Abstract
The Banach—Tarski paradox is one of the most shocking results of mathematics. In this chapter we show how tilings of the hyperbolic plane can help us visualize the paradox. The images shown here display three congruent subsets of the hyperbolic plane. In the left image, the congruence is evident. The right image changes the viewpoint a little and changes the green to a blue shade; it is evident that the red set is congruent to its complement. Thus these sets are, simultaneously, one half and one third of the hyperbolic plane.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Wagon, S. (2010). The Banach–Tarski Paradox. In: Wagon, S. (eds) Mathematica in Action. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75477-2_20
Download citation
DOI: https://doi.org/10.1007/978-0-387-75477-2_20
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-75366-9
Online ISBN: 978-0-387-75477-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)