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\title[Números que brillan]{Números que \texorpdfstring{{\hspace{-0pt}\includegraphics[width=46pt]{./images/brillan.png}}}
\ :\\ {\small matemáticas en imágenes cotidianas}}
\institute{Instituto de Ciencias Matemáticas}
\date{5 de noviembre de 2024}
\author[Fernando Chamizo]{Fernando Chamizo {\footnotesize (UAM-ICMAT)}}
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\begin{document}


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\titlepage
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% \tableofcontents
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\vspace{15pt}

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
%{\footnotesize\color{gray}\textbf{Abstract.} 

\begin{itemize}\setlength{\itemsep}{10pt}
 \item 
  {\color{blue}Las ideas [matemáticas], como los colores o las palabras, deben combinarse de manera armoniosa.}
 {\footnotesize [G. H. Hardy, matemático (1877--1947)]}. 
 \item 
 {\color{blue}Una imagen vale más que mil palabras.}
 {\footnotesize [Atribuida a A. Brisbane, editor e inversor (1864--1936)]}. 
 \item 
 {\color{blue}La aritmética es capaz de contar hasta 20 sin quitarte los zapatos.}
 {\footnotesize [Atribuida a Mickey Mouse, dibujo animado (1928--$\infty$)]}. 
\end{itemize}

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\section{Cardioide}
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Matemáticas en el fondo de una taza
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\vfill
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\frametitle{Matemáticas en el fondo de una taza}


\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
Acercando una linterna al borde una taza o iluminándola bajo un foco surge un curioso patrón de reflejos.
\end{miblock}



\begin{center}
\begin{tabular}{c}
 \begin{tcolorbox}[width=140pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 \includegraphics[height=95pt]{./images/cup.jpg}
 \end{tcolorbox}
\end{tabular}
\end{center}


\begin{miblock}[halign=flush left,colframe=ct0,boxrule=1pt, coltitle=white]
Se parece a los que se ven en el fondo de las piscinas, pero no es aleatorio.
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\

\vspace{10pt}

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
Aparece algo similar en objetos cotidianos o en experimentos y simulaciones de ciencia y tecnología.
\end{miblock}

\vspace{10pt}

\begin{center}
\begin{tabular}{cc}
 \begin{tcolorbox}[width=140pt,halign=center,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 \begin{tabular}{c}
   \includegraphics[height=95pt]{./images/watch.jpg}
  \\
  {\scriptsize Reloj de pulsera}
 \end{tabular}
 \end{tcolorbox}
 &
 \begin{tcolorbox}[width=143pt,halign=center,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 \begin{tabular}{c}
   \includegraphics[height=95pt, origin=c, angle=90]{./images/mic.png}
  \\
  {\scriptsize Atenuación \hspace{-1pt}de \hspace{-1pt}un \hspace{-1pt}micrófono}
 \end{tabular}
 \end{tcolorbox}
\end{tabular}
\end{center}


\vspace{5pt}


{\tiny Fuente: Wikipedia: cardioid, microphone}

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\

La figura aparece también como una curva mecánica:

\vspace{-25pt}

\hspace{50pt}\begin{tikzpicture}[scale=0.8, rotate=-90, transform shape]
\draw[step=1cm,help lines,opacity=0] (-5.35,-4) grid (5.35,4);
\begin{scope}[scale=1.3]
%%%%FIJO
    \filldraw[draw=gray!80,fill=white,opacity=.8,even odd rule] (0,0) circle (.33cm) (0,0) circle (.165cm);
    \filldraw[draw=gray!80,fill=white,even odd rule] (0,0) circle (1cm) (0,0) circle (.33cm);
    \begin{scope}[even odd rule]
        \clip (0,0) circle (.33cm) (0,0) circle (1cm);
        \foreach \r in {45,135,225,315} \foreach \i [evaluate = {\s=30;}] in {0,2,...,30}
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    \end{scope}
    \draw[gray!60,opacity=.4,line width=.07cm] (0,0) circle (.4cm);
%%%%
\def\p{90}
\foreach \time/\g in {1/0,2/-30,3/-60,4/-90,5/-120,6/-150,7/-180,8/-210,9/-240,10/-270,11/-300,12/-330,13/-360}{
\only<\time>{%
    \def\ang{\p+\g}
%%%%MÓVIL
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        \filldraw[draw=gray!80,fill=white,opacity=.8,even odd rule] (0,0) circle (.33cm) (0,0) circle (.165cm);
        \filldraw[draw=gray!80,fill=white,even odd rule] (0,0) circle (1cm) (0,0) circle (.33cm);
        \begin{scope}[even odd rule]
            \clip (0,0) circle (.33cm) (0,0) circle (1cm);
            \foreach \r in {45,135,225,315} \foreach \i [evaluate = {\s=30;}] in {0,2,...,30}
                \fill[black, fill opacity=1/50] (0,0) -- (\r+\s-\i:1) arc (\r+\s-\i:\r-\s+\i:1)  -- cycle;%
        \end{scope}
    \draw[gray!60,opacity=.4,line width=.07cm] (0,0) circle (.4cm);
%%%%
    %\draw[rojo1] (0,0)--(\p:-1);
    %\draw (0,0)--(\p-\g:-1);
    \filldraw[red] (\p:-1) circle (1pt);
    \end{scope}
    %\draw[rojo1] (0,0)--(\p:1);
    %\draw (0,0)--(\ang:1);
    %\filldraw[rojo1] (\p:1) circle (1pt);
    \begin{scope}[shift={(0,1)}]
        \draw[thick,color=red,domain=\p:\ang,samples=40,smooth] plot (xy polar cs:angle=\x,radius= {2-2*sin(\x)});
    \end{scope}%
}}
\end{scope}
\end{tikzpicture}

