\documentclass{article} \usepackage{amsmath, amssymb, booktabs} \usepackage{geometry} \geometry{a4paper, margin=1in} \title{Novel Geometric Ratios of the Cardioid Solid of Revolution} \author{} \date{} \begin{document} \maketitle \begin{abstract} This note presents novel scaling laws and volume decomposition ratios derived from the solid of revolution generated by the cardioid $r = a(1 - \cos\theta)$ revolved about the polar axis. Using the \textbf{Parametric Radius} ($a$) as the fundamental unit of scaling, we establish a set of simple integer and fractional constants for the 1D, 2D, and 3D properties of the curve and its solid. Most notably, we quantify the internal asymmetry of the solid's volume, revealing a clean 1:15 distribution. \end{abstract} \section{The Parametric Scaling Laws} The cardioid is defined in polar coordinates by $r = a(1 - \cos\theta)$, where $a$ is the \textbf{Parametric Radius}. All geometric properties of the shape are multiples of this radius and s...