Exact Angle Trisection with Straightedge and Compass
by Secondary Geometry

Tran Dinh Son

doi:https://doi.org/10.14445/22315373/IJMTT-V69I5P502

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 69 | Issue 5 | Year 2023 | Article Id. IJMTT-V69I5P502 | DOI : https://doi.org/10.14445/22315373/IJMTT-V69I5P502

Exact Angle Trisection with Straightedge and Compass by Secondary Geometry

Tran Dinh Son

Received	Revised	Accepted	Published
22 Mar 2023	29 Apr 2023	05 Oct 2023	22 May 2023

Citation :

Tran Dinh Son, "Exact Angle Trisection with Straightedge and Compass by Secondary Geometry," International Journal of Mathematics Trends and Technology (IJMTT), vol. 69, no. 5, pp. 9-24, 2023. Crossref, https://doi.org/10.14445/22315373/IJMTT-V69I5P502

Abstract

Angle trisection is a classical problem of straightedge and compass construction from the ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass. There are three classical problems in the ancient Greek mathematics which were extremely influential in the development of geometry. These problems were those of squaring the circle, doubling the cube and trisecting an angle. This Thesis focuses on the problem of trisecting an arbitrary angle. It is possible to trisect certain angles, e.g. a right angle. It is difficult to give an accurate date as to when the problem of trisecting an angle first appeared. Result of this research paper is an exact solution for the thousand-year challenge “Trisecting the Angle” by a construction with only a straightedge and a compass by means of the secondary Geometry.

Keywords

Angle trisection, Divide angle into 3 equally small angle, Divide angle with straightedge and compass, Divide angle by secondary geometry, Trisecting the angle.

References

[1] Andrew M. Gleason, “Angle Trisection, the Heptagon, and the Triskaidecagon,” The American Mathematical Monthly, vol. 95, no. 3, pp. 185-194, 2018.
[CrossRef] [Google Scholar] [Publisher Link]
[2] M.L. Wantzel, “Research on the Means of Knowing Whether a Problem in Geometry can be Solved with Ruler and Compass,” pp. 366–372, 1837.
[Publisher Link]
[3] Underwood Dudley, The Trisectors, The Mathematical Association of America, 1994.
[Google Scholar]
[4] Robert C.Yates, The Trisection Problem, First Published in 1942.
[Google Scholar] [Publisher Link]
[5] Harold Scott Macdonald Coxeter, Introduction to Geometry (2nd Ed.), John Wiley & Sons, 1967.
[6] J.L. Heiberg, Euclid's Elements, Cambridge University Press, 1883.
[7] S. Lang, Geometry: Euclid and Beyond, Springer Science & Business Media, 1997.
[8] R.B. Nelsen, Proofs without Words II, Mathematical Association of America, 2001.
[9] W. Dunham, The Mathematical Universe, John Wiley & Sons, 2007.
[10] M. Field, and J. Jones, “Impossible Constructions,” Mathematical Gazette, vol. 56, no. 397, 15-22, 1972.
[11] J.B. Kennedy, “Angle Trisection, The Heptagon, and The Triskaidecagon,” American Mathematical Monthly, vol. 77, no. 8, pp. 851- 869, 1970.
[12] Mario Livio, The Golden Ratio: The Story of Phi, The World's Most Astonishing Number, Broadway Books, 2003.
[13] S.A. Stahl, “The Impossible Trisection Problem,” The College Mathematics Journal, vol. 24, no. 2, pp. 100-110, 1993.
[14] W. Szmielew, The Problem of Trisecting an Angle, Mathematica Slovaca, vol. 44, no. 3, pp. 321-342, 1994.
[15] Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001.
[16] B.L. Van der Waerden, A History of Algebra, Springer Science & Business Media, 1991.
[17] I.M. Yaglom, and V.G. Boltyanskii, Geometric Transformations II, Random House, 1962.
[18] J.K. Yu, Golden Section: Nature's Greatest Secret, World Scientific, 2005.
[19] H.J.M. Bos, and J. Smit, “Euclides Triangles,” New Archives for Mathematics, vol. 9, no. 4, pp. 227-235, 2008.
[20] Florian Cajori, History of Elementary Mathematics, The Macmillan Company, 1898.
[Google Scholar]
[21] H.W. Guggenheimer, Differential Geometry, Dover Publications, 1997.
[22] M. Hazewinkel, Encyclopedia of Mathematics, Springer Science & Business Media, 2001.
[23] W.R. Knorr, The Ancient Tradition of Geometric Problems, Dover Publications, 1986.
[24] Eli Maor, Trigonometric Delights, Princeton University Press, 1998.