Geometric Proof Research Articles

Comprehending geometry proofs: A study on prospective mathematics teachers' perceptions and preferences of proof formats

The explicative aim of this research is to investigate the optics of proof format preferences held by prospective mathematics teachers (PMTs), in an effort to assess the effect that such preferences may have on their understanding of geometric proofs. The researchers employed a qualitative case study approach, in which six PMTs participated in Focus Group Discussions (FGDs) on the three styles of proof: paragraph proof, two-column proof and flow-chart proof. The methodology employed also focused on the formatting models by exploring the participants’ suggestions regarding each format in order to determine their cognitive responses as well as their preferences. The study exposes different preferences among PMTs which are a result of their personal cognitive style and experience with proofs. Some of the PMTs preferred the two-column format due to its systematic and logical nature, other PMTs preferred the storytelling nature of the paragraph proof format sponsored also flow-chart representation. These preferences bring out the requirement for the development of pedagogical strategies that employ different proof formats in order to meet the needs of different types of learners for effective understanding. In summary, this research demonstrates the need for customizing proof instruction to different students’ needs which can hopefully enhance PMTs’ responses and interactions with the concept of mathematical proofs. The practical outcomes show that teachers should not be afraid to think in more flexible witnessing forms of proof in order to aid students learning while minimizing the cognitive load in line with the principles of constructivist learning. It would be worthwhile however to extend this research and examine what these preferences would mean for the future and how those would fit within the context of online education with the overall aim of creating a better mathematics curriculum that is more effective and has a wider reach.

A geometric proof of the binomial identity

We give a geometric proof of the binomial identity for all natural n and real a,b. This work was inspired by the book [1], where the binomial identity for n = 3 and a,b > 0 is proved by breaking a cube C of size (a + b) × (a + b) × (a + b) into eight rectangular boxes and counting their volumes as follows.

The Geometric Scale of an Urban Space—Neuro-perceptual Limitations for Face-to-Face Driver Behavior

For the last century, the roadway design paradigm has been grounded in the physics of point masses, adjusted for human limitations. However, this is inadequate to assure pedestrian safety in complete streets where vehicle kinematics no longer control. Instead, social and psychological factors manage behavior. Unfortunately, the appropriate design critical psychological principles have yet to be elucidated. We posit that interpersonal perception governs drivers’ behavior, attentiveness and speed in streets. This is bounded by previously delineated human neurological and perceptual propensities. Using these perceptual limitations as postulates in a geometric style proof, we derive the person perception panorama (PPP): a window of interactivity around the moving driver that is continuously monitored for human presence, roughly 60 to 90 feet wide. In this area interpersonal interaction is implicit, functional, and has an impact on driver behavior. Validating evidence, additional governing principles, and an initial speed prediction formula are also included.

Mirrors to Del Pezzo Surfaces and the Classification of $T$-Polygons

We give a new geometric proof of the classification of $T$-polygons, a theorem originally due to Kasprzyk, Nill and Prince, using ideas from mirror symmetry. In particular, this gives a completely geometric proof that any two toric $\mathbb{Q}$-Gorenstein degenerations of a smooth del Pezzo $X$ surface are connected via trees of rational curves in the moduli space of $X$.

A Natural Geometric Proof of the Length-Curvature Identity

Summary This note offers a natural geometric proof for the length-curvature identity for a C 2 function of two variables.

Dirac Eigenvalue Optimisation and Harmonic Maps to Complex Projective Spaces

Abstract Consider a Dirac operator on an oriented compact surface endowed with a Riemannian metric and spin structure. Provided the area and the conformal class are fixed, how small can the $k$-th positive Dirac eigenvalue be? This problem mirrors the maximization problem for the eigenvalues of the Laplacian, which is related to the study of harmonic maps into spheres. We uncover the connection between the critical metrics for Dirac eigenvalues and harmonic maps into complex projective spaces. Using this approach we show that for many conformal classes on a torus the first nonzero Dirac eigenvalue is minimised by the flat metric. We also present a new geometric proof of Bär’s theorem stating that the first nonzero Dirac eigenvalue on the sphere is minimised by the standard round metric.

