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http://dx.doi.org/10.7858/eamj.2020.012

A Study on the Existence of the Solution in the Isoperimetric Problem  

Lee, Hosoo (Department of Mathematics Education, Teachers College (Elementary Education Research Institute) Jeju National University)
Choi, Keunbae (Department of Mathematics Education, Teachers College (Elementary Education Research Institute) Jeju National University)
Publication Information
East Asian mathematical journal / v.36, no.2, 2020 , pp. 131-146 More about this Journal
Abstract
The isoperimetric problem is a well-known optimization problem from ancient Greek. Among plane figures with the same perimeter, which is the largest area surrounded? The answer to the question is circle. Zenodorus and Steiner's pure geometric proofs, which left a lot of achievements in this matter, looked beautiful with ideas at that time. But there was a fatal flaw in the proof. The weakness is related to the existence of the solution. In this paper, from a view of the existence of the solution, we investigate proofs of Zenodorus and Steiner and get educational implications.
Keywords
polygon; isoperimetric problem;
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1 우정호 (2006). 수학 학습-지도 원리와 방법. 서울대학교출판부, 서울.
2 최근배 (2011). 초등 영재 교수.학습을 위한 평면에서의 등주문제 내용구성 연구-기하적인 방법을 중심으로-, 한국수학교유학회지 시리즈 A <수학교육> 50(4), 435-460.
3 Anton, H. (1981). Elementary linear algebra. John Wiley & Sons, Inc. USA.
4 Blasjo, V. (2005). The Evolution of the Isoperimetric Problem, The American Mathematical Monthly, 112, pp. 526-566.   DOI
5 Edler, F. (1882). Vervollstandigung der Steinerschen elementargeometrischen Beweise fur den Satz, dass der Kreis grosseren Flacheninhalt besitzt, als jede andere ebene Figur gleich grossen Umfanges, Nachrichten von der Konigl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universitat zu Gottingen 73-80.
6 Gandz, S. (1940). Isoperimetric problems and the origin of quadratic equations, Isis 32 103-115.   DOI
7 Geiser, C. F. (1874). Zur Erinnerung an Jakob Steiner, Verhandlungen der Schweizerischen Naturforschenden Gesellschaft 56 215-251.
8 Hales, T. C. (2001). The Honeycomb Conjecture. Discrete and Computational Geometry, 25 (1): 1-22.   DOI
9 Heath, T. L. (1921) A History of Greek Mathematics, vol. 2, Clarendon Press, Oxford.
10 Kline, M. (1985), Mathematics for the Nonmathematician, Dover, New York.
11 Munkres, J. R. (1975), Topology: a first course, Prentice-Hall, Inc. Englewood Cliffs, New Jersey.
12 Perron, O. (1913). Zur Existenzfrage eines Maximums oder Minimums, Jahresber. Deutsch. Math.-Verein. 22 140-144.
13 Spivak, M. (1979), A Comprehensive Introduction to Differential Geometry, vol. 4, 2nd ed., Publish or Perish, Berkeley, CA.
14 Steiner, J. (1838). Einfache Beweise der isoperimetrischen Hauptsatze, J. Reine Angew. Math. 18 281-296.
15 Steiner, J. (1842). Sur le maximum et le minimum des figures dans le plan, sur la sphere et dans l'espace engeneral. premier memoire, J. Reine Angew. Math. 24 93-162.
16 Thomas, I. (1941). Selections Illustrating the History of Greek Mathematics, vol. 2, Harvard University Press, Cambridge.
17 Toomer, G. J. (1984). Ptolemy's Almagest, Gerald Duckworth & Co. Ltd, London.
18 Virgil (Edited by Anthony Uyl). (2016). The Aeneid, Woodstock, Ontario, Canada.
19 Weierstrass, K. Mathematische Werke, vol. 7, Mayer & Muller, Berlin, 1927.