Taxicab Geometry: Exploring Non-Euclidean Shapes and Distance

Taxicab Geometry Presented by: Whitney Chadwick, Zachary Dickinson, & Guadalupe Esquivel

Agenda • History • Definitions • Circles/Shapes • The Basics • Euclid’s Axioms • Distance • Applications • Conclusion

History • Hermann Minkowski (1864-1909) • Karl Menger - 1952 • Taxicab Geometry: An Adventure in Non-Euclidean Geometry - 1975

Definitions • Since the distance between two points is now measured by horizontal and vertical lines, it has some implications of other basic elements of geometry. • In Euclidean, there was one shortest distance between two points. In taxicab geometry, there are many…

Circles • The Euclidean definition of a circle is a set of points with a fixed distance from a point. Taxicab circles hold the same property, but look a little different. • Also, pi is no longer equal to 3.14, but 4.

Taxicab Shapes

The Basics • The Distance Formula • The City-Dweller’s Shortest Distance • The Distance Impact

Euclid’s Axioms • Review • Failed Axiom

How do you find the distance between point to line? • In Euclidean: The distance between A and D • Measured by taking a perpendicular line to the BC through point D

In Taxicab Geometry, it’s measured with vertical lines

Midset of Two Points • In Euclidean Geometry, the midset is the perpendicular bisector of the line connecting the two points

Midset of Two Points • Case 1 • Case 2 0<|m|<1

Midset of Two Points • Case 3 |m|>1 • Case 4 |m|=1

Applications • Finding the shortest distance between two places • 2D video games

Conclusion • Why is it significant? • How does it relate to our class? ?

The End! Thank you for your time!

References • http://strikes.blogs.uny.ac.id/2015/10/ • http://jwilson.coe.uga.edu/MATH7200/TaxiCab/TaxiCab.html • http://www.math.iastate.edu/thesisarchive/MSM/JanssenMSMSS07.pdf • http://taxicabgeometry.net/

Taxicab Geometry: Exploring Non-Euclidean Shapes and Distance