A Hacker News post explores representing inequalities through geometry, moving beyond algebra. The author, inspired by a 1985 image, develops animations to visualize the HM-AM-GM-QM inequality chain. This fundamental concept, taught in schools, compares different means: the Harmonic Mean (HM) for average speeds, the Geometric Mean (GM) for growth rates, the Arithmetic Mean (AM) for simple averages, and the Quadratic Mean (QM) or Root Mean Square.

The post demonstrates these using circles and a semicircle, showing how the GM (a leg) is always less than the AM (hypotenuse) in a right triangle formed by two circles, and how the QM (hypotenuse) exceeds the AM unless the numbers are equal. The HM emerges as the altitude within the semicircle. These visualizations provide a concrete geometric intuition for abstract algebraic concepts.