Physical Sciences and Mathematics Commons^™

Voronoi Diagrams, Michael Mumm

The Mathematics Enthusiast

Suppose we have a finite number of distinct points in the plane. We refer to these points as sites. We wish to partition the plane into disjoint regions called cells, each of which contains exactly one site, so that all other points within a cell are closer to that cell's site than to any other site.

Tme Volume 1, Number 2

The Mathematics Enthusiast

No abstract provided.

Author Information

The Mathematics Enthusiast

No abstract provided.

Analogies And Mathematics: What Is The Connection? Book Review Of Mathematical And Analogical Reasoning Of Young Learners, Bharath Sriraman

The Mathematics Enthusiast

Lyn English (Ed). Mathematical and Analogical Reasoning of Young Learners. New Jersey: Lawrence Erlbaum & Associates, 2004. ISBN 0-8058-4945-9.

In the last decade and a half mathematics education literature has shown a rapid increase in books and articles that focus on the social and cultural issues related to mathematics learning and teaching. Although the social and cultural dimensions are important and relevant, the cognitive dimension of mathematical learning is equally important and received less attention. Mathematical and Analogical Reasoning of Young Learners takes us back to the very roots of learning and investigates foundational questions on the nature of …

Editorial, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

The Morley Trisector Theorem, Grant Swicegood

The Mathematics Enthusiast

This paper deals with an unannounced theorem by Frank Morley that he originally published amid a collection of other, more general, theorems. Having intrigued mathematicians for the past century, it is now simply referred to as Morley’s trisector theorem:

The three intersections of the angles of a triangle, lying near the three sides respectively, form an equilateral triangle.

Regular Polytopes, Jonathan Comes

The Mathematics Enthusiast

In the last proposition of the Elements Euclid proved that there are only five regular polyhedra, namely the tetrahedron, octahedron, icosahedron, cube, and dodecahedron. To show there can be no more than five he used the fact that in a polyhedra, the sum of the interior angles of the faces which meet at each vertex must be less than 360.

The Nature Of Proof In Today's Classroom. Book Review Of The Nature Of Proof, Erica Lane

The Mathematics Enthusiast

Harold Fawcett. The Nature of Proof. New York City: Bureau of Publications, Columbia University, 1938. Re-printed by the National Council of Teachers of Mathematics in 1995.

Throughout the world, children are taught new ideas and concepts in a variety of ways, and the advocates of the different approaches claim that their method is the most effective and profitable for the students. In addition to the different teaching styles, each individual learns the material in a distinct manner. Teachers, therefore, must be aware of the diverse techniques of both teaching and learning in an effort to optimize the learning experience …

Radius, Diameter, Circumference, Pi, Geometer's Sketchpad, And You!, Scott Edge

The Mathematics Enthusiast

I truly believe learning mathematics can be a fun experience for children of all ages. It is up to us, the teachers, to present math as an interesting application. The addition of computers into our ever-changing world has given us an important tool, which can assist us on our journey to teach math in new fun and interesting ways. The Program Geometer’s Sketchpad© is one of many mathematic programs we as teachers can use to better help kids understand different geometric concepts. I would like to use Geometer’s Sketchpad© in my classroom to help teach my students about circles and …

Exploring Perimeter And Area With 4th Graders, Amber Lieberg Winkler

The Mathematics Enthusiast

I am going to teach basic introductory geometry skills to 4th graders using Geometry’s Sketchpad. At this age, children are only beginning to learn about geometry in their math classes, and I would like for the students to understand these basic concepts using technology. This problem is so important for the students to learn early, and learn correctly. These skills will involve finding the area and perimeter of regular polygons, basic skills using Geometry’s Sketchpad, and activities that will apply these introductory concepts; directly correlating within the national geometry standards of mathematics. Children use geometry everyday, even when they don’t …

Teaching Symmetry In The Elementary Curriculum, Christy Knuchel

The Mathematics Enthusiast

Symmetry is a fundamental part of geometry, nature, and shapes. It creates patterns that help us organize our world conceptually. We see symmetry every day but often don’t realize it. People use concepts of symmetry, including translations, rotations, reflections, and tessellations as part of their careers. Examples of careers that incorporate these ideas are artists, craftspeople, musicians, choreographers, and not to mention, mathematicians.

It is important for students to grasp the concepts of geometry and symmetry while at the elementary level as a means of exposing them to things they see everyday that aren’t obviously related to mathematics but have …

Author Information

The Mathematics Enthusiast

No abstract provided.

Understanding Polygons And Polyhedrons Using Flexagons, Aaron Tekulve

The Mathematics Enthusiast

The goal of this paper is to help students understand simple polygons and simple polyhedrons. First the project within this paper involves having students look at polygons. Though much of this information should have been learned in the fourth grade it is still important to review this material. Having the students define certain shapes illustrates their true understanding of the subject. The second part to the project within this paper is to use the student’s knowledge of polygons and build polyhedrons. In this paper the students will only have to concern themselves with squares and equilateral triangles. Here they will …

Tme Volume 1, Number 1

The Mathematics Enthusiast

No abstract provided.

Editorial, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

A Quasi-Linear Manifolds And Quasi-Linear Mapping Between Them, Aki̇f Abbasov

Turkish Journal of Mathematics

In this article a special class of Banach manifolds (called QL-manifolds) and mapping between them (QL-mappings) are introduced and some examples are given.

