On drawing curves using the Pythagorean means

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.