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: q 2 2 dE (x, y) = (x1 − x2) + (y1 − y2) .
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 What is Taxicab Geometry? 5
6
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When Does a Circle Look Like a Square? Michael B. Scott Activities 6
Let’s do some problems
When does a circle look like a square? Applications of Taxicab geometry Alice and Bruno problem and Ideal City Police dept. High school district boundary problem Further investigations Congruence of triangles in Taxicab geometry Distance between a point and a line in Taxicab geometry
When Does a Circle Look Like a Square? Michael B. Scott When does a circle look like square? (Solution) 7
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When Does a Circle Look Like a Square? Michael B. Scott When does a circle look like square? (Solution) 7
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When Does a Circle Look Like a Square? Michael B. Scott Applications of Taxicab Geometry: Part 1a (Solution) 8
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When Does a Circle Look Like a Square? Michael B. Scott Applications of Taxicab Geometry: Part 1a (Solution) 8
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When Does a Circle Look Like a Square? Michael B. Scott Applications of Taxicab Geometry: Part 1b (Solution) 9
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When Does a Circle Look Like a Square? Michael B. Scott Applications of Taxicab Geometry Part 2a (Solution) 10
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When Does a Circle Look Like a Square? Michael B. Scott Applications of Taxicab Geometry Part 2a (Solution) 11
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When Does a Circle Look Like a Square? Michael B. Scott Applications of Taxicab Geometry Part 2a (Solution) 12
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When Does a Circle Look Like a Square? Michael B. Scott Applications of Taxicab Geometry Part 2a (Solution) 13
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When Does a Circle Look Like a Square? Michael B. Scott Applications of Taxicab Geometry Part 2a (Solution) 14
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When Does a Circle Look Like a Square? Michael B. Scott Applications of Taxicab Geometry Part 2B (Solution) 15
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When Does a Circle Look Like a Square? Michael B. Scott Applications of Taxicab Geometry Part 2c (Solution) 16
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When Does a Circle Look Like a Square? Michael B. Scott Applications of Taxicab Geometry Part 2c (Solution) 17
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When Does a Circle Look Like a Square? Michael B. Scott Congruence of Triangles 18
6 C A0 B0 £ £ £ £ £ £ AB £ £ £ £ Not Congruent! £ £ - C 0
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When Does a Circle Look Like a Square? Michael B. Scott Congruence of Triangles 18
6 C A0 B0 £ £ £ £ £ £ AB £ £ £ £ Not Congruent! £ £ - C 0
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When Does a Circle Look Like a Square? Michael B. Scott Congruence of Triangles 18
6 C A0 B0 £ £ £ £ £ £ AB £ £ £ £ Not Congruent! £ £ - C 0
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When Does a Circle Look Like a Square? Michael B. Scott Congruence of Triangles 18
6 C A0 B0 £ £ £ £ £ £ AB £ £ £ £ Not Congruent! £ £ - C 0
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When Does a Circle Look Like a Square? Michael B. Scott Distance from a point to a line 19
6 A L ¡ ¡ d (A, L) = 6.5 ¡ E s T ¡ s ¡ D ¡ s ¡ ¡C ¡ ¡ s - ¡ ¡ ¡ ¡ ¡ ¡ ¡ B ¡ s ¡ ¡ ?
When Does a Circle Look Like a Square? Michael B. Scott Distance from a point to a line (non-unique) 20 L 6 A
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When Does a Circle Look Like a Square? Michael B. Scott Follow Up 21
Notice it’s easier to compute Taxicab distance using graph as opposed to plugging numbers into the formula Relevant to the Kansas state curriculum standard in geometry. Geometric Figures and Their Properties Measurement and Estimation Transformational Geometry Geometry from an Algebraic Perspective Printable work sheets at: http://www.math.ksu.edu/∼mbscott/taxicab/ If you have any questions or concerns regarding the mathematics of Taxicab geometry, then please contact me.
When Does a Circle Look Like a Square? Michael B. Scott Contact Information
Current: Department of Mathematics Kansas State University Manhattan, KS 66506 Email: [email protected] Webpage: http://www.math.ksu.edu/∼mbscott/
Spring 2005 and later: Department of Mathematics and Statistics California State University, Monterey Bay 100 Campus Center Seaside, CA 93955-8001 Email: michael b [email protected] Webpage: http://home.csumb.edu/s/scottmichaelb/world/
When Does a Circle Look Like a Square? Michael B. Scott