When Does a Circle Look Like a Square? - Kansas State University

When Does a Circle Look Like a Square? An Exploration of Taxicab Geometry

Michael B. Scott

Department of Mathematics Kansas State University

100th Annual KATM Conference, Washburn University Topeka, KS October 21-23, 2004

When Does a Circle Look Like a Square? Michael B. Scott

Introduction

1

To fully understand Euclidean geometry need some contact with non-Euclidean geometry.

Taxicab geometry is non-Euclidean geometry that has: Applications to urban geography Can be understood by anyone who has some understanding of Euclidean geometry

Following exercises adapted from Taxicab Geometry: An Adventure in Non-Euclidean Geometry by Eugene Krause (list price $5.95)

When Does a Circle Look Like a Square? Michael B. Scott

Objectives

2

Discover the effect of the "shape" of a circle when we change how we compute distance.

Discover how replacing the standard (Euclidean) distance formula with the Taxicab distance formula effects familiar axioms in Euclidean geometry. For example:

Does the SSS condition imply two triangles are congruent? Is the distance between a point and line the length of the perpendicular line segment passing through the point and the line?

Problem solving involving non-Euclidean geometry with real-life applications to urban geography.

When Does a Circle Look Like a Square? Michael B. Scott

When would I teach this?

3

Middle school and secondary students just after covering coordinate geometry and the distance formula.

When does a circle look like a square? Applications of Taxicab geometry

Students in or who have had a standard secondary geometry course

Congruence of triangles Distance from a point to a line

When Does a Circle Look Like a Square? Michael B. Scott

What is Taxicab Geometry?

4

Euclidean distance between two points (x1, y1) and (x2, y2) given by:

dE (x, y ) = (x1 - x2)2 + (y1 - y2)2.

Formula is derived from Pythagorean Theorem Taxicab geometry differs from Euclidean geometry by how we compute the distance between two points. Taxicab distance between two points (x1, y1) and (x2, y2) given by:

dT (x, y ) = |x1 - x2| + |y1 - y2|.

When Does a Circle Look Like a Square? Michael B. Scott

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