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\def\DDirac{\not\hspace{-1.25mm}D}

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\begin{document}

\message{ !name(formulas.tex) !offset(-3) }



\noindent{1. \it Primes and the zeta function:} 
$$ \prod_p \frac{1}{1-\frac{1}{p^s}} = 
\sum_{n=1}^{\infty}\frac{1}{n^s}$$


\vs
\noindent{2. \it Archimedes. Volume of Sphere:}
\begin{figure}[htp]
\centering \includegraphics[width=1.5in]{archimedes.eps}
\end{figure}
$$ v = \frac{2}{3}V$$

\vs

\noindent{3. \it Yang-Baxter Equation}
$$R_{12}R_{23}R_{12} = R_{23}R_{12}R_{23}$$
\begin{figure}[htp]
\centering \includegraphics[width=2in]{braid.eps}
\end{figure}

\vs

\noindent{4. \it Gauss. Quadratic reciprocity:}
$$ \left (~\frac{p}{q}~\right )\left ( ~\frac{q}{p}~\right ) 
= (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$$

\vs

\noindent{5. \it Completing the square:}
$$ax^2 + bx + c = a\left (x +\frac{\ds b}{\ds 2a}\right )^2  - \frac{\ds b^2-4ac}{\ds 4a}$$

\vs

\noindent{6. \it Newton's Law: }
$${\bf F} = m{\bf a}$$

\vs
\noindent{7. \it Newton's Law of Gravitation:}
$$ F = \frac{\ds G m_1m_2}{\ds r^2}$$

\vs
%{\it diagram with three 2x2 tangent circles inscribed in a circle}

\noindent{8. \it Pythagoras' Theorem with no-word proof:} 

$$c^2 = a^2 + b^2 $$

\begin{figure}[ht]
\centering \includegraphics[width=3in]{pythagoras.eps}
\end{figure}


\vs


\noindent{9. \it Pascal's Hexagon Theorem:}
\begin{figure}[ht]
\centering \includegraphics[width=3in]{pascal.eps}
\end{figure}

\vs

\noindent{10. \it Euler's Equation:}
$$ e^{\ds i\pi} + 1 = 0$$

\vs

\noindent{11. \it Schr\"{o}dinger's Equation:}
$$ i\hbar \frac{\ds \partial\psi}{\ds \partial t} = 
-\frac{\ds \hbar^2}{\ds 2m} \nabla^2\psi + V\psi$$

\vs

\noindent{12. \it Maxwell's Equations in Vacuum imply Wave Equation:}

\begin{figure}[htp]
\centering \includegraphics[width=4in]{light.eps}
\end{figure}

$$\begin{array}{l}

\nabla \cdot {\bf B}  = 0 \\
\nabla \times {\bf B}  = \frac{\ds 1}{\ds c}\frac {\ds \partial {\bf E}}
{\ds \partial t}\\
\nabla \cdot {\bf E}  = 0 \\
\nabla \times {\bf E}  = -\frac{\ds 1}{\ds c}\frac{\ds \partial {\bf B}}
{\ds \partial t}\\
\end{array} ~~~~ \Rightarrow ~~~~\nabla^2 {\bf E} = \frac{\ds 1}{\ds c^2}\frac {\ds \partial^2 {\bf E}}       
{\ds \partial t^2}$$




\vs 



\noindent{13. \it Einstein's Equation:}
$$E = mc^2$$

\vs

\noindent{14. \it Einstein's General Relativity Equation: }
$$R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu} + \Lambda g_{\mu\nu} = 8 \pi T_{\mu\nu}$$

\vs

\noindent{15. \it Einstein quotation from Fine Hall Common Room Fireplace:}
\vs

\centerline
{\large \frakfamily Raffiniert ist der Herr Gott, aber boshaft ist er nicht.}


\vs

\noindent{16. \it Heisenberg Uncertainty Principle:}
$$\Delta x \Delta p \geq \frac{\ds \hbar}{\ds 2}$$

\vs


\noindent{17. \it Dynkin diagram of $E_8$:}
\begin{figure}[htp]
\centering \includegraphics[width=2in]{dynkin.eps}
\end{figure}

\vs

\noindent{18. \it Quotation from Harish-Chandra:}
\vs

\centerline{Why has God made the exceptional groups?} 



\vs
\newpage

\noindent{19. \it Limiting ratio of Fibonacci numbers = golden mean  
= partial fraction expansion:}
$$ \lim_{n\rightarrow\infty}\frac{\ds F_{n+1}}{\ds F_n}
= \frac{\ds 1 + \sqrt{5}}{\ds 2} = {1}+ \frac{\ds 1}{1+\frac{1}{1+ 
\frac{1}{1+ \dots } }    }$$

\begin{figure}[htp]
\centering \includegraphics[width=2in]{spiral.eps}
\end{figure}
\vs