Tropical Refined Curve Counting with Descendants

We prove a q-refined tropical correspondence theorem for higher genus descendant logarithmic Gromov–Witten invariants with a λg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _g$$\\end{document} class in toric surfaces. Specifically, a generating series of such logarithmic Gromov–Witten invariants agrees with a q-refined count of rational tropical curves satisfying higher valency conditions. As a corollary, we obtain a geometric proof of the deformation invariance of this tropical count. In particular, our results give an algebro-geometric meaning to the tropical count defined by Blechman and Shustin. Our strategy is to use the logarithmic degeneration formula, and the key new technique is to reduce to computing integrals against double ramification cycles and connect these integrals to the non-commutative KdV hierarchy.

Geometric diagrams as an effective notation

AbstractIn what way does a mathematical proof depend on the notation used in its presentation? This paper examines this question by analysing the computational differences, in the sense of Larkin and Simon's ‘Why a diagram is (sometimes) worth 10,000 words’, between diagrammatic and sentential notations as a means for presenting geometric proofs. Wittgenstein takes up the question of mathematical notation and proof in Section III of Remarks on the Foundations of Mathematics. After discussing his observations on a proof's ‘characteristic visual shape’ in Section III with respect to arithmetical proofs, the paper shows how the notion of a characteristic visual shape illuminates the special effectiveness of diagrammatic notation in geometry.

On the positivity of B-spline Wronskians

A proof that Wronskians of non-zero B-splines are positive is given, using only recursive formulas for B-splines and their derivatives. This could be used to generalize the de Boor–DeVore geometric proof of the Schoenberg–Whitney conditions and total positivity of B-splines to Hermite interpolation. For Wronskians of maximal order with respect to a given degree, positivity follows from a simple formula.

A Geometric Proof of the Polygonal Isoperimetric Inequality

A Geometric Proof of the Polygonal Isoperimetric Inequality

The Integration of Digital Technology in Task-Design on Eye-Tracking Studies in Geometry

Eye-tracking (ET) method provides a promising channel for educational researchers to connect learning outcomes to cognitive processes. The main principle of ET is that our gaze and our focus of attention are connected. Due to the advent of digital technologies, eye tracking studies are increasingly growing in different fields and in mathematics education. We assume that the provided task is one of the ways to analyse deeply cognitive processes, and we wonder how ET studies are dealing with the integration of digital technology in task-design in geometry. Starting from Strohmaier et al. (2020) we found three studies, one focused on geometric proofs and two based on construction of geometric objects. We highlight the contributions of integrating digital technology in learning and instruction, and in research in geometry.

A Unified Geometric Proof of Aristarchus’ Inequality

Summary We present a refined, simplified, and unified proof of Aristarchus’ inequality, without resorting to trigonometry, calculus or extended geometric arguments.

Symplectic Fourier–Deligne Transforms on G/U and the Algebra of Braids and Ties

Abstract We explicitly identify the algebra generated by symplectic Fourier–Deligne transforms (i.e., convolution with Kazhdan–Laumon sheaves) acting on the Grothendieck group of perverse sheaves on the basic affine space $G/U$, answering a question originally raised by A. Polishchuk. We show it is isomorphic to a distinguished subalgebra, studied by I. Marin, of the generalized algebra of braids and ties (defined in Type $A$ by F. Aicardi and J. Juyumaya and generalized to all types by Marin), providing a connection between geometric representation theory and an algebra defined in the context of knot theory. Our geometric interpretation of this algebra entails some algebraic consequences: we obtain a short and type-independent geometric proof of the braid relations for Juyumaya’s generators of the Yokonuma–Hecke algebra (previously proved case-by-case in types $A, D, E$ by Juyumaya and separately for types $B, C, F_{4}, G_{2}$ by Juyumaya and S. S. Kannan), a natural candidate for an analogue of a Kazhdan–Lusztig basis, and finally an explicit formula for the dimension of Marin’s algebra in Type $A_{n}$ (previously only known for $n \leq 4$).

