believes that teachers can ask their students to construct proofs for these theorems. The Appendix contains proof outlines for each theorem.

Dynamic geometry software (DGS) such as Cabri and Geometers’ Sketchpad has been regularly used worldwide for teaching and learning Euclidean geometry for a long time. The DGS with its inductive nature allows students to learn Euclidean geometry via explorations. However, with respect to non-Euclidean geometries, do we need to introduce them to students in a deductive manner? Do students have quite different experiences in non-Euclidean environment? This study addresses these questions by illustrating the student mathematics teachers’ actions in dynamic spherical geometry environment. We describe how student mathematics teachers explore new conjectures in spherical geometry and how their conjectures lead them to find proofs in DGS.

In “The Surfer Problem: A ‘Whys’ Approach” (Mathematics Teacher 100, no. 1 [August 2006]: 14–19), Larry Copes and Jeremy Kahan introduced a variety of proofs to one of the classic geometry theorems: The sum of distances from all points in the interior of an equilateral triangle to the three edges is the same. The authors went on to prove that this constant distance is the height of the triangle. Although they presented a range of methods for proving this theorem, I offer one additional method that makes use of a lemma concerning isosceles triangles, introduced below. I will then extend the theorem to points in the exterior of the triangle. Finally, I will replace the equilateral triangle with any regular polygon and show that the revised statement is true.

We provide several radically different proofs of the following unique continuation result: The Oseen eigenvalue problem with over-determined homogeneous Cauchy data for the velocity field on the boundary implies the zero solution, at least if the equilibrium solution is sufficiently 'small.' In particular, this unique continuation result from over-determined boundary data holds true for the Stokes problem.

This paper investigates the geometry of the expansion RQ of the real field R by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusinski–Lion–Rolin). To this end, we study non-standard models R of the universal diagram T of RQ in the language L augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation theorem, upon which the majority of fundamental results in analytic geometry rely, but which is unavailable in the general quasianalytic geometry. The basic tools applied here are transformation to normal crossings and decomposition into special cubes. The latter method, developed in our article [32], combines modifications by blowing up with a suitable partitioning. Via an analysis of L-terms and infinitesimals, we prove the valuation property for functions given by L-terms, and next the exchange property for substructures of a given model R. Our proofs are based on the concepts of analytically independent as well as active and non-active infinitesimals, introduced in this article. Further, quantifier elimination for T is established through model-theoretic compactness. The universal theory T is thus complete and o-minimal, andRQ is its prime model. Under the circumstances, every definable function is given piecewise by L-terms, and therefore the previous results concerning L-terms generalize immediately to definable functions. In this fashion, we obtain the valuation property and preparation theorem for quasi-subanalytic functions. Finally, a quasi-subanalytic version of Puiseux’s theorem with parameter is demonstrated. AMS Classification: Primary 32S45, 14P15, 32B20; Secondary 03C10, 26E10, 03C64.

Here we give a mathematical proof that a current method that calculates dominant poles is locally convergent. Moreover, the convergence is at least quadratic.

International audience; Numerous brain regions encode variables using spatial distribution of activity in neuronal maps. Their specific geometry is usually explained by sensory considerations only. We provide here, for the first time, a theory involving the motor function of the superior colliculus to explain the geometry of its maps. We use six hypotheses in accordance with neurobiology to show that linear and logarithmic mappings are the only ones compatible with the generation of saccadic motor command. This mathematical proof gives a global coherence to the neurobiological studies on which it is based. Moreover, a new solution to the problem of saccades involving both colliculi is proposed. Comparative simulations show that it is more precise than the classical one.

The purpose of this paper is to provide examples of ‘non-traditional conjectures’ in a dynamic geometry environment that can be used to stimulate students to provide proofs. The author believes that teachers can ask their students to find out if these conjectures are true; if the conjectures are true, then the students can come up with their own proofs. The appendix contains brief descriptions (pertaining to each conjecture) explaining why these conjectures are true or false.

In the theory of beam bending, transverse sections of a beam are assumed to remain plane during bending. In torsion analysis, the radii of shafts of circular cross-sections are assumed to remain straight during twisting. Some popular textbooks present mathematical proofs for these statements. This article shows that these proofs are incorrect and that these statements should be treated as assumptions.

The proof of the existence and of the uniqueness of groups of J3-type by G. Higman and J. McKay is based on the fact that a group of J3-type is a faithful completion of an amalgam of J3-type, see [11]. In this paper here, we provide a direct reference for that fact. The proofs in this paper are elementary and we do not use any character theory.