\noindent{20. \it Wilson loop average w.r.t Chern-Simons action
gives knot invariant:}

\begin{figure}[htp]
\centering \includegraphics[width=2in]{trefoil.eps}
\end{figure}
$$ V_K(q) \,=\, q + q^3 - q^4 $$
\vs



$$ V_K\!\left( q = {\rm e}^{\!{{2\pi i}\over{k+2}}} \right) \,=\, {1\over Z}    
\int_{\mathcal A} \Tr_{\bf 2} P\exp{\!\left(-\oint_K A \right)} \, {\rm         
e}^{\!{{i k}\over{4 \pi}} \, {\rm CS}(A)} {\mathcal D} A$$  
\vs


\newpage
\noindent{21. \it Gauss-Bonnet for spherical triangle:}\vs

\begin{figure}[htp]
\centering \includegraphics[width=2in]{bonnet-2x.eps}
\end{figure}

$$\alpha + \beta + \gamma = \pi + \frac{\ds A}{\ds R^2}$$

\vs

\noindent{22. \it Stokes' Theorem}
$$\int_{M}d\omega=\int_{\partial M}\omega$$

\newpage
\noindent{23. \it Uniformization of a surface of genus $2$:}
\begin{center}
\mbox{
\beginpicture
\setplotarea x from -100 to 100, y from -100 to 100

\circulararc 360 degrees from  0 -100  center at 0 0
\circulararc 45 degrees from  0 -59  center at 0 -100
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\circulararc -45 degrees from  0 -59  center at 0 -100
\circulararc 45 degrees from   -59 0 center at  -100 0
\circulararc -45 degrees from  -59   0 center at  -100 0
\circulararc 45 degrees from  59 0  center at  100 0
\circulararc -45 degrees from  59 0 center at  100 0
\circulararc 45 degrees from  42 -42  center at  71 -71
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\arrow <3pt> [1,3] from 59  2 to 59 -2
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\arrow <6pt> [1,2] from 145 70  to 145 10
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center at 120 30
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\vs

\noindent{24. \it Atiyah-Singer Theorem for Twisted Dirac Operator:} 

%$$\Dirac :  \Gamma ({\mathbb S}_+\otimes E) \to \Gamma ({\mathbb S}_-\otimes E)$$

$$\dim \ker  (\Dirac) - \dim \ker  (\Dirac^*)  = \int_{M^{4k}} \hat{A}(M) \smile ch (E)$$
\vs

\noindent{25. \it Generalized Gauss-Bonnet Formula:} 
$$
\chi (M^{2n})= \frac{1}{(8\pi)^n n!}
\int_{M}\underbrace{R^{ij}_{ab}\cdots R^{k\ell}_{cd}}_n
\varepsilon^{ab\cdots cd}
\varepsilon_{ij\cdots k\ell}
~d\mu 
$$
\vs

\noindent{26. \it Classical Gauss-Bonnet Theorem:} 



\bigskip


$$
\chi (M^{2})= \frac{1}{2\pi}
\int_{M}K 
~dA 
$$




\begin{center}
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\setplotarea x from 0 to 350, y from -5 to 70
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\vs


\noindent{27. \it Chern-Simons Action:}

$$ S_{CS}=\frac{k}{4\pi}\int_M{\rm Tr}\left(A\wedge d A+\frac{2}{3}A\wedge A\wedge A\right)$$
\vs

\noindent{28. \it Witten's AdS/CFT prescription:}

$$\left\langle e^{\int_{S^d}\varphi_0 {\cal O}}\right\rangle_{CFT_d}=Z_{AdS_{d+1}}[\varphi_0]$$ 
\vs

\noindent{29. \it Snell's (Pascal's) Law:}
\begin{figure}[htp]
\centering \includegraphics[width=2in]{snell.eps}
\end{figure}
$$\sin i = n \sin r$$
\vs

\noindent{30. \it Veneziano amplitude:}
$$S(k_1;k_2;k_3;k_4)=\frac{2ig_o^2}{\alpha'}(2\pi\delta)^{26}\left 
(\sum_i k_i\right )[B(-\alpha(s),-\alpha(t))+ B(-\alpha(s),-\alpha(u))+B((-\alpha(t),-\alpha(u))]$$\vs

\noindent{31. \it Liouville equation:}
$$\partial_z\partial_{\bar z}\varphi=\frac{1}{2} e^{\varphi}$$
\vs

\noindent{32. \it Virasoro algebra:}
$$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}  $$
\vs

\noindent{33. \it Poincar\'{e}'s epigraph to ``Sur le probl\`{e}me              
    des trois corps ...'' }\vs

\centerline{Nunquam                  
    praescriptos transibunt sidera fines.}
\centerline{(Never will the stars cross their prescribed limits.)}
\vs