\vspace{-40pt}

{\tiny\textbf{Curiosidad:} El círculo que gira da dos vueltas.}
\end{frame}





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\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-7);
    \node[text width=10.5cm,align=flush left,inner sep=0pt] at (5,-1)
        {La explicación es que los rayos de luz se reflejan en el borde circular dando lugar a rayos reflejados que son tangentes a la curva. A lo largo de ella, se produce una acumulación e interferencia de rayos que muestra una zona especialmente iluminada.};
    \begin{scope}[shift={(3,-4)}]%%%%Cardioide
        \def\r{.5}
        \fill[white,opacity=.5,even odd rule,domain=0:360,samples=40,smooth] (-\r,0) circle (3*\r cm) plot (xy polar cs:angle=\x,radius= {2*\r*(1-cos(\x))});
        \draw (-\r,0) circle (3*\r cm);
        \draw[thick,color=red,domain=0:360,samples=40,smooth] plot (xy polar cs:angle=\x,radius= {2*\r*(1-cos(\x))});
        \foreach \ang in {-90,-70,-50,-30,-10,10,30,50,70,90}{
            %\filldraw[blue] ({-\r+3*\r*cos(\ang))},{3*\r*sin(\ang)}) circle (.5pt);
            %\filldraw[rojo1] ({2*\r*(1-cos(\ang))*cos(\ang)},{2*\r*(1-cos(\ang))*sin(\ang)}) circle (.5pt);
            %\filldraw[green] ({\r+3*\r*cos(2*\ang))},{3*\r*sin(2*\ang)}) circle (.5pt);
            \draw[ct2,shorten >=-.5cm] (-\r*4,0) -- ({-\r+3*\r*cos(\ang))},{3*\r*sin(\ang)}) -- ({2*\r*(1-cos(\ang))*cos(\ang)},{2*\r*(1-cos(\ang))*sin(\ang)});}
    \end{scope}
    \begin{scope}[shift={(7.5,-4)}]%%%%Nefroide
        \def\r{.375}
        \fill[white,opacity=.5,even odd rule,domain=90:-90,samples=40,smooth] (0,0)--(90:4*\r) arc (90:-90:4*\r)--cycle plot ({3*\r*cos(\x)-\r*cos(3*\x)},{3*\r*sin(\x)-\r*sin(3*\x)});
        \draw (0,0) circle (4*\r cm);
        \draw[thick,color=red,domain=90:-90,samples=40,smooth] plot ({3*\r*cos(\x)-\r*cos(3*\x)},{3*\r*sin(\x)-\r*sin(3*\x)});
        \foreach \ang in {45,30,15,0,-15,-30,-45}{
            %\filldraw[blue] ({4*\r*cos(\ang)},{4*\r*sin(\ang)}) circle (.5pt);
            %\filldraw[rojo1] ({3*\r*cos(\ang)-\r*cos(3*\ang)},{3*\r*sin(\ang)-\r*sin(3*\ang)}) circle (.5pt);
            %\filldraw[green] ({4*\r*cos(3*\ang)},{4*\r*sin(3*\ang)}) circle (.5pt);
            \draw[ct2,shorten >=-.5cm] (-4*\r-.25,{4*\r*sin(\ang)}) -- ({4*\r*cos(\ang)},{4*\r*sin(\ang)}) -- ({3*\r*cos(\ang)-\r*cos(3*\ang)},{3*\r*sin(\ang)-\r*sin(3*\ang)});}
    \end{scope}
    \node[inner sep=0pt] at (2.5,-6.2) {\structure{\begin{tabular}{c}
    Luz en el borde
    \\[3pt]
    \textbf{\color{green!20!black}Cardioide}
    \end{tabular}}};
%    \node[inner sep=0pt] at (7.5,-6) {\structure{Luz lejana}};
    \node[inner sep=0pt] at (7.5,-6.2) {\structure{\begin{tabular}{c}
    Luz lejana
    \\[3pt]
    \textbf{\color{green!20!black}Nefroide}
    \end{tabular}}};
\end{tikzpicture}