This article is concerned with the mathematical proof that distillation boundaries correspond to locally longest residue curves. Perturbation theory and the implicit function theorem are used to provide an analytical relationship between neighboring trajectories while local congruence and a variety of limiting arguments are used to establish the relative distances between neighboring residue curves. The mathematical conditions on which this analysis is based include (1) The assumptions that trajectories are twice continuously differentiable and do not cross. (2) The comparison of neighboring trajectories that converge to the same stable node. (3) The existence and placement of a proper focal point. (4) Local congruence. While all of these assumptions and conditions are important, it is the existence and proper placement of a focal point and local congruence that are the keys to establishing relative distances between neighboring residue curves and, in turn, the main result that distillation boundaries are defined by locally longest trajectories from any unstable node to all reachable stable nodes. Wherever possible, geometric illustrations are used to clarify key concepts. 2007 American Institute of Chemical Engineers AIChE J, 53: 1779– 1788, 2007

According to the NCTM Standards (2000), conjecturing and proving should be central activities throughout students' mathematical education. However, the question of how we can help students generate proofs, especially the formal proofs expected at the high school level, is still a difficult one. In this article, we argue that one cause of students' troubles with proof is that they are not accustomed to making the important, but difficult, connections between their intuitive sensemaking and the formal proofs they are supposed to produce. Thus as teachers, we should provide opportunities for students to make these connections.

This result could have been proved by the Greeks more than two thousand years ago, so we certainly did not expect it to be a new theorem. However, since we found it (1996) we have asked a large number of experts, we have searched the literature, and we have posted queries on the internet, all without finding anyone who has heard of the theorem. We thus feel comfortable in concluding that the theorem is, if not unknown, at least not well known. If a reader of this note knows a reference to the theorem or to related facts, we would appreciate being informed. Before we give a mathematical proof of the theorem, we will explain in principle how the theorem was discovered experimentally. An extremely short light pulse is emitted in a wide range of directions from a point A. The duration of the light pulse is about 1 picosecond = 10−12 seconds, which produces a light pulse of length about 0.3 mm. As the light hits an arbitrary fixed plane π , it is scattered in all directions. At a point B, different from A, a “photograph” is taken with extremely short exposure time. The outcome of the experiment is that a bright circle is seen on the photographic plate. In other words, a bright curve—which is actually an ellipse—is seen on the plane π , and this curve looks circular as viewed from B. (The technique used to carry out such experiments is described in [2].) How can we interpret this experiment as supporting our theorem? Well, let us assume that photons are emitted precisely at time t = 0 from the point A and that the “photograph” is taken at time t = t1 > 0. Then photons originating at A can contribute to the picture only if their travel time is exactly t1, and hence the distance that they have travelled is exactly s = ct1, where c is the speed of light. This means that for any photon that has made one bounce at some point P on the plane π on its way from A to B, the sum of the distances AP and P B must be equal to s. What is the locus of the set of all such points P? If for a moment we forget about the plane π , the locus of all the points P in space such that the sum of the distances AP and P B is equal to s is a rotational ellipsoid E with axis of symmetry through A and B and focal points at A and B. Accordingly the set of bounce points P on the plane π is the curve in which π intersects E . This is the bright curve that is viewed by our camera located at the point B. It is an elementary fact that this curve is an ellipse, and as already explained, the assertion of the theorem is that this curve looks circular when viewed from B.

This research explored students’ views of geometric objects through the implementation of a curriculum module that allowed them to explore the relationships between transformational geometry and linear algebra. The majority of the students were middle and secondary mathematics education majors enrolled in a one-semester geometry course that is aimed at prospective teachers. A preponderance of the evidence suggests that the participating students, for the most part, viewed isometries operationally and viewed geometric objects (triangle, etc.) as “perceived.” Results also suggest that these two views influenced the students’ abilities to understand and to construct geometric proofs in transformational geometry.

We show that many algebraic actions of higher-rank abelian groups on zero-dimensional groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov--Rokhlin formula for half-space entropies., Comment: 21 page latex

In this paper we survey the historical and contemporary connections in mathematics between classical “conceptual” tools versus modern computing tools. In this process we highlight the interplay between the inductive and deductive, experimental and theoretical, and propose the notion of Situated proofs as a didactic tool for the teaching of geometry in the 21st century.

Four potential modes of interaction with diagrams in geometry are introduced. These are used to discuss how interaction with diagrams has supported the customary work of ‘doing proofs’ in American geometry classes and what interaction with diagrams might support the work of building reasoned conjectures. The extent to which the latter kind of interaction may induce tensions on the work of a teacher as she manages students’ mathematical work is illustrated.

The recent development of powerful new technologies such as dynamic geometry software (DGS) with drag capability has made possible the continuous variation of geometric configurations and allows one to quickly and easily investigate whether particular conjectures are true or not. Because of the inductive nature of the DGS, the experimental-theoretical gap that exists in the acquisition and justification of geometrical knowledge becomes an important pedagogical concern. In this article we discuss the implications of the development of this new software for the teaching of proof and making proof meaningful to students. We describe how three prospective primary school teachers explored problems in geometry and how their constructions and conjectures led them "see" proofs in DGS. © National Science Council. 2 3 339 352 Cited By :14

We consider the dynamics of finite systems of spherical particles where a fraction of the kinetic energy may be lost during collisions. We show that the set of initial configurations leading to infinitely many collisions in finite time can have positive measure, contrary to the hard-collision (energy conservation) case, in which this particular set is claimed to be empty. We also show that after sufficient time a system will decouple into maximal subsystems. This generalizes proofs that in the hard-sphere case there can be at most finitely many collisions for almost all initial configurations.

We investigate the adiabatic invariance of the action variable of one degree of freedom time-dependent Hamiltonian systems when trajectories cross a separatrix and give relatively simple, geometrically motivated proofs.