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\newcommand{\half}{\frac{1}{2}}
\begin{document}
%Viete's formula
$$\frac{2}{\pi} = \sqrt{\frac{1}{2}}\sqrt{\frac{1}{2}-\half\sqrt{\half}}\sqrt{\half-\half\sqrt{\frac{1}{2}-\half\sqrt{\half}}} \dotsm$$
\newpage
%Lord Brouncker's Formula
$$4/\pi = 1 + \cfrac{1}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cfrac{7^2}{2+\cfrac{9^2}{\ddots}}}}}$$
\newpage
$$\frac{\pi}{2} =
1+\frac{1}{3}\left(1+\frac{2}{5}\left(1+\frac{3}{7}\left(1+\frac{4}{9}\left(1+\dotsb\right)\right)\right)\right)$$
%Beeler
\newpage
$$\pi = \sum_{n=0}^\infty\left(2n \atop n\right)^3
\frac{42n+5}{2^{12n+4}}$$
%Ramanujan
\newpage
\begin{align*}
\frac{\pi}{4} & = \sum_{i=1}^\infty\arctan{(1/F_{2i+2})} \\ &
=\arctan{\frac{1}{3}} + \arctan{\frac{1}{8}} + \arctan\frac{1}{21}+
\dotsb
%http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpi.html
\end{align*}
\newpage
\begin{align*}
&\pi = 11.\\
&00100100001111110110101010001000100\\
&00101101000110000100011010011000100\\
&11000110011000101000101110000000110\\
&11100000111001101000100101001000000\\
&10010011100000100010001010011001111\\
&10011000111010000000010000010111011\\
&11101010011000111011000100111001101\\
&10010001001010001010010100000100001\\
&11100110001110001101000000010011011\\
&10111101111100101010001100110110011\\
&11001101001110100100001100011011001\\
&10000001010110000101001101101111100\\
&10010111110001010000110111010011111\\
&11000010011010101101101011011010101\\
&00011100001001000101111001001000010\\
&11011010101110110011000100101111001\\
&11111011000110111101000100110001000\\
&01011101001101001100011011111101101\\
&011010110000101111111111010111_{(2)}\dotsc
%&101101111010000000110101101111_{(2)}\dotsc
%&01111011100011100001101011111110110\\
%&10110101000100110011111101001011010\\
%&111010011111001001000001000101111100010010110001111111100110010010010010100001100110010100011110110011100100010110110011110111000010000000000111110010111000101000010110001110111111000001011001100011011010010010000011011000011100010
\end{align*}
\newpage
$$\pi = \lim_{n^2 \rightarrow
\infty}\frac{4}{n^2}\sum_{j=0}^n\sqrt{n^2-j^2}$$
\newpage
\begin{align*}\begin{split}\pi &= 6\Bigl(\frac{1}{2} +
\frac{1}{2\cdot3\cdot2^2} + \frac{1\cdot 3}{2 \cdot 4 \cdot 5 \cdot
2^4} \\ & \quad + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 7
\cdot 2^8}+\dotsb\Bigr)\end{split} \end{align*}
%newton 22 terms, 16 d places, arcsin expansion
\newpage
$$\pi=\frac{1}{2 i}\oint_\gamma \frac{dx}{x+iy} + \frac{i \,
dy}{x+iy}$$ \centerline{ where $\gamma$ is any simple closed}
\centerline{counterclockwise path in $\mathds{C}$ containing the
origin}
\newpage
$$\frac{6}{\pi^2}$$ is the limit, as $n$ goes to infinity, of the probability
 that two random
integers selected uniformly from $1,\dotsc, n$, are not relatively
prime
\newpage
the symbol $\pi$ was first used to denote the ratio of a circle's
circumference to its radius around 1706, by William Jones, in his
\emph{Synopsis Palmariorum Matheseos}
%Previously, the famous ratio, which is approximately 3.14 but cannot
%be calculated to perfect precision, was called the Ludolphian
%number, after Ludolph van Ceulen, a German mathematician
\newpage
$$1 - \langle e^{\sqrt{163}\pi} \rangle < 10^{-12}$$
where $\langle x \rangle$ denotes the fractional part of $x$
\newpage
$$\frac{1}{\pi}$$
is the probability that a needle of length $\ell$ will land on a
line, given a floor with equally spaced parallel lines a distance
$\ell$ apart
%buffon
\newpage
$$2\sqrt{\pi}$$
is the minimum length of fence needed to completely enclose one
square unit of area
\newpage
{\center{\small
``Why, $\pi$! Stop, $\pi$! Weird anomalies do behave badly! \\
You, madly conjured, imperfect, strange, numerical, \\ Why do you
maintain this facade? \\In finite time you are barbaric!\\ You do
wonders, mesmerize minds! \\O, do elements numerous have a beautiful
meaning- \\A system isolating all mysteries, solutions for puzzles,
chaos, a \\O snafu apparent in O Universal Concept from believing
lies? \\That there, obstinate in you, O Strange Constant, \\A Divine
Sign O exists is unlikely unless \\Is O revealed Something
Brilliant, negating belief! \\In formulas, O, you show yourself in
Greek and math as a $\pi$ forever-- \\O hidden wonders absconded,
infinite, in a tiny constant, \\ O, sneakily, rather? \\Never, I
say!"\\}}