\noindent{34. \it Yang-Mills equations in two notations:}

$$F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + 
f_{abc}A^b_\mu A^c_\nu$$
$$\nabla^{\mu}F_{\mu\nu}^a = 0$$     

(or)
$$\partial^\mu F_{\mu\nu} -ig \left[ A^{\mu} , F_{\mu\nu} \right] = 0$$ 
\vs

$$F = dA + A \wedge A$$
$$\nabla \wedge \star F =0$$
\vs

\noindent{35. \it Yang-Mills Lagrangian}
$$\mathcal{L} = -\frac{1}{4 g^2} \textrm{Tr} \, F^2$$ 

\noindent{36. \it Hodge Theorem:}                  
$$H^p(M) = \{ \varphi \in \Omega^p(M)~|~ d\varphi = 0, ~d\star \varphi = 0\}$$




\noindent{37. \it Dirac Equation:}

$$
\begin{array}{rcl}
\left(-i\hbar \gamma^\mu \nabla_\mu +mc\right)  \psi&=&0\\
\gamma^\mu\gamma^\nu+ \gamma^\nu\gamma^\mu&=&2g^{\mu\nu}
\end{array}
$$

\noindent{38. \it Dirac Equation (another):}
$$\begin{array}{rcl}
%(\slash ?\! \! \! \! {D}+im)\psi& =& 0\\
(\DDirac + im)\psi& =& 0\\
D&\equiv&\partial+ieA
\end{array} $$
\vs

\noindent{39. \it Cauchy's Integral Formula:}

$$f(z) = \frac{\ds 1}{\ds 2\pi i}\oint_C \frac{\ds f(\zeta)}{\ds \zeta - z}~d\zeta$$
\vs

\noindent{40. \it Hodge Decomposition Theorem:}

$$H^n(X,\mathbb C)=\oplus_{p+q=n} H^{p,q}(X)$$  
\vs




\noindent{42. \it Navier-Stokes equation:}
$$  \p_t v_i + v_j \p_j v_i = -\p_i p + \nu \p_j \p_j v_i  $$
\vs



\noindent{43. \it Kolmogorov Law:}                  
$$   E(k) \sim \varepsilon^{2/3} k^{-5/3} $$
\vs

\noindent{44. \it Hubbard model:}
$$  H=-t \sum_{\langle i,j\rangle,\sigma}\left(
c^\dagger_{i,\sigma}c_{j,\sigma} + h.c.\right) +
U \sum_i n_{i\uparrow} n_{i\downarrow} $$  
\vs

\noindent{41. \it Pentagon identity of dilogarithm:}

\begin{figure}[h]
\centering \includegraphics[width=3in]{pentagon.eps}
\end{figure}

$$Li_2(x) = \int_{0}^{1} \frac{dt}{t} \log( 1 - x t )$$ 
\vs

\noindent{45. \it Renormalization:}
$$ \beta(g) = \mu \frac{\partial g}{\partial \mu} $$
\vs

\noindent{46. \it "no comment"}
$$ SU(3) \times SU(2) \times U(1) $$ 
\vs

\noindent{47. \it The entropy formula:}
$$ S = -k_B \sum_i p_i \log p_i  $$ 
\vs

\noindent{48. \it CKM matrix:}
$$\left [\begin{array}{ccc}
|V_{ud}|&|V_{us}|&|V_{ub}|\\
|V_{cd}|&|V_{cs}|&|V_{cb}|\\
|V_{td}|&|V_{ts}|&|V_{tb}|
\end{array}
\right ] = \left [ \begin{array}{lll}
0.9742 &0.2257 &0.0036 \\
0.2256 &0.9733 & 0.0415\\
0.0087 &0.0407 &0.9991 
\end{array} \right ] $$
\vs

\noindent{49. \it Exact homotopy sequence of Stiefel fibration:}

$$\cdots 
\rightarrow \pi_{i+1}S^n
\rightarrow \pi_iV_{k-1}{\mathbb R}^n
\rightarrow \pi_iV_k{\mathbb R}^{n+1}
\rightarrow \pi_iS^n
\rightarrow \cdots $$

\noindent{50. \it Ricci flow:}
$$\frac{dg_{t}}{dt} = -2Ric(g_{t})$$ 
\vs

\noindent{51. \it Apollonian fractal:}
\begin{figure}[htp]
\centering \includegraphics[width=5in]{scott_apollonian.eps}
\end{figure}
\newpage
\noindent{52. \it Sierpinski gasket:}
\begin{figure}[htp]
\centering \includegraphics[width=5in]{scott_sierpinski.eps}
\end{figure}
\newpage
\noindent{53. \it Pascal's triangle, Fibonacci numbers:}
\begin{figure}[htp]
\centering \includegraphics[width=3in]{triangle.eps}
\end{figure}
\vs


