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\begin{document}

\noindent I. {\it Jones Polynomial; Witten's Path-integral Formulation}

$$V_K(t) = (t+t^3-t^4)(t^{\frac{1}{2}}+t^{-\frac{1}{2}})$$


$$V_K(e^{\frac{2\pi i}{k+2}}) = -\frac{1}{Z}\int_{\mathcal A} 
\mbox{Tr} \left (  P \exp \oint_KA\right )e^{\frac{ik}{4\pi}CS(A)}\,\mathcal{D}A $$


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\noindent II. {\it Commutation relation}


$$C_{ijk}\eta^{kl}C_{lmn} = C_{mjk}\eta^{kl}C_{lin}$$
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\noindent III. {\it Braid Relation}

 
$$R_{12}R_{23}R_{12} = R_{23}R_{12}R_{23}$$
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\noindent IV. {\it Equations for Lorenz attractor:} 
$$\frac{\ds dx}{\ds dt} = \sigma(y-x)$$
$$\frac{\ds dy}{\ds dt} = x(\rho-z) -y$$
$$\frac{\ds dz}{\ds dt} = xy - \beta z$$
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\noindent V. {\it Black hole radius}


$$r_S=2Gm/c^2$$\vs





\noindent VI. {\it Platonic Solids} 
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\noindent VII. {\it Golden mean  
= partial fraction expansion:}
$$\lim_{n\rightarrow\infty}\frac{F_{n+1}}{F_n} = {1}+ \frac{\ds 1}{1+\dots} $$
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\noindent VIII. {\it Babylonian calculation of $\sqrt{2}$}

$$1;24;51;10 = 1.414213 $$
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\newpage
\noindent IX. {\it Pythagorean Theorem}

$$c^2 = a^2 + b^2$$
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\noindent X. {\it Archimedes' calculation of volume, area of sphere} 

$$v = \frac{2}{3}V$$\vs

$$a = \frac{2}{3}A$$
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\noindent XI. {\it Euler characteristic, Gauss-Bonnet Theorem} 




$$2\pi\chi = \int_M K~dA$$
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\noindent XII. {\it Aharonov-Bohm Effect} 


%$$\int_{C_1}\vec{A}\cdot d\vec{\ell} - \int_{C_2}\vec{A}\cdot d\vec{\ell} 
%= \Phi$$

$$\int_{C_1}{\bf A}\cdot {\bf d\ell} - \int_{C_2}{\bf A}\cdot {\bf d\ell}
= \Phi$$
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\noindent XIII. {\it Supergravity potential}

$${\mathcal L}=R-\bar{\psi}_{\mu}\gamma^{\mu\rho\sigma}D_{\rho}\psi_{\sigma}$$
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\noindent XIV. {\it Navier-Stokes Equation for Fluid Flow}
$$\rho(\p_t v_i + v_j \p_j v_i) $$

$$= -\p_i p + \mu \p_j \p_j v_i$$

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\end{document}