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\begin{document}

\title{Session 10 -- The number $\varphi$}
\subtitle{European section -- Season 2}
\date{}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{People you will hear in the recording}

\large

\begin{itemize}
\item Simon Singh, author who has specialised in writing about mathematical and scientific topics in an accessible manner, 
\item Ian Stewart, professor of mathematics at the University of Warwick, England, and a widely known popular-science writer.
\item Robin Wilson, Math historian at the Open University.
\item Adam Spencer, Australian radio DJ with a penchant for pure mathematics
\item Ron Knott, University of Surey, specialist about the Fibonacci numbers
\end{itemize}

\end{frame}

\begin{frame}{The seven parts of the recording}


{\bf Part I --} The Golden Ratio (Simon Singh, Ian Stewart, Robin Wilson).\\
{\bf Part II -- } Places where the Golden Ratio can be found (Simon Singh,  Ian Stewart, Adam Spencer).\\
{\bf Part III --} Properties of the number (Simon Singh, Robin Wilson, Ron Knott).\\
{\bf Part IV --} The Fibonacci numbers (Adam Spencer)\\
{\bf Part V --�} Fibonacci numbers in parking meters (Simon Singh and Ron Knott)\\
Fibonacci numbers in sunflowers (Ian Stewart)\\
Fibonacci numbers in pineaples (Simon Singh)\\
{\bf Part VI --} Fibonacci numbers and the Golden Ratio (Simon Singh)

\large

\end{frame}


\begin{frame}{What does Ian Stewart call the Platonist concept of the ideal world ?}

\ \pause

They sought the perfect circle, the perfect line, and saw the Golden Ratio as a kind of perfect ratio.

\begin{center}
\includegraphics[width=3.5cm]{images/plato.eps} \\
Plato
\end{center}



\end{frame}

\begin{frame}{How did the Ancient Greek define the number $\pi$ ?}

\ \pause

The Ancient Greek defined the number the number $\pi$ as the ratio between the circumference of a circle and its diameter.

\begin{center}
\begin{tabular}{cc}
\includegraphics[width=4cm]{images/pi2.eps} &
\includegraphics[width=6cm]{images/pi.eps} \\
\end{tabular}
\end{center}




\end{frame}

\begin{frame}{What was the preferred way of the Ancient Greek to talk about ``strange'' numbers such as $\pi$ or $\varphi$ ?}

\ \pause


The Ancient Greek defined these numbers as ratios of two lengths.

\begin{center}
\includegraphics[scale=0.3]{images/greek.eps}\\
Euclid's Elements
\end{center}




\end{frame}

\begin{frame}{What is an irrational number ?}

\ \pause


An irrational number is a real number that is not an exact fraction, such as $\sqrt{2}$, $\pi$ or $\varphi$.

\begin{center}
\includegraphics[scale=0.3]{images/sq2.eps}\\
$\sqrt{2}$ is irrational.
\end{center}

\end{frame}

\begin{frame}{What approximate value to 6DP of $\varphi$ is given by Ian Stewart ?}

\ \pause

{\Huge
$$\varphi\simeq1.618034$$
}

\end{frame}

\begin{frame}{What are the other names of the Golden Ratio ?}

\ \pause

The Golden Ratio is also called the Golden Mean or the Divine Ratio.

\begin{center}
\includegraphics[scale=0.4]{images/pentagram.eps}\\
The pentagram.
\end{center}


\end{frame}

\begin{frame}{What did the Ancient Greek regard as the perfect rectangle ?}

\ \pause

The perfect rectangle was the Golden Rectangle, with one side $\varphi$ times longer then the other side.

\begin{center}
\includegraphics[scale=0.25]{images/rectangle.eps}\\
The golden rectangle.
\end{center}

\end{frame}

\begin{frame}{Why was the rectangle built using the Golden Ratio considered perfect ?}

\ \pause

The Golden Rectangle was considered perfect because it was not too squarish, and not too long and thin. 

\begin{center}
\includegraphics[scale=0.18]{images/rectangles.eps}\\
John Searles, {\sl Nine rectangles}
\end{center}

