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When Does a Circle Look Like a Square?An Exploration of Taxicab Geometry
Michael B. Scott
Department of MathematicsKansas State University
100th Annual KATM Conference,Washburn University
Topeka, KSOctober 21-23, 2004
When Does a Circle Look Like a Square?
Michael B. Scott
Introduction 1
To fully understand Euclidean geometry need some contactwith non-Euclidean geometry.
Taxicab geometry is non-Euclidean geometry that has:
Applications to urban geographyCan be understood by anyone who has some understanding ofEuclidean geometry
Following exercises adapted fromTaxicab Geometry: An Adventure in Non-Euclidean Geometryby 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 changehow we compute distance.
Discover how replacing the standard (Euclidean) distanceformula with the Taxicab distance formula effects familiaraxioms 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 theperpendicular line segment passing through the point and theline?
Problem solving involving non-Euclidean geometry withreal-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 coveringcoordinate 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 geometrycourse
Congruence of trianglesDistance 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 wecompute 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
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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 geometryDistance 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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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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Applications of Taxicab Geometry: Part 1a (Solution) 8
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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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Applications of Taxicab Geometry Part 2a (Solution) 12
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Applications of Taxicab Geometry Part 2a (Solution) 13
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Applications of Taxicab Geometry Part 2a (Solution) 14
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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?
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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
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When Does a Circle Look Like a Square?
Michael B. Scott
Congruence of Triangles 18
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When Does a Circle Look Like a Square?
Michael B. Scott
Congruence of Triangles 18
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When Does a Circle Look Like a Square?
Michael B. Scott
Congruence of Triangles 18
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When Does a Circle Look Like a Square?
Michael B. Scott
Distance from a point to a line 19
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When Does a Circle Look Like a Square?
Michael B. Scott
Distance from a point to a line (non-unique) 20
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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 asopposed to plugging numbers into the formula
Relevant to the Kansas state curriculum standard in geometry.
Geometric Figures and Their PropertiesMeasurement and EstimationTransformational GeometryGeometry from an Algebraic Perspective
Printable work sheets at:http://www.math.ksu.edu/∼mbscott/taxicab/If you have any questions or concerns regarding themathematics of Taxicab geometry, then please contact me.
When Does a Circle Look Like a Square?
Michael B. Scott
Contact Information
Current:Department of MathematicsKansas State UniversityManhattan, KS 66506Email: mbscott@math.ksu.eduWebpage: http://www.math.ksu.edu/∼mbscott/
Spring 2005 and later:Department of Mathematics and StatisticsCalifornia State University, Monterey Bay100 Campus Center Seaside, CA 93955-8001Email: michael b scott@csumb.eduWebpage: http://home.csumb.edu/s/scottmichaelb/world/
When Does a Circle Look Like a Square?
Michael B. Scott
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