Constructions of squaring the circle, doubling the cube and angle trisection

  • Veselin M. Rmuš
DOI: https://doi.org/10.5937/vojtehg65-13404
Keywords: construction, squaring the circle, doubling the cube, angle trisection, coordinate system, unit circle, rotation, proportion,

Abstract


The constructions of three classical Greek problems (squaring the circle, doubling the cube and angle trisection) using only a ruler and a compass  are considered unsolvable. The aim of this article is to explain the original methods of construction of the above-mentioned problems, which is something new in geometry. For the construction of squaring the circle and doubling the cube the Thales' theorem of proportional lengths has been used, whereas the angle trisection relies on a rotation of the unit circle in the Cartesian coordinate system and the axioms of angle measurement. The constructions are not related to the precise drawing figures in practice, but the intention is to find a theoretical solution, by using a ruler and a compass, under the assumption that the above-mentioned instruments are perfectly precise.


Author Biography

Veselin M. Rmuš

Teacher of mathematics and descriptive geometry, educational consultant, social and cultural worker, humanist.Veselin Rmuš was born on 12 May 1945 in Berane, Montenegro where he finished elementary and high school. He graduated from the Faculty of Sciences and Mathematicsin Priština. After graduation, he taught mathematics and descriptive geometry at “Panto Mališić” high school. From1979 to 2004 he worked at the Educational and Pedagogica lInstitute of the Ministry of Education and Science ofMontenegro as an educational consultant, supervisor andinspector for Mathematics teachers in the upper grades ofelementary school for municipalities in the northern Montenegro. He also taught Mathematics at the Vocational school Berane until his retirement. From 1988 to 2010, as a volunteer, he was a president of the local branch of the Red Cross in Berane. He has also been involved in scientific research in the field of mathematics. Veselin Rmuš held the following lectures (elaborations): “One formula for calculating the number π” at the Ninth Congress of Mathematiciansin Petrovac (1995) and “Areas and volumes of regular polyhedrons” at the Tenth Congress of Mathematicians in Belgrade (2001). His most notable published papers are the following: Simpson’s rule taught in regular and extramathematics classes, One proof of the Pythagorean theorem, Squaring a polygon, etc. He was a member of theCommission on curriculum development of Mathematicsfor primary schools. He made a major contribution to thepromotion of the Movement “Youth Science” and for this work he received the Golden plaque. He is the author of the book “The Most Beautiful Chess Games of Anto Rmuš” published in 2012, in which themost important chess games of his prematurely deceased son Anto, an international chess master, were collected.

References

Courant, R., Robbins, H., 1973. What is Mathematics? An Elementary Approach to Ideas and Methods. Oxford University Press, London.

Dolićanin, Ć., 1984. Маthematics. Institute for Textbooks and Educational Materials, Priština.

Ostojić, О., 1980. Theoretical Basis of Initial Mathematics Teaching, Handbook for Math Teachers, Republic Institute for the Improvement of Education, Titograd.

Published
2017/06/29
Issue
Vol. 65 No. 3 (2017): July – September
Section
Original Scientific Papers

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