\end{frame}

\begin{frame}{Where did Leonardo Da Vinci see the Golden Ratio ?}

\ \pause

Leonardo Da Vinci thought that the Golden Ratio defined perfect proportion in the human body. 

\begin{center}
\includegraphics[scale=0.5]{images/davinci.eps}\\
Leonardo Da Vinci, {\sl Vitruvian Man} sketch.
\end{center}


\end{frame}

\begin{frame}{Which modern painter used repeatedly the Golden Ratio ?}

\ \pause

Piet Mondrian repeatedly used the Golden Ratio in his geometrical art.

\begin{center}
\includegraphics[scale=1.3]{images/mondrian.eps}\\
Piet Mondrian, {\sl Composition with Yellow, Blue, and Red}
\end{center}

\end{frame}

\begin{frame}{What famous Greek building is referred to in this program ? Why ?}

\ \pause

The Parthenon, in Athens, is referred to in this program because it has golden rectangles within it.

\begin{center}
\includegraphics[scale=1]{images/parthenon.eps}\\
The Parthenon, in Athens.
\end{center}

\end{frame}

\begin{frame}{What is the danger of looking for the Golden Ratio everywhere ?}

\ \pause

In any building, there are thousands and thousands of measurements. If you start comparing them, you will always find something close to the Golden Ratio.

\begin{center}
\includegraphics[scale=0.8]{images/architecture.eps}
\end{center}


\end{frame}

\begin{frame}{Which famous modern architect used the Golden Ratio extensively ?}

\ \pause

Le Corbusier deleberately used the Golden Ratio a lot, as he thought it was the perfect proportion for desigining human-size buildings.

\begin{center}
\includegraphics[scale=0.5]{images/citeradieuse.eps}\\
La Cit� Radieuse, Marseille
\end{center}

\end{frame}

\begin{frame}{Why can we hear a heartbeat in the program ?}

\ \pause

Because it seems that the ventricles in the heart reset themselves at the golden ratio point in the heart's rythmic cycle.

\begin{center}
\includegraphics[scale=0.8]{images/heartbeat.eps}
\end{center}

\end{frame}

\begin{frame}{How is the DNA spiral involving the Golden Ratio ?}

\ \pause

Divide the pitch of the DNA spiral by its diameter, and you get roughly the Golden Ratio.

\begin{center}
\includegraphics[scale=0.75]{images/dna.eps}
\end{center}

\end{frame}

\begin{frame}{What figure is created by the rectangles introduced by Adam Spencer ?}

\ \pause

The figure is created by this series of golden rectangles is called the Fibonacci spiral or Golden spiral. It's mistakenly called spiral of Archimedes in the program.

\begin{center}
\includegraphics[scale=0.25]{images/spiral.eps}
\end{center}

\end{frame}

\begin{frame}{Where is this figure appearing in nature ?}

\ \pause

The spiral of Archimedes can be found in and snailshells and crustaceans.

\begin{center}
\includegraphics[scale=0.25]{images/nautilus.eps}\\
Cutaway of a nautilus shell
\end{center}

\end{frame}

\begin{frame}{What do you get if you square the Golden Ratio ? What if you take its reciprocal ?}

\ \pause

{\Huge
\begin{eqnarray*}
\varphi^2 &=& \varphi+1\simeq 2.618\\
\frac{1}{\varphi} &=& \varphi-1\simeq 0.618
\end{eqnarray*}
}

\end{frame}

\begin{frame}{Why does the Golden Ratio have this property ?}

\ \pause

The Golden Ratio has this property because it satisfies the quadratic equation

{\Huge
$$x^2=x+1$$
}

\end{frame}

\begin{frame}{What process described by Ron Knott ends up with the Golden Ratio ?}

\ \pause

The process described by Ron Knott is :  Take any number, add one to it, compute its reciprocal, add one to the result, compute its reciprocal, and so on.

