UMass Lowell CyberEd
92.419 Computer Algebra with Mathematica
Kenneth M. Levasseur
Department of Mathematical Sciences
University of Massachusetts Lowell
Lowell, MA 01852

Taxicab Geometry

Subject

Mathematics

Topic

Taxicab Geometry - an alternative geometry based on a distance function different from the usual Eucidean distance.

Taxicabs in a city often must travel along a grid of squares (city streets), making distances between points different from the Euclidean "straight line" distance. Other grid systems can also be considered. One possiblity is a grid of triangles. Sets of points that generalize classicals sets such as circles and hyperbolas have interesting properties.

Reference(s)

Sowell, Katye O., "Taxicab Geometry - A New Slant," Mathematics Magazine , 62(1989) 238-248.
Krause, Eugene F., Taxicab Geometry , Addison-Wesley, Menlo Park, 1975.
Gardner, Martin, "Mathematical Games," Scientific American , 243(Nov. 1980) 18-30.
Polya, G, Induction and Analogy in Mathematics , Princeton U Press: Princeton NJ, 1954.

Project Idea(s)

Sowell, in [1], already displays certain sets of constant distance including the iso-taxi ellipse, iso-taxi hyperbola and iso-taxi hyperbola. To quote Sowell, "Other taxicab geometric sets might also be explored.." You might attempt "traffic center problem" [4, page 145] in this geometry and use the same graphical capabilities for visualization.

Geometry on grid of hexagons could be explored. Sowell mentions this possibility but gives no specific reference. [2] or [3] may have more information.

Prerequisite Mathematics

Geometry and the ability to generalize geometric notions to arbitary distance functions.

Required Programming Level

Moderate. The key may be to represent the grid of triangles in an efficient way.

Key Words

Geometry, Distance, Graphics

Reviewer

K. M. Levasseur ( Kenneth_Levasseur@uml.edu )