등주문제에서 해의 존재성 고찰 = A Study on the Existence of the Solution in the Isoperimetric Problem

다국어 초록 (Multilingual 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.

참고문헌 (Reference)

1 최근배, "초등 영재 교수․학습을 위한 평면에서의 등주문제 내용구성 연구 - 기하적인 방법을 중심으로 -" 한국수학교육학회 50 (50): 441-466, 2011

2 우정호, "수학 학습-지도 원리와 방법" 서울대학교출판부 2006

3 Perron, O., "Zur Existenzfrage eines Maximums oder Minimums" 22 : 140-144, 1913

4 Geiser, C. F, "Zur Erinnerung an Jakob Steiner" 56 : 215-251, 1874

5 Edler, F, "Vervollständigung der Steinerschen elementargeometrischen Beweise für den Satz, daß der Kreis grösseren Flächeninhalt besitzt, als jede andere ebene Figur gleich grossen Umfanges, Nachrichten von der Königl"

6 Munkres, J. R, "Topology: a first course" Prentice-Hall, Inc 1975

7 Hales, T. C., "The Honeycomb Conjecture" 25 (25): 1-22, 2001

8 Blasjo, V, "The Evolution of the Isoperimetric Problem" 112 : 526-566, 2005

9 Virgil, "The Aeneid" 2016

10 Steiner, J., "Sur le maximum et le minimum des figures dans le plan, sur la sph`ere et dans l’espace engeneral. premier memoire" 24 : 93-162, 1842

11 Thomas, I, "Selections Illustrating the History of Greek Mathematics, vol. 2" Harvard University Press 1941

12 Toomer, G. J., "Ptolemy’s Almagest" Gerald Duckworth & Co. Ltd 1984

13 Weierstrass, K, "Mathematische Werke, vol. 7" Mayer & Muller 1927

14 Kline, M., "Mathematics for the Nonmathematician" Dover 1985

15 Gandz, S, "Isoperimetric problems and the origin of quadratic equations" 32 : 103-115, 1940

16 Anton, H., "Elementary linear algebra" John Wiley & Sons, Inc 1981

17 Steiner, J, "Einfache Beweise der isoperimetrischen Hauptsatze" 18 : 281-296, 1838

18 Heath, T. L., "A History of Greek Mathematics, vol. 2" Clarendon Press 1921

19 Spivak, M, "A Comprehensive Introduction to Differential Geometry, vol. 4" Publish or Perish 1979