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\title{MAT 4040 Project: Transformations}
\author{Sarah J Greenwald}

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 \frametitle{$$\mbox{MAT 4040 Project: Transformations}$$ }


\begin{center} 
Sarah J Greenwald


\includegraphics[width=3.3 in]{Fig1.jpg}   \\
A D'Arcy Thompson Transformation Grid [12]
%On Growth and Form
%Redrawn from Thompson 1917 in Palaeo-math 101	
%Norman MacLeod
%Palaeontology Department
%The Natural History Museum, London
%http://www.palass.org/modules.php?name=palaeo_math&page=26

 \end{center}
 
 %various disciplines
 \begin{itemize}
\item Connects geometry, algebra, linear algebra and analysis 


\item Applications in statistics, physics, computer science, architecture, art, astronomy and optics
\end{itemize}
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\begin{frame}

 \frametitle{$$\mbox{Experience with Transformations}$$ }
 
%You can adjust vertical spacing up or down:
 \vspace*{-1cm}

\begin{Definition}
In general a transformation changes some aspect while at the same time preserving some type of structure.  
\end{Definition}

\begin{enumerate}[a)]
\item Representations using coordinates, vectors, functions, matrices and groups
\item Matrix transformations in 2-D and 3-D in MAT 2240
\item Symmetry groups in MAT 3110
\item Symmetries of  polyhedra in MAT 3610
\end{enumerate}



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\begin{frame}
 \frametitle{$$\mbox{Criteria Reminder}$$ }


\begin{itemize}
\item Definitions, examples, important results, proofs... (from prior experiences) 
\end{itemize}

\begin{example}
\[
A=
\left(\begin{array}{ccc}
\cos{\theta} & -\sin{\theta}& 0\\
\sin{\theta}& \cos{\theta}& 0\\
0& 0& 1\\
\end{array}\right)
\]  represents a counterclockwise rotation by $\theta$ in the $x-y$ plane with the $z$ coordinate fixed.
\end{example}

\begin{itemize}
\item Historical and recent progress 
%in the area [into at least the 20th century] with at least 5 references, in addition,  at least 2 scholarly articles from the last 20 years
\end{itemize}

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 \frametitle{$$\mbox{Rich Early History of Transformations}$$ }



\vspace*{.4cm}

\begin{itemize}
\item Aesthetically pleasing design patterns
\item \color{cyan}Aristotle's \color{black} objections
\item \color{cyan}Euclid of Alexandria\color{black} : rotations
%superposition
\item \color{cyan}Archimedes of Syracuse\color{black} :  axial motions
\item \color{cyan}Appolonius of Perga\color{black} :  inversion
\item \color{cyan}Marcus Vitruvious\color{black} : projections
\item \color{cyan}Omer Khayyam's \color{black}  objections
\item \color{cyan}Thabit ibn Qurra \color{black}  and his grandson \color{cyan}Ibhrahim ibn Sinan\color{black} :  affine transformations of the plane

\end{itemize}

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 \frametitle{$$\mbox{A Few Generalizations Related to Classes}$$ }

\vspace*{-.9cm}
\begin{itemize}
%\item Alexis Clairaut and Leonard Euler:  general affine transformations
\item \color{cyan}Sir Isaac Newton\color{black}: transformations of coordinate systems
%\item Girard Desargues, Edward Waring and Gaspard Monge:  projective transformations
\item \color{cyan}August Mobius\color{black}:  homogeneous coordinates 
%of affine and projective transformations
\item \color{cyan}Carl Friedrich Gauss\color{black}:  linked to linear algebra
\item \color{cyan}Gotthold Eisenstein \color{black} and \color{cyan}Charles Hermits\color{black}:  addition and multiplication of linear transformations
\item \color{cyan}Arthur Cayley\color{black}:  composition of transformations
%\item James Sylvester:  properties preserved under transformations and nullity
\end{itemize}

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\begin{frame}
 \frametitle{$$\mbox{Exs of 19th Century Work and Apps in the 20th Century}$$ }

\begin{itemize}
%\item Luigi Cremona studied birational transformations [Cremona transformations in study of algebraic functions and integrals]
%\item Henri Poincare:  fractional linear transformations [hyperbolic geometry]
\item \color{cyan} Sophus Lie\color{black} :  continuous transformation groups and differential equation [Lie groups in physics]
\item \color{cyan} James Maxwell\color{black} : equations were invariant under Lorentz transformations 
%named for Hendrik Lorentz 
[geometry of space-time and beginning of relativity theory]
\end{itemize}
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 \frametitle{$$\mbox{Beginning of the 20th Century:}$$ }

\vspace*{-.7cm}
\begin{center} \includegraphics[width=1.7 in]{Klein_6.jpg}   \\
Felix Klein [11]
 \end{center}
%MacTutor http://www.gap-system.org/~history/PictDisplay/Klein.html
\begin{itemize}
\item \color{cyan} Felix Klein's \color{black}  \textit{Erlanger Programm}
\item Transformation group to understand a space 
\item Link with algebra revolutionized geometry
\end{itemize}





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 \frametitle{$$\mbox{One Application at the End of the 20th Century}$$ }

\begin{center} \includegraphics[width=1.7 in]{orosz_the_well.jpg}   \\
\emph{The Well} by Istv�n Orosz (1998) 
 \end{center}

Translations, rotations and reflections do not distort, but dilations and inversions do.  
Invert the image in a cylindrical mirror in the well.
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\begin{center} \includegraphics[width=1.7 in]{orosz_the_well_alt.jpg}   \\
Cylindrical inversion of Istvn Orosz's \emph{The Well} [9]


 \end{center}

\pause \color{cyan}M.C. Escher\color{black}, who was known for transformations

\begin{itemize}
\item {\small Escher remarked:  \emph{I express transformations in reality. I often do this by having a mirror in my picture reflect a scene that does not fit at all. Sometimes I make subtle changes to my picture that by the other side makes it completely different, a flock of birds changes into a city scape.} [Metropolitan Mathematician's Journal, 1948]

\item Challenge by \color{cyan}Bruno Ernst\color{black}}

%Escher was challenged by Bruno Ernst, to create a scene in which a mirror was placed to show the surrounding areas otherwise hidden from view.  Escher's health failed and he never completed this.  
\end{itemize}

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