Ten Points on a Cubic

Abstract

The 16-year old Blaise Pascal found an incidence relation that holds when six points lie on a conic. A century later, Braikenridge and Maclaurin extended Pascal’s result to a straightedge construction that characterizes when six points lie on a conic. Nearly 400 years later, we develop a straightedge construction to check whether ten points lie on a cubic curve.

MSC:

• 14H50
• 51A20

Acknowledgment

We thank Bernd Sturmfels for suggesting the problem to us and Mike Roth for helpful discussions. We are grateful to J. Chris Fisher for suggesting an alternate approach to Construction 2 and for encouraging feedback. The computer algebra system MAGMA [4] was extremely helpful and all figures in the paper were produced using GeoGebra [16], which is a great resource for developing geometric intuition. We thank the members of the Editorial Board of the MONTHLY for their suggestions and advice. We also thank the anonymous referees for carefully reading the manuscript and providing many helpful suggestions.

Notes

1 Apparently, Hadamard was paraphrasing Paul Painlevé [19], the French mathematician and statesman who served twice as Minister of War and twice as Prime Minister of France.

2 There seems to be some controversy about the spelling of Steiner’s first name. The authoritative version of his collected works [23] gives the author’s name spelled with a k and the subject’s name spelled with a c. Perhaps this confusion is common among people whose work is important enough to be translated into many languages.

3 https://www.cut-the-knot.org/pythagoras/PPower.shtml

Additional information

Notes on contributors

Will Traves

WILL TRAVES(https://orcid.org/0000-0002-8115-1243) nearly failed freshman physics at Queen’s University but went on to become a professor and past chair of the Naval Academy mathematics department. His research interests include geometry, data science, and operations research.

United States Naval Academy, Mail Stop 9E, Annapolis, MD 21402, USA

[email protected]

David Wehlau

David Wehlau(https://orcid.org/0000-0002-0272-8404) received his Ph.D. from Brandeis University and is a professor and past head of the Department of Mathematics and Computer Science at the Royal Military College of Canada. He is also a professor at Queen’s University and enjoys being able to work with mathematics students at both institutions.

Royal Military College of Canada, PO Box 17000 Stn Forces, Kingston, ON, K7K 7B4, Canada

[email protected]