Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-04T21:32:13.204Z Has data issue: false hasContentIssue false
  • English
  • Français

The quartic equation: alignment with an equivalent tetrahedron

Published online by Cambridge University Press:  23 January 2015

R. W. D. Nickalls
R. W. D. Nickalls*
Affiliation:
5 Elm Bank Drive, Mapperley Park, Nottingham, NG3 5AL, UK, e-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

The lower polynomials are inextricably linked to the symmetries of polyhedra and Platonic solids [1, 2, 3], and the quartic is no exception; its alter ego is the regular tetrahedron [4]. In this article we present a solution to the problem of aligning the vertices of a tetrahedron with the roots of a particular quartic. After establishing the size of a quartic-equivalent tetrahedron, we derive a triple-angle expression for the alignment rotation, analogous to that for the cubic [5].

Type
Articles
Information
The Mathematical Gazette , Volume 96 , Issue 535 , March 2012 , pp. 49 - 55
Copyright
Copyright © The Mathematical Association 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Klein, F., Lectures on the icosahedron and the solution of equations of the fifth degree (2nd revised edition of the 1888 English translation by Morrice, G. G.), Dover Publications (1956).Google Scholar
2. Toth, G., Glimpses of algebra and geometry, Springer (2002).Google Scholar
3. Connes, A., Symmetries, Pour la Science, no. 292, (February 2002) [French]. English translation in: European Mathematical Society Newsletter, No. 54 (December 2004) pp. 11–18. www.ems-ph.org/joumals/newsletter/pdf/2004-12-54.pdf Google Scholar
4. Chalkley, R., Quartic equations and tetrahedral symmetries, The Mathematics Magazine 48 (1975) pp. 211–215.Google Scholar
5. Nickalls, R. W. D., A new approach to solving the cubic: Cardan's solution revealed, Math. Gaz. 77 (November 1993) pp. 354–359.Google Scholar
6. Nickalls, R. W. D., The quartic equation: invariants and Euler's solution revealed, Math. Gaz. 93 (March 2009), pp. 66–75.Google Scholar