encodes the first 144 digits of $\pi$: the number of letters in each
word gives the sequence of digits, and O stands for 0

\newpage

\tiny {
%Viete's formula
$$\frac{2}{\pi} = \sqrt{\frac{1}{2}}\sqrt{\frac{1}{2}-\half\sqrt{\half}}\sqrt{\half-\half\sqrt{\frac{1}{2}-\half\sqrt{\half}}} \dotsm$$
\\
{\textbf{Not $\pi$.} \texttt{Change the - symbols to + signs and you
get Vi\`ete's formula, the first (1593) infinite series for $\pi$.
}}
\\
%Lord Brouncker's Formula
$$4/\pi = 1 + \cfrac{1}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cfrac{7^2}{2+\cfrac{9^2}{\ddots}}}}}$$

{\textbf{$\pi$.} \texttt{This formula comes from Brouckner (circa
1650), and was derived from Wallis' formula. }}
\\ \\
$$\frac{\pi}{2} =
1+\frac{1}{3}\left(1+\frac{2}{5}\left(1+\frac{3}{7}\left(1+\frac{4}{9}\left(1+\dotsb\right)\right)\right)\right)$$
{\textbf{$\pi$.} \texttt{Beeler et al, 1972. Converges at one bit
per term.}}
%Beeler
$$\pi = \sum_{n=0}^\infty\left(2n \atop n\right)^3
\frac{42n+5}{2^{12n+4}}$$ {\textbf{Not $\pi$.} \texttt{Change $\pi$
to $1/\pi$ and you get an identity of Ramanujan's.}}
%Ramanujan
\\ \\
\newpage
\begin{align*}
\frac{\pi}{4} & = \sum_{i=1}^\infty\arctan{(1/F_{2i+2})} \\ &
=\arctan{\frac{1}{3}} + \arctan{\frac{1}{8}} + \arctan\frac{1}{21}+
\dotsb
%http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpi.html
\end{align*}
{\textbf{Not $\pi$.} \texttt{Change $F_{2i+2}$ to $F_{2i+1}$ and the
equality is true.}}
\\ \\
\begin{align*}
&\pi = 11.\\
&00100100001111110110101010001000100\\
&001011010001100001000110100110_{(2)}\dotsc
\end{align*}
\\ {\textbf{$\pi$.}
\texttt{Computed using Maple.}}\\ \\
$$\pi = \lim_{n^2 \rightarrow
\infty}\frac{4}{n^2}\sum_{j=0}^n\sqrt{n^2-j^2}$$ \\ \\
{\textbf{$\pi$.} \texttt{This represents four times the area of a
quarter-circle, converted from integral form.}}
\begin{align*}\begin{split}\pi &= 6\Bigl(\frac{1}{2} +
\frac{1}{2\cdot3\cdot2^2} + \frac{1\cdot 3}{2 \cdot 4 \cdot 5 \cdot
2^4} \\ & \quad + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 7
\cdot 2^8}+\dotsb\Bigr)\end{split} \end{align*} \\
{\textbf{Not $\pi$.} \texttt{ Change $3, 5, 7, \dotsc,$ to $2, 4, 8,
\dotsc,$ and you obtain an identity of Newton's derived from the
arcsine power series. In 22 terms it gave 16 decimal places. }}
\\ \\
$$\pi=\frac{1}{2 i}\oint_\gamma \frac{dx}{x+iy} + \frac{i \,
dy}{x+iy}$$ \centerline{ where $\gamma$ is any simple closed}
\centerline{counterclockwise path in $\mathds{C}$ containing the
origin} \\
{{\textbf{$\pi$.} \texttt{This is a rewriting of the definition of
the winding number from complex analysis.}}\\ \\