\bigskip

{\Large
An example : 5 $\mapsto$ 6 $\mapsto$ 0.167 $\mapsto$ 1.167 $\mapsto$ 0.857 $\mapsto$ 1.857 $\mapsto$ 0.538 $\mapsto$ 1.538 $\mapsto$ 0.65 $\mapsto$ 1.65 $\mapsto$ 0.606 $\mapsto$ 1.606 $\mapsto$ 0.623 $\mapsto$ 1.623 $\mapsto$ 0.616 $\mapsto$ 1.616 $\mapsto$ 0.618 $\mapsto$ {\bf 1.618} \par}

\end{frame}

\begin{frame}{Who was Fibonacci ?}

\ \pause

Fibonacci was a mathematician around 1180, called Leonardo da Pisa.

\begin{center}
\begin{tabular}{cc}
\includegraphics[width=3.5cm]{images/fibonacci.eps} & 
\includegraphics[width=3.5cm]{images/fibpage.eps}\\
Leonardo da Pisa AKA Fibonacci & Page 1 of his {\sl Liber quadratorum}
\end{tabular}

\end{center}


\end{frame}

\begin{frame}{How is the Fibonacci sequence built ?}

\ \pause

Start with the two numbers 0 and 1. Add them together to get 1. Take the last two numbers of the list to get 2. Keep adding the last two numbers of the list to generate the next one.

\begin{center}
\includegraphics[scale=0.8]{images/fibtiling.eps}\\
A tiling with squares whose sides are successive Fibonacci numbers
\end{center}

\end{frame}

\begin{frame}{Initially, what phenomenon were the Fibonacci numbers modelled on ?}

\ \pause

The Fibonacci numbers were originally modelled on a hypothetical population of rabbits.

\begin{center}
\includegraphics[scale=0.4]{images/rabbits.eps}\\
The Fibonacci rabbits.
\end{center}

\end{frame}

\begin{frame}{Why are the Fibonacci numbers so important in mathematics ?}

\ \pause

The Fibonacci numbers are so important because the crop up in many different areas of mathematics.

\begin{center}
\includegraphics[scale=0.35]{images/pascal.eps}\\
The Fibonacci numbers in Pascal's triangle.
\end{center}


\end{frame}

\begin{frame}{What is the link between Fibonacci numbers and car-parks ?}

\ \pause

If you need to pay only with 1-pound and 2-pounds coins, the number of ways to pay a certain amount is a Fibonacci number.

\newcommand{\onep}{\includegraphics[width=1cm]{images/1pound.eps}}
\newcommand{\twop}{\includegraphics[width=1cm]{images/2pounds.eps}}

\begin{center}
\begin{tabular}{cccc}
\onep &\onep &\onep &\onep \\
\onep &\onep &\twop & \\
\onep &\twop &\onep & \\
\twop &\onep &\onep & \\
\twop &\twop & & \\
\end{tabular}
\end{center}



\end{frame}

\begin{frame}{What is the link between Fibonacci numbers and sunflowers ?}

\ \pause

The numbers of clockwise and anticlockwise seeds spirals on a sunflower are Fibonacci numbers.

\begin{center}
\includegraphics[scale=0.35]{images/sunflower.eps}
\end{center}

\end{frame}

\begin{frame}{What is the link between Fibonacci numbers and pineapples ?}

\ \pause

The numbers of clockwise and anticlockwise losange spirals on a pineapple are Fibonacci numbers.

\begin{center}
\includegraphics[scale=0.6]{images/pineapple.eps}
\end{center}



\end{frame}

\begin{frame}{What is the relation between Fibonacci numbers and the Golden Ratio ?}

\ \pause

The ratio between to consecutive Fibonacci numbers approaches the Golden Ratio.

\begin{center}
\begin{tabular}{cc}
\includegraphics[width=3.6cm]{images/pressure.eps} &
\includegraphics[width=3.6cm]{images/galaxy.eps} \\
A low pressure area over Iceland & The Whirlpool Galaxy \\
\includegraphics[width=3.6cm]{images/romanesco.eps} &
\includegraphics[width=3.6cm]{images/hand.eps} \\
Romanesco broccoli & Fibonacci numbers in fingers \\
\end{tabular}

\end{center}

\end{frame}


\end{document}