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\


\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
Multiplicando por dos conseguimos aproximar el patrón. Esto se sigue de sencillas leyes de la óptica.
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\begin{center}
 \begin{tcolorbox}[width=150pt,halign=center,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 \includegraphics[height=105pt]{./images/refl1.png}
 \end{tcolorbox}
 \begin{tcolorbox}[width=150pt,halign=center,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 \includegraphics[height=105pt]{./images/refl2.png}
 \end{tcolorbox}
\end{center}

\

{\scriptsize Receta universal de los espejos: ángulo de incidencia\,$=$\,ángulo de reflexión.}

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\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
Con más rayos conseguimos mejores aproximaciones. 
\end{miblock}


\begin{center}
\begin{tabular}{cc}
 \begin{tcolorbox}[width=140pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 \begin{center}
   \includegraphics[height=98pt]{./images/card40.png}
   \\
   {\scriptsize 40 puntos}
 \end{center}
 \end{tcolorbox}
 &
 \begin{tcolorbox}[width=140pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 \begin{center}
 \includegraphics[height=98pt]{./images/card100.png}
   \\
   {\scriptsize 100 puntos}
 \end{center}
 \end{tcolorbox}
\end{tabular}
\end{center}


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\

Y con muchísimos, algunos fenómenos extraños\dots

\begin{center}
\begin{tabular}{cc}
 \begin{tcolorbox}[width=140pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 \begin{center}
 \includegraphics[height=95pt]{./images/card200.png}
   \\
   {\scriptsize 200 puntos}
 \end{center}
 \end{tcolorbox}
 &
 \begin{tcolorbox}[width=140pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 \begin{center}
 \includegraphics[height=95pt]{./images/card600.png}
   \\
   {\scriptsize 600 puntos}
 \end{center}
 \end{tcolorbox}
\end{tabular}
\end{center}


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\section{Moiré}
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\vskip2cm
\begin{tcolorbox}[enhanced,center,width=10.0cm,boxsep=10pt,halign=center,%
                  fontupper=\Large,colupper=ct0,%
                  colback=white!97!ct2,colframe=ct2!70!black,%
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No lleves a la tele camisa de rayas
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\vfill
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\frametitle{No lleves a la tele camisa de rayas}

\begin{tikzpicture}[scale=1.2]
%\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-7);
    \node[text width=10.5cm,align=flush left,rounded corners,fill=white,draw=ct2,line width=1pt] at (5,-1)
        {Imprimamos un montón de sectores muy finos de un círculo grande en un lámina transparente:};
    \node at (5,-3.5) {\includegraphics[width=3cm]{./images/moire_s}};
    \node[text width=10.5cm,align=flush left,inner sep=0pt] at (5,-6)
        {Ya en esta diapositiva se pueden ver unas curvas raras cerca del centro.};
\end{tikzpicture}
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\begin{tikzpicture}[scale=1.2]
%\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-7);
    \node[text width=10.5cm,align=flush left,rounded corners,fill=white,draw=ct2,line width=1pt] at (5,-1)
        {Según acercamos los centros de las imágenes en horizontal, se producen interferencias más nítidas.};
\foreach \time/\x in {1/.25,2/.5,3/.75,4/1,5/1.25,6/1.35,7/1.45}{
\only<\time>{%
    \node at (3.5+\x,-3.5) {\includegraphics[width=3cm]{./images/moire_s}};
    \node at (6.5-\x,-3.5) {\includegraphics[width=3cm]{./images/moire_s}};
    }}
    \node[text width=10.5cm,align=flush left,rounded corners,fill=white,draw=ct2,line width=1pt] at (5,-6)
        {Si nos apartamos un poco de la horizontal o giramos las imágenes, se producen curiosas sensaciones de movimiento.};
\end{tikzpicture}
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\begin{miblock}[halign=flush left,colframe=red!60!black,boxrule=1pt, coltitle=white]
{\color{red!60!black}$\bullet$} \ El efecto moiré es un efecto a evitar para la correcta visualización de imágenes estáticas, como las producidas por impresoras, o dinámicas, como las emitidas por televisión.
\end{miblock}

