diff --git a/math/butterfly.tex b/math/butterfly.tex
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+\documentclass{standalone}
+\usepackage{tikz}
+\usepackage{pgfplots}
+\pgfplotsset{width=7cm,compat=1.8}
+\usepgfplotslibrary{polar}
+
+\title{Butterfly curve}
+\begin{document}
+	\begin{tikzpicture}
+		\def\a{1}
+		\def\b{4}
+		\def\c{12}
+		\def\A{2}
+		\def\B{5}
+		\def\pig{3.14}
+		%
+		\begin{scope}
+			\node at (2.5,0) {The Butterfly curve: Fay's polar equation};
+		\end{scope}
+		%
+		\begin{scope}[shift={(0,-6.5)}]
+			\begin{polaraxis}[grid=none,hide y axis]
+				\addplot+[mark=none,domain=0:3600,samples=2000] {e^cos(\a * x)-\A * cos(\b * x) + (sin(x / \c))^\B};
+			\end{polaraxis}
+			\node at (2.5,-1) {$\rho = e^{\cos \vartheta} - 2 \cos (4 \theta) + \sin^5 (\vartheta / 12)$};
+		\end{scope}
+		%
+		\begin{scope}[shift={(0,-15)}]
+			\node at (2.5,6.5) {The Butterfly curve: Geum and Kim's equation};
+			\begin{polaraxis}[grid=none,hide y axis]
+				\addplot+[mark=none,domain=0:3600,samples=2000,color=red] {e^sin(\a * x)-\A * cos(\b * x) + (sin((2*x - \pig)/ (2*\c)))^\B};
+			\end{polaraxis}
+			\node at (2.5,-1) {$\rho = e^{\sin \vartheta} - 2 \cos (4 \theta) + \sin^5 \left ( \frac{2\vartheta -\pi}{24} \right )$};
+		\end{scope}
+	\end{tikzpicture}
+\end{document}
\ No newline at end of file
diff --git a/math/pdf/butterfly.pdf b/math/pdf/butterfly.pdf
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