Why is the Legendre Transformation Involutive?

TL;DR

This paper provides a simple, symmetric pictorial explanation for why the Legendre transformation is an involution, clarifying its fundamental mathematical property.

Contribution

It introduces a visual, intuitive argument demonstrating the involutive nature of the Legendre transformation, enhancing understanding of its symmetry.

Findings

The Legendre transform is involutive due to a symmetric pictorial property.

A simple geometric argument explains the involution.

The approach clarifies the mathematical symmetry of the transformation.

Abstract

The question posed in the title is answered in terms of a simple pictorial argument that is manifestly symmetric between the two functions that are Legendre transform of each other.