A geometric proof for the root-independence of the greedoid polynomial of Eulerian branching greedoids

We define the root polytope of a regular oriented matroid, and show that the greedoid polynomial of an Eulerian branching greedoid rooted at vertex v0 is equivalent to the h⁎-polynomial of the root polytope of the dual of the graphic matroid.As the definition of the root polytope is independent of the vertex v0, this gives a geometric proof for the root-independence of the greedoid polynomial for Eulerian branching greedoids, a fact which was first proved by Swee Hong Chan, Kévin Perrot and Trung Van Pham using sandpile models. We also obtain that the greedoid polynomial does not change if we reverse every edge of an Eulerian digraph.

Constant term functors with $\mathbb{F}_{p}$-coefficients

We study the constant term functor for \mathbb{F}_{p} -sheaves on the affine Grassmannian in characteristic p with respect to a Levi subgroup. Our main result is that the constant term functor induces a tensor functor between categories of equivariant perverse \mathbb{F}_{p} -sheaves. We apply this fact to get information about the Tannakian monoids of the corresponding categories of perverse sheaves. As a byproduct we also obtain geometric proofs of several results due to Herzig on the mod p Satake transform and the structure of the space of mod p Satake parameters.

How Schwarzschild could have discovered and fixed the problem with his metric

Karl Schwarzschild's landmark paper revealed the first ever exact solution to Einstein's gravitational field equation. This was a major scientific achievement. However, his solution for the metric of a spherically symmetric spacetime is different than the one found in textbooks today—because Schwarzschild assumed, like in Newtonian gravity, that only one singularity could exist. Given the newness of Einstein's general theory of relativity and the existing paradigm of Newtonian gravity, his assumption was natural. While other authors have previously pointed out this shortcoming and corrected it, the contribution of this paper is to show how Schwarzschild might have discovered it himself. A simple geometric proof indicates how Schwarzschild, with a different assignment of a single constant of integration, would have arrived at the metric found in textbooks today.

Exploring decision-making prospective mathematics teacher in solving geometric proof problems

Facing the current digital era, correct and accurate decisions are needed to deal with the problems. This research aimed to explore the decision-making abilities of prospective mathematics teacher students in solving geometric proof problems. This research was a qualitative descriptive study with research subjects 3 out of 32 students of the 2nd semester Mathematics Education study program. The research instrument is the researcher himself, who is assisted with geometric proof problem tests and interview guidelines. Data analysis was including reduction, presentation, and conclusion. The validity of the research data was tested using methods triangulation. The research results show that subjects with a proof ability of K2 can build and clarify geometric concepts/principles and can assess the reasonableness of ideas even though the subject feels unsure about their choice. Subjects with a proof ability of K3 can build and clarify geometric concept/principle ideas but cannot assess the reasonableness of proof ideas. Subjects with a proof ability of K4 can build ideas of geometric concepts/principles, can clarify even though they are incomplete, and cannot assess the reasonableness of proof ideas. Students need to be equipped with decision-making skills in solving proof problems as preparation to become mathematics teachers in the future.

A Rigorous Euclidean Geometric Proof of the Cube Duplication Impossibility

A Rigorous Euclidean Geometric Proof of the Cube Duplication Impossibility

An update on Haiman’s conjectures

Abstract We revisit Haiman’s conjecture on the relations between characters of Kazdhan–Lusztig basis elements of the Hecke algebra over $S_n$ . The conjecture asserts that, for purposes of character evaluation, any Kazhdan–Lusztig basis element is reducible to a sum of the simplest possible ones (those associated to so-called codominant permutations). When the basis element is associated to a smooth permutation, we are able to give a geometric proof of this conjecture. On the other hand, if the permutation is singular, we provide a counterexample.

Wittgenstein on Proof and Concept-Formation

Abstract In his Remarks on the Foundations of Mathematics, Wittgenstein claims, puzzlingly, that ‘the proof creates a new concept’ (RFM III-41). This paper aims to contribute to clarifying this idea, and to showing how it marks a major break with the traditional conception of proof. Moreover, since the most natural way to understand his claim is open to criticism, a secondary goal of what follows is to offer an interpretation of it that neutralizes the objection. The discussion proceeds by analysing a well-known geometrical proof.