\

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
{\color{ct1}$\bullet$} \ También se utiliza en sentido positivo para efectuar mediciones ópticas con cierta precisión y alguna vez se ha propuesto como medida de seguridad en tarjetas bancarias.
\end{miblock}

\

Su explicación proviene de las matemáticas de la interferencia de ondas, aunque hay casos que admiten una explicación sencilla. 

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\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]

\begin{center}
 \begin{tabular}{cccc}
  \scalebox{-1}[1]{\includegraphics[height=55.0pt, angle=90]{./images/moirl1}}
  &
  \scalebox{-1}[1]{\includegraphics[height=55.0pt, angle=90]{./images/moirl2}}
  &
  \scalebox{-1}[1]{\includegraphics[height=55.0pt, angle=90]{./images/moirl3}}
  &
  \scalebox{-1}[1]{\includegraphics[height=55.0pt, angle=90]{./images/moirl4}}
  \\[-6pt]
  {\scriptsize$\alpha = 5^\circ$} &
  {\scriptsize$\alpha = 10^\circ$} &
  {\scriptsize$\alpha = 15^\circ$} &
  {\scriptsize$\alpha = 20^\circ$} 
 \end{tabular}
\end{center}
\end{miblock}

\

\textbf{Problema:}
\medskip

\begin{tabular}{c}
 \includegraphics[height=65.0pt]{./images/mdiam1}
\end{tabular}
\qquad
\begin{tabular}{l}
 Dada la separación $s$,
 \\
 ¿cómo varía $d$
 \\
 con el ángulo $\alpha$?
\end{tabular}

\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}

\vskip17pt

\begin{miblock}[halign=flush left,colframe=ct2,boxrule=1pt, coltitle=white, title=Un pequeño argumento matemático]
\begin{center}
 \begin{tabular}{c}
  \includegraphics[height=50.0pt]{./images/mdiam2}
 \end{tabular}
\end{center}


\begin{center}
\begin{tabular}{c}
 \begin{tcolorbox}[width=195pt,halign=flush left,colframe=ct11,boxrule=0.5pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 \begin{tabular}{c}
  Circunferencia completa$\rightarrow$ $2\pi R$
  \\
  \phantom{{\tiny(regla de tres)}} $\Downarrow$ {\tiny(regla de tres)}
  \\
  Arco de ángulo $\frac{\alpha}{2}${$\rightarrow$ }
  $2\pi R\cdot \frac{\alpha/2}{360}$
 \end{tabular}
 \end{tcolorbox}
\end{tabular}
\end{center}

Tomando $R=d$ y aproximando $\ell\approx s$, se tiene 
\[
 s
 \approx 2\pi d\cdot \frac{\alpha/2}{360}
 \quad\Rightarrow\quad 
 \boxed{d\approx \frac{360s}{\pi\alpha}}
\]

\vspace{-1pt}

\end{miblock}


\end{frame}


%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\section{Ilusiones}
\begin{frame}
\vskip2cm
\begin{tcolorbox}[enhanced,center,width=10.0cm,boxsep=10pt,halign=center,%
                  fontupper=\Large,colupper=ct0,%
                  colback=white!97!ct2,colframe=ct2!70!black,%
                  sharp corners=downhill,drop fuzzy shadow=ct2!70!black]
¡Qué ilusión!
\end{tcolorbox}
\vfill
\end{frame}



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% SLIDE    %
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\begin{frame}
\frametitle{¡Qué ilusión!}


\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
\small
K. Sugihara es un matemático y artista famoso por sus ilusiones ópticas.
Un vídeo suyo muy difundido muestra paradojas en un espejo.
\end{miblock}


\begin{tikzpicture}[node distance=2cm]
\node (A) at (0, 1) {\begin{tabular}{c}
  \includegraphics[height=60pt]{./images/sug_v1.jpg}
 \end{tabular}};
\node (B) at (3, -2) {\phantom{\scriptsize ¿El truco?}\begin{tabular}{c}
  \includegraphics[height=60pt]{./images/sug_v2.jpg}
 \end{tabular}{\scriptsize ¿El truco?}};
\node (C) at (6, 1) {\begin{tabular}{c}
  \includegraphics[height=60pt]{./images/sug_v3.jpg}
 \end{tabular}};

