\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 simple, fixed constants. \textbf{Discovery 1 (Parametric Scaling Laws):} When normalized against the Parametric Radius ($a$) or its squared/cubed area/volume equivalents, the properties of the cardioid are governed by the simple scaling factors $8$, $\frac{3}{2}$, $2$, and $\frac{32}{5}$. \begin{center} \begin{tabular}{lccc} \toprule \textbf{Property} & \textbf{Formula} & \textbf{Constant Ratio} & \textbf{Value} \\ \midrule Arc Length (1D Perimeter) & $L = 8a$ & $L / a$ & $8$ \\ Area (2D Enclosed) & $A = \dfrac{3\pi a^2}{2}$ & $A / (\pi a^2)$ & $\dfrac{3}{2}$ (1.5) \\ Volume of Solid (3D) & $V = \dfrac{8\pi a^3}{3}$ & $V / \left(\dfrac{4}{3}\pi a^3\right)$ & $2$ \\ Surface Area of Solid (3D) & $S = \dfrac{32\pi a^2}{5}$ & $S / (\pi a^2)$ & $\dfrac{32}{5}$ (6.4) \\ \bottomrule \end{tabular} \end{center} \section{Internal Volume Decomposition} The solid of revolution exhibits a pronounced asymmetry when partitioned by the $y$-$z$ plane, dividing the domain at $\theta = \pi/2$. We define the regions: \begin{itemize} \item \textbf{Trench (Near Cusp):} $0 \le \theta \le \pi/2$ \item \textbf{Bulge (Away from Cusp):} $\pi/2 < \theta \le \pi$ \end{itemize} \textbf{Discovery 2 (1:15 Volume Split):} The volume of the cardioid solid of revolution is distributed between the Trench and Bulge sections in an exact 1:15 ratio, confirming a sharp geometric asymmetry near the cusp. The total volume of the solid is $V_{\text{Total}} = \dfrac{8\pi a^3}{3}$. \begin{center} \begin{tabular}{lccc} \toprule \textbf{Region} & \textbf{Exact Volume ($a=1$)} & \textbf{Fraction of $V_{\text{Total}}$} & \textbf{Percentage} \\ \midrule Trench ($V_T$) & $\dfrac{\pi}{6}$ & $\mathbf{\dfrac{1}{16}}$ & $6.25\%$ \\ Bulge ($V_B$) & $\dfrac{5\pi}{2}$ & $\mathbf{\dfrac{15}{16}}$ & $93.75\%$ \\ \midrule \textbf{Sum} & $\dfrac{8\pi}{3}$ & $\mathbf{1}$ & $100\%$ \\ \bottomrule \end{tabular} \end{center} This demonstrates that $V_B = 15 \times V_T$, a direct result of the geometry being heavily weighted away from the origin-aligned cusp. \end{document}