Summary
It is known that the three classical geometric construction problems introduced by the ancient Greeks: trisecting an angle, squaring a circle, and doubling a cube, cannot be solved using the Euclidean tools. However, ancient Greek mathematicians solved these three problems using other means. We present solutions to the doubling-the-cube problem using ideas that go beyond the Euclidean tools, and we consider generalizations to higher dimensions.
Acknowledgments
We would like to thank Jeffrey Hatley and Karl Zimmermann for helpful discussions about this paper and for making useful comments on an earlier draft. We also thank two anonymous referees and the editor for their helpful suggestions.
Notes
1 Note that the online version of this article has color diagrams.
Additional information
Notes on contributors
Julius Barbanel
Julius Barbanel received his Ph.D. from the State University of New York, Buffalo, in 1979. He spent almost all of his academic career at Union College, from which he retired in 2015. He began his mathematical research in set theory and later studied fair division. He also developed interests in ancient Greece, and in particular their mathematics. He enjoys cycling and cross-country skiing, and is presently trying to learn ancient Greek.