\draw [->, line width=2pt] (A) edge (B) (B) edge (C);

\end{tikzpicture}


\end{frame}



%%%%%%%%%%%%
% SLIDE    %
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\begin{frame}


\vskip9pt


\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]

\begin{tabular}{cp{3.2cm}|cp{3.2cm}}
 \begin{tabular}{c}
  \includegraphics[height=90pt]{./images/s_cylinder.png}
 \end{tabular}
 & 
 \vspace{-45pt}
 \qquad\begin{tabular}{c}
 \includegraphics[height=30pt]{./images/scirc.png}
 \end{tabular}

 {\footnotesize Una curva pintada en un cilindro transparente desde arriba se ve como una circunferencia.
 
 }
 &
 \begin{tabular}{c}
  \includegraphics[height=90pt]{./images/s_prism.png}
 \end{tabular}
 & 
 \vspace{-45pt}
 \qquad\begin{tabular}{c}
 \includegraphics[height=30pt]{./images/ssqua.png}
 \end{tabular}


 {\footnotesize Para este ortoedro, una curva se vería cuadrada desde arriba.
 
 
 }
\end{tabular}
\end{miblock}

\vspace{7pt}

\begin{tabular}{c}
 \begin{tcolorbox}[width=290pt,halign=center,colframe=ct1,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
\begin{tabular}{cp{3.2cm}r}
 \begin{tabular}{c}
  \hspace{-15pt}\includegraphics[height=80pt]{./images/s_cross.png}
 \end{tabular}\hspace{-10pt}
 & 

  \vspace{-30pt}

 {\footnotesize La curva intersección se verá por un lado como circunferencia y por el otro, como cuadrado.}
 &
 \begin{tabular}{c}
 \hspace{-15pt}\includegraphics[height=22pt]{./images/s_curve1.png}
 \end{tabular}
\end{tabular}
 \end{tcolorbox}
\end{tabular}



\end{frame}



%%%%%%%%%%%%
% SLIDE    %
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\begin{frame}

\begin{scriptsize}
 La curva intersección puede ser complicada, pero hallarla es {\sl solo matemáticas}\dots
 \[
  \begin{array}{l}
   c= c_+\cup c_{-}
   \\
   c_{\pm}(t)
   =
   \big(t, \pm f_+(t^*), \pm f_-(t^*)\big)
  \end{array}
  \qquad
  \begin{array}{l}
   t^*= t\cdot \text{sgn}(t),\ -1\le t\le 1
   \\
   f_{\pm}(t)
   =
   \frac12 \big(1-t\pm \sqrt{1-t^2}\big)
  \end{array}
 \]
\end{scriptsize}




\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white, title= Geometría de un espejo]

\begin{tabular}{cc}
\begin{tabular}{c}
 \includegraphics[height=90pt]{./images/gmirr.png}
 \end{tabular}
 &
\begin{tabular}{c}
\includegraphics[height=90pt, trim = {5.5cm 1cm 6.5cm 2cm}, clip]{./images/sug_v1.jpg}
 \end{tabular}
\end{tabular}




\end{miblock}
 \end{frame}


 


%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}



El caso de un triángulo-cuadrado es matemáticamente más simple porque la curva es una línea poligonal (no plana).
\medskip

Si quieres intentarlo, aquí tienes una plantilla:
\medskip


\begin{tabular}{c}
 \includegraphics[height=85pt]{./images/sugi_00.png}
 \includegraphics[height=85pt]{./images/sugi_01.jpg}
 \end{tabular}

{\footnotesize Una vez construido, hay que ajustar un poco las esquinas hasta que se vea un triángulo reflejado como un cuadrado al mirar con un ángulo de unos $45^\circ$.  Al darle la vuelta, tendrás un cuadrado reflejado como un triángulo. 

}
 
\ 
 
{\tiny
\textbf{Misterio}:
Por alguna razón que no sé explicar, en la práctica se ve mejor el triángulo reflejado como cuadrado que el cuadrado reflejado como triángulo. 

}
\end{frame}

 
 

%%%%%%%%%%%%
% SLIDE    %
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\begin{frame}
\frametitle{¡Y en 3D!}

\vspace{-3pt}

\begin{miblock}[halign=flush left,colframe=ct00,