On Graded Primary Ideals, Mashhoor Refai, Khaldoun Al-Zoubi

Turkish Journal of Mathematics

Let G be a group and R be a G-graded commutative ring, i.e., R = \oplus_{g \in G} R_g and R_gR_h \subseteq R_{gh} for all g, h \in G. In this paper, we study the graded primary ideals and graded primary G-decomposition of a graded ideal.

Determination Of A Fractional-Linear Pencil Of Sturm-Liouville Operators By Two Of Its Spectra, R. T. Pashayev

Turkish Journal of Mathematics

In this paper we consider the Sturm-Liouville equations on a finite interval which is fractional-linear in the spectral parameter. The inverse spectral problem consisting of the recovering of the operator from the two spectra is investigated and a uniqueness theorem for solution of the inverse problem is proved.

Perelman's Monotonicity Formula And Applications, Natasa Sesum

Turkish Journal of Mathematics

This article relies on [15] that the author wrote with Gang Tian and Xiaodong Wang. In view of Hamilton's important work on the Ricci flow and Perelman's paper on the Ricci flow where he developes the techniques that he will later use in completing Hamilton's program for the geometrization conjecture, there may be more interest in the area. We will also discuss the author's theorem which says that the curvature tensor stays uniformly bounded under the unnormalized Ricci flow in a finite time, if the curvatures are uniformly bounded. We will prove that in the case of a Kähler-Ricci flow …

Flops Of Crepant Resolutions, Anda Degeratu

Turkish Journal of Mathematics

Let G be a finite subgroup of SL(3, \mathcal{C}) acting with an isolated singularity on \mathcal{C}^3. A crepant resolution of \mathcal{C}^3/G comes together with a set of tautological line bundles associated to each irreducible representation of G. In this note we give a formula for the triple product of the first Chern class of the tautological bundles in terms of both the geometry of the crepant resolution and the representation theory of G. From here we derive the way these triple products change when we perform a flop.

The Cross Curvature Flow Of 3-Manifolds With Negative Sectional Curvature, Bennett Chow, Richard S. Hamilton

Turkish Journal of Mathematics

We consider the cross curvature flow, an evolution equation of metrics on 3-manifolds. We establish short time existence when the sectional curvature has a sign. In the case of negative sectional curvature, we obtain some monotonicity formulas which support the conjecture that after normalization, for initial metrics on closed 3-manifolds with negative sectional curvature, the solution exists for all time and converges to a hyperbolic metric. This conjecture is still open at the present time.

Surgery Diagrams For Contact 3-Manifolds, Fan Ding, Hansjörg Geiges, Andras I. Stipsicz

Turkish Journal of Mathematics

In two previous papers, the two first-named authors introduced a notion of contact r-surgery along Legendrian knots in contact 3-manifolds. They also showed how (at least in principle) to convert any contact r-surgery into a sequence of contact (\pm 1)-surgeries, and used this to prove that any (closed) contact 3-manifold can be obtained from the standard contact structure on S^3 by a sequence of such contact (\pm 1)-surgeries. In the present paper, we give a shorter proof of that result and a more explicit algorithm for turning a contact r-surgery into (\pm 1)-surgeries. We use this to give explicit surgery …

Quasipositivity Problem For 3-Braids, Stepan Yu. Orevkov

Turkish Journal of Mathematics

A braid is called quasipositive if it is a product of conjugates of standard generators of the braid group. We present an algorithm deciding if a given braid with three strings is quasipositive or not. The complexity (the time of work) of our algorithm is O(n^{k+1}) where n is the length of the word in standard generators representing the braid and k is the algebraic length of the braid. The algorithm is based on the Garside normal form. The problem of quasipositivity in braid groups is motivated by the topology of plane real algebraic curves (16th Hilbert's problem). In particular, …

Solution Of The Word Problem In The Singular Braid Group, Stepan Yu. Orevkov

Turkish Journal of Mathematics

Singular braids are isotopy classes of smooth strings which are allowed to cross each other pairwise with distinct tangents. Under the usual multiplication of braids, they form a monoid. The singular braid group was introduced by Fenn-Keyman-Rourke as the quotient group of the singular braid monoid. We give a solution of the word problem for this group. It is obtained as a combination of the results by Fenn-Keyman-Rourke and some simple geometric considerations based on the mapping class interpretation of braids. Combined with Corran's normal form for the singular braid monoid, our algorithm provides a computable normal form for the …

Is Mathematics Education Taking A Step Backward?, Frances Kuwahara Chinn

Humanistic Mathematics Network Journal

This paper considers the recent history of mathematics teaching.

Using Humanistic Content And Teaching Methods To Motivate Students And Counteract Negative Perceptions Of Mathematics, Roger Haglund

Humanistic Mathematics Network Journal

This paper examines the following questions: How is math commonly taught, why is it taught this way, and what are the outcomes? Who are some of the voices calling for change and what are they saying? Can a humanistic approach produce positive results in students who have learned to dislike math and have not been successful in a traditional classroom?

Taxicab Geometry As A Vehicle For The Journey Toward Enlightenment, Neil Greenspan

Humanistic Mathematics Network Journal

No abstract provided.

Tesselland: A Mathematical Oddment, Martin Glover

Humanistic Mathematics Network Journal

No abstract provided.

Bridging To Infinity, Mike Pinter

Humanistic Mathematics Network Journal

The author's own experiences as a mathematics student and teacher have influenced how he thinks about the infinite. Author Madeleine L'Engle has also shaped his thinking with her writing. The author offers some thoughts that connect some of L'Engle's writing with his experience.

Man's Cards And God's Dice: A Conceptual Analysis Of Probability For The Advanced Student, Elie Feder

Humanistic Mathematics Network Journal

No abstract provided.