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Volume 36, Number 3, pages 231-239 (2025)
https://doi.org/10.26830/symmetry_2025_3_231
ON DRAWING CURVES USING THE PYTHAGOREAN MEANS
Luděk Spíchal
Czech Forestry Academy Trutnov, Lesnická 9, Trutnov, Czech Republic
Email: spichal@clatrutnov.cz
Abstract: This article explores a novel geometric construction in which the Pythagorean means – arithmetic, geometric, and harmonic – serve as a foundation for generating a class of lemniscate-like curves. By interpreting classical numerical means geometrically along a circle's diameter, the author derives parametric and implicit equations for the resulting curves. These curves exhibit bilateral symmetry and bear visual similarity to the lemniscate of Bernoulli but differ fundamentally in their algebraic structure and geometric behaviour. The analysis includes transformations into polar coordinates, inversion with respect to the unit circle, and computations of area and arc length. Though lacking a traditional locus-based interpretation, the studied curves provide a compelling example of how ancient mathematical concepts can inspire new forms in algebraic geometry, with potential applications in visualization, mathematical art, and education.
Keywords: Lemniscate-like curve, Pythagorean means, lemniscate of Bernoulli
MSC 2010: 14H50, 26E60, 51M25
References:
Eves, H. (2003) Means Appearing in Geometric Figures. Mathematics Magazine, 76, 4, 292–294. https://doi.org/10.1080/0025570X.2003.11953195
Høibakk, R., Lukkassen, D., Meidell, A., and Persson, L.-E. (2018) Geometric construction of some Lehmer means. Mathematics, 6, 11, Article 251. https://doi.org/10.3390/math6110251
Langer, J. C., Singer, D. A. (2010) Reflections on the lemniscate of Bernoulli: The forty-eight faces of a mathematical gem. Milan Journal of Mathematics, 78, 1, 643–682. https://doi.org/10.1007/s00032-010-0124-5
Spíchal, L. (2023) The geometric constructions of the Greek means. Symmetry: Culture and Science, 34, 4, 407–416. https://doi.org/10.26830/symmetry_2023_4_407
Wikipedia contributors, "Lemniscate of Bernoulli", Wikipedia, The Free Encyclopaedia, (accessed June 8, 2025) https://en.wikipedia.org/w/index.php?title=Lemniscate_of_Bernoulli&oldid=1288908900.
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