Figure 1: Zindler curve. Any of the chords of equal length cuts the curve and the enclosed area into halves.Figure 2: Examples of Zindler curves with a = 8 (blue), a = 16 (green) and a = 24 (red).
A Zindler curve is a simple closed plane curve with the defining property that:
(L) All chords which cut the curve length into halves have the same length.
The most simple examples are circles. The Austrian mathematician Konrad Zindler discovered further examples, and gave a method to construct them. Herman Auerbach was the first, who used (in 1938) the now established name Zindler curve.
Auerbach proved that a figure bounded by a Zindler curve and with half the density of water will float in water in any position. This gives a negative answer to the bidimensional version of Stanislaw Ulam's problem on floating bodies (Problem 19 of the Scottish Book), which asks if the disk is the only figure of uniform density which will float in water in any position (the original problem asks if the sphere is the only solid having this property in three dimension).
Zindler curves are also connected to the problem of establishing if it is possible to determine the direction of the motion of a bicycle given only the closed rear and front tracks.[1]
^Bor, Gil; Levi, Mark; Perline, Ron; Tabachnikov, Sergei (2018). "Tire Tracks and Integrable Curve Evolution". International Mathematics Research Notices. 2020 (9): 2698–2768. arXiv:1705.06314. doi:10.1093/imrn/rny087.
A Zindlercurve is a simple closed plane curve with the defining property that: (L) All chords which cut the curve length into halves have the same length...
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Science Letters 126, 275-287 (1994). Bard E, Hamelin B, Fairbanks RG, Zindler A. Calibration of the 14C timescale over the past 30,000 years using mass...
1007/BF03015310 "Osservazioni sulla «Réclamation de priorité» del sig. Zindler." Annali di Matematica Pura ed Applicata (1898–1922) 19, no. 1 (1912):...
.90..367L. doi:10.1007/BF00384715. S2CID 140663216. Hart, Stanley R.; Zindler, Alan (1986). "In search of a bulk-Earth composition". Chemical Geology...
Asch became a professor of economics at Rutgers University, married Ruth Zindler and had two sons, Eric and David. Peter died of heart failure at age 52...
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