Deepening the Understanding of Mathematics with Geometric Intuition

Geometric intuition is one of the core concepts introduced by the new mathematical curriculum standards. It aims to use intuition and intuitive materials to deepen the understanding of mathematics in mathematical cognition activities. It does not only play a role in the learning of “graphics and geometry,’ but its’ irreplaceable role also involves the whole process of mathematics education. Therefore, if teachers can skillfully use geometric intuition in the teaching process, classroom efficiency will be greatly improved.

The shape of higher-dimensional state space: Bloch-ball analog for a qutrit

Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum system, a two-level system (or qubit). However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more representative for a general quantum system than a qubit. In this work we demonstrate that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter. Besides being of indisputable theoretical value, this opens the door to a new type of representation, thus extending our geometric intuition beyond the simplest quantum systems.

FROM DANTE’S UNIVERSE TO CONTEMPORARY COSMOLOGY

A reading of the Divina Commedia with the eyes of a modern scientist reveals that Dante devoted great attention to the description of a wide variety of natural phenomena, particularly those involving astronomy, optics and geometry. A remarkable case is the structure of the cosmos emerging from the Paradiso, which foreshadows a non-Euclidean geometrical structure with remarkable similarities to Einstein’s 1917 static cosmological solution. Such model, however, as well as other solutions with positive spatial curvature, are ruled out by current astrophysical observations. Here I discuss Dante’s geometric intuition and show its close analogy with the shape of the observable cosmic space-time in the standard CDM expanding model, fully supported by present-day cosmological data.

Frege on the Foundation of Geometry in Intuition

I investigate the role of geometric intuition in Frege’s early mathemat- ical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappen- den, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arith- metization and to the Riemannian conceptual/geometrical tradition at Göttingen. Concurring with Tappenden, I hold that Frege’s logicism should not be understood as a continuing a project of reductionist arith- metization. However, Frege does not quite take up the Riemannian ban- ner either. His logicism supports a hierarchical understanding of the structure of mathematical knowledge, according to which arithmetic is applicable to geometry but not vice versa because the former is more general, as revealed by the strictly logical nature of its objects in com- parison to the intuitional nature of geometric objects. I suggest, in par- ticular, that Frege intended that foundational work would show the use of geometric intuition in complex analysis, a source of error for Rie- mann that Weierstrass was proud to have uncovered, to be inessential.

Hauteur asymptotique des points de Heegner

Geometric intuition suggests that the Néron–Tate height of Heegner points on a rational elliptic curve E should be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross–Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin–Selberg convolution of E with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twisted L-function of E by a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with these L-series and L-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves.

A Deviation-Function Method for the Design of New Conjugate Kinematic Pairs

In this paper a method for generating new conjugate kinematic pairs is developed. This method, called the “Deviation-Function” method, is to use deviation functions to reshape the original pitch pairs so that the desired profiles of generated pairs can be obtained. A deviation function is defined as the distance between contacts and their corresponding instant centers of two conjugate pairs. In other words, a deviation function measures the amount of deviation of a point on the generated profile from its corresponding point on the original pitch profile. As demonstrated, this new method is superior to the existing methods for conjugate pair generation in three ways: 1. it is applicable to any pitch pairs including circular or noncircular, identical or non-identical; 2. it enables the design of geometric intuition into the generated profiles; and 3. it enables the design of special analytical properties to the generated profiles.

Activities: Patterns and Puzzles for Pyramids and Prisms

Teacher's Guide: The construction projects, puzzles, and experiments presented here provide hands-on experiences in spatial visualization and problem solving for an important class of three-dimensional figures: pyramids and prisms. Students will enjoy creating physical models and performing experiments with the models, but the main goals of the activities are to develop students' geometric intuition and build a concrete foundation on which abstract principles can be grounded.

Visual Thinking with Translations, Half-Turns, and Dilations

Geometry is too important to be confined to one year of the high school curriculum. It should be used whenever it can provide visual imagery to help students understand analytical definitions and formal reasoning. In many instances, geometric intuition should precede or parallel analytic formulation of mathematical ideas. This approach is certainly true in vector geometry, which deserves an important place in the last two years of the high school curriculum.

Do All Graphs Have Points with Integral Coordinates?

The development of the general equation of a line (ax + by = c) is usually based on geometric intuition and the concept of slope. Students are given a point, (a, b), whose coordinates are integers, and a slope, m, and asked to write the equation of the line and graph it.