Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T12:55:38.616Z Has data issue: false hasContentIssue false
  • English
  • Français

Inscribed squares and square-like quadrilaterals in closed curves

Part of: Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  26 February 2010

Walter Stromquist
Walter Stromquist
Affiliation:
Daniel H. Wagner, Associates, Station Square Two, Paoli, PA 19301, U.S.A..
Get access

Abstract

We show that for every smooth curve in Rn, there is a quadrilateral with equal sides and equal diagonals whose vertices lie on the curve. In the case of a smooth plane curve, this implies that the curve admits an inscribed square, strengthening a theorem of Schnirelmann and Guggenheimer. “Smooth” means having a continuously turning tangent. We give a weaker condition which is still sufficient for the existence of an inscribed square in a plane curve, and which is satisfied if the curve is convex, if it is a polygon, or (with certain restrictions) if it is piecewise of class C1. For other curves, the question remains open.

MSC classification

Secondary: 55M99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 36 , Issue 2 , December 1989 , pp. 187 - 197
Copyright
Copyright © University College London 1989

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.Klee, Victor. Some unsolved problems in plane geometry. Math. Magazine, 52 (1979), 131–145.CrossRefGoogle Scholar
2.Klee, Victor and Wagon, Stan. New and old unsolved problems in plane geometry and number theory (Mathematical Association of America, to appear).Google Scholar
3.Emch, A.. Some properties of closed convex curves in a plane. Amer. J. of Math., 35 (1913), 407–412.CrossRefGoogle Scholar
4.Emch, A.. On the medians of a closed convex polygon. Amer. J. of Math., 37 (1915), 19–28.CrossRefGoogle Scholar
5.Schnirelmann, L. G.. O nekotoryh geometriceskih svoistvah zamknutyh krivyh (On certain geometrical properties of closed curves). Sbornik rabot matematičeskogo razdela sekcii estestvennyh i točnyh nauk Komakademii, Moskva (1929); Uspehi Matematičeskih Nauk, 10 (1944), 34–44.Google Scholar
6.Guggenheimer, H.. Finite sets on curves and surfaces. Isreal J. Math., 3 (1965), 104–112.CrossRefGoogle Scholar
7.Jerrard, R. P.. Inscribed squares in plane curves. Trans. Amer. Math. Soc, 98 (1961), 234–241.CrossRefGoogle Scholar
8.Fenn, Roger. The table theorem. Bull. London Math. Soc, 2 (1970), 73–76.CrossRefGoogle Scholar
9.Tucker, A. W.. Some topological properties of disk and sphere. In Proc. First Canadian Mathematical Congress, Montreal, 1945 (Univ. Toronto Press, 1946), pp. 285–309.Google Scholar
10.Lefschetz, S.. Introduction to Topology (Princeton University Press, 1949).CrossRefGoogle Scholar
11.Meyerson, M.. Convexity and the table theorem. Pacific J. Math., 97 (1981), 167–169.CrossRefGoogle Scholar
12.Meyerson, M.. Remarks on Fenn's “The Table Theorem” and Zaks' “The Chair Theorem”. Pacific J. Math., 110 (1984), 167–169.CrossRefGoogle Scholar
13.Meyerson, M.. Balancing acts. Topology Proceedings, 6 (1981), 59–75.Google Scholar