$$\frac{6}{\pi^2}$$ is the limit, as $n$ goes to infinity, of the probability
 that two random
integers selected uniformly from $1,\dotsc, n$, are not relatively
prime \\ {\textbf{Not $\pi$.} \texttt{ Remove the word, ``not'' and
this is true. It can be proved by using the fact that the sum of the
reciprocals of the square numbers is $\pi^2/6$. }}\\ \\ \\ the
symbol $\pi$ was first used to denote the ratio of a circle's
circumference to its diameter around 1706, by William Jones, in his
\emph{Synopsis Palmariorum Matheseos} \\{\textbf{$\pi$.} \texttt{
Previously, the famous ratio was called the Ludolphian number, after
Ludolph van Ceulen, a German mathematician. The use of the symbol
was popularized by Euler later that century.}}
\\ \\
$$1 - \langle e^{\sqrt{163}\pi} \rangle < 10^{-12}$$
where $\langle x \rangle$ denotes the fractional part of $x$ \\
{\textbf{$\pi$.} \texttt{ Actually, Martin Gardner once falsely
claimed that Ramanujan proved $e^{\sqrt{163}\pi}$ to be an integer.
(It was the April 1st issue.) For this reason the number is called
the Ramanujan constant.}}\newpage
$$\frac{1}{\pi}$$
is the probability that a needle of length $\ell$ will land on a
line, given a floor with equally spaced parallel lines a distance
$\ell$ apart
\\
{\textbf{Not $\pi$.} \texttt{ The left hand side should be $2/\pi$.
 This is Buffon's needle problem.}}
%buffon
\\ \\
$$2\sqrt{\pi}$$
is the minimum length of fence needed to completely enclose one
square unit of area \\
{\textbf{$\pi$.} \texttt{ This relates to the isoperimetric problem:
among all closed curves with a given perimeter, the circle has the
largest area.}}
\\ {\center{
``Why, $\pi$! Stop, $\pi$! Weird anomalies do behave badly! \\
You, madly conjured, imperfect, strange, numerical, \\ Why do you
maintain this facade? \\In finite time you are barbaric!\\ You do
wonders, mesmerize minds! \\O, do elements numerous have a beautiful
meaning- \\A system isolating all mysteries, solutions for puzzles,
chaos, a \\O snafu apparent in O Universal Concept from believing
lies? \\That there, obstinate in you, O Strange Constant, \\A Divine
Sign O exists is unlikely unless \\Is O revealed Something
Brilliant, negating belief! \\In formulas, O, you show yourself in
Greek and math as a $\pi$ forever-- \\O hidden wonders absconded,
infinite, in a tiny constant, \\ O, sneakily, rather?
\\Never, I say!"\\}}

encodes the first 144 digits of $\pi$: the number of letters in each
word gives the sequence of digits, and O stands for 0 }
\\{\textbf{Not $\pi$.} \texttt{ There are 112 words.}}

\end{document}
