Geometry, e.g., the shape of the environment, can be used by numerous animal species to orientate, but data concerning the mouse are lacking. We addressed the question of whether mice are capable of using geometry for navigating. To test whether aging could affect searching strategies, we compared adult (3- to 5-mo old) and aged (20- to 21-mo old)…

There are four Euclidean centres of a triangle--the circumcentre, the centroid, the incentre and the orthocentre. In this article, the authors prove the following: if the centre is the incentre (resp. orthocentre) then there exists a triangle with given distances of its vertices from its incentre (resp. orthocentre). They also consider uniqueness…

Geometric constructions have previously been shown that can be interpreted as rays of light trapped either in polygons or in conics, by successive reflections. The same question, trapping light in closed Fermat curves, is addressed here. Numerical methods are used to study the behaviour of the reflection points of a triangle when the degree of the…

During the 17th century, Baroque decoration used anamorphism to combine actual architectural elements with illusionistic painting. When viewed from a particular point in space, the architecture blends with painting to form a combined image. In this article, Julian Beever, a leading anamorphic pavement artist, explains to the author the principles…

Mathematical historians place Heron in the first century. Right-angled triangles with integer sides and area had been determined before Heron, but he discovered such a "non" right-angled triangle, viz 13, 14, 15; 84. In view of this, triangles with integer sides and area are named "Heron triangles." The Indian mathematician Brahmagupta, born in…

The author has always been fascinated by the title identity. It's charming and simple, as well as easy to believe after pressing a few calculator keys. Several fine proofs have appeared in the literature, including several proofs without words. His own earlier proof is trigonometric, and he has often been dissatisfied with not being able to…

Technology is a powerful tool in assisting students in problem solving by allowing for multiple representations. The vignette offered in this article provides insight into ways to solve open-ended problems using multiple technologies.

This article describes two alternative coordinate systems and their use in graphing conic sections. This alternative graph paper helps students explore the idea of eccentricity using the definitions of the conic sections.

From a geometry course for prospective teacher, several methods for dividing a circle into three parts using only Euclidean geometry are explored.

Building on their knowledge of the three possible outcomes of solving 2x2 systems of equations, students use three-dimensional geometric figures to investigate the eight possible outcomes for solving 3x3 systems of equations.

Real numbers are often a missing link in mathematical education. The standard working assumption in calculus courses is that there exists a system of "numbers", extending the rational number system, adequate for measuring continuous quantities. Moreover, that such "numbers" are in one-to-one correspondence with points on a "number line". But…

The authors hope to show how a geometric insight can add to the richness of our students' experiences when they first encounter the solutions to two equations in two unknowns.

The current mantra in education is "technology, technology, technology." Many teachers and prospective teachers become frustrated with their lack of knowledge regarding the "appropriate" use of technology in the classroom. Prospective teachers need training in their education to understand how technology can be used "appropriately" in the…

There are over 400 proofs of the Pythagorean Theorem. Some are visual proofs, others are algebraic. This paper features several proofs of the Pythagorean Theorem in different cultures--Greek, Chinese, Hindu and American. Several interactive websites are introduced to explore ways to prove this beautiful theorem. (Contains 8 figures.)

This article offers an introductory activity for the limit concept with a geometrical and historical foundation. A connection among Geometry, Measurement and Calculus is highlighted with the help of technology. The geometrical drawing, measurement and graphing capabilities of both TI-89 and Geometer's Sketchpad make it possible for students to…