\noindent{54. \it Heat Kernel Version of the Index Theorem:}

\[
\operatorname{index}(\slashed{D}) = \operatorname{Trace}[(-1)^Fe^{-\beta H}] 
\]

\noindent{55. \it Aharonov-Bohm Effect:}
{\Large $$e^{iq\oint {\vec A}\cdot d{\vec\ell}}=e^{ i2\pi F}$$}
\vs
\newpage
%\noindent{. \it Ising Model:}



\noindent{56. \it Group law on cubic:}

\begin{figure}[htp]
\centering \includegraphics[width=3in]{cubic.eps}
\end{figure}
$${\bf\tt a}\cdot {\bf\tt b} \cdot {\bf\tt c} = 1$$

\noindent{57. \it Riemann-Roch-Hirzebruch:}
$$
\sum_{k=0}^n (-1)^k{\rm dim} \, H^k(X,E)\ \ =\ \ \int_X ch\,E \cup Todd(X)
$$

\noindent{58. \it Platonic solids, Euler characteristic:}
\begin{figure}[htp]
\centering \includegraphics[width=1in]{tetrahedron.eps}
\centering \includegraphics[width=1in]{cube.eps}
\centering \includegraphics[width=1in]{octahedron.eps}
\centering \includegraphics[width=1in]{dodecahedron.eps}
\centering \includegraphics[width=1in]{icosahedron.eps}
\end{figure}
$$ V - E + F = 2$$
\vs

\newpage
\noindent{59. \it Lorenz attractor:}
\begin{figure}[htp]
\centering \includegraphics[width=4in]{scott-lorenz.eps}
\end{figure}
\vs

\noindent{60. \it Black hole entropy}

$$S_{BH}=\frac{A}{4}$$ 
\newpage

\noindent{61. \it Mandelbrot Set:}
\begin{figure}[htp]
\centering \includegraphics[width=4in]{mandelbrot-fake-sm.eps}
\end{figure}
\vs


\noindent{64. \it Prime number Theorem:}
$$ \pi(n)~\sim~\frac{n}{\ln(n)}$$
\vs


\noindent{62. \it Sonnet by Edna St. Vincent Millay}

Euclid alone has looked on Beauty bare.

Let all who prate of Beauty hold their peace,

And lay them prone upon the earth and cease

To ponder on themselves, the while they stare

At nothing, intricately drawn nowhere

In shapes of shifting lineage; let geese

Gabble and hiss, but heroes seek release

From dusty bondage into luminous air.

O blinding hour, O holy, terrible day,

When first the shaft into his vision shone

Of light anatomized! Euclid alone

Has looked on Beauty bare. Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone. 
\vs

\noindent{63. \it Eratosthenes' mesurement of rdius of Earth:}
\begin{figure}[htp]
\centering \includegraphics[width=4in]{milnor-eratos.ps}
\end{figure}
\vs

\noindent{65. \it Supergravity:}
$${\mathcal L}=R-\bar{\psi}_{\mu}\gamma^{\mu\rho\sigma}D_{\rho}\psi_{\sigma}$$
\vs

\noindent{66. \it Fourier transform:}


$$\hat{f}(\xi) = \int f(x) e^{2\pi i x\cdot\xi} dx \hspace{1in} f(x) = \int \hat{f}(\xi) e^{-2\pi i x\cdot\xi} d\xi$$ 
\vs


\noindent {67. \it Euler's summation for $\zeta(2)$:}

$$ 1 + \frac{1}{4} + \frac{1}{9} + \cdots = \frac{\ds \pi^2}{\ds 6}$$
\vs
\newpage

\noindent{68. \it Kepler's laws:}

\begin{figure}[htp]
\centering \includegraphics[width=4in]{milnor-keppic.ps}
\end{figure}
%\centerline{Kepler's Laws:\qquad $r\;d\theta/dt={\rm constant}\,,
%\qquad \oint dt \sim s^{3/2}$}
\vs

\noindent{69. \it Prime number theorem:}
$$ \pi(n)~\sim~{n\over \ln(n)}  $$ 
\vs


\end{document}

$$W(K) = \frac{\ds 1}{\ds Z}\int_{\mathcal A} 
\mbox{tr} P\int_K A\cdot dt ~e^{i\int_M CS(A)} dA$$
$$W_R(C) \,=\, \Tr_R P\exp{\!\left(-\oint_C A\right)}
\langle W_R(C) \rangle = V(q)
q = \exp{\!\left(2\pi i/(k+2)\right)}.
$$ S_{\rm CS}(A) \,=\, {k \over {4\pi}} \, \int_M \Tr\!\left(A \wedge d A       
\,+\, {2 \over 3} A \wedge A \wedge A\right) $$  

\message{ !name(formulas.tex) !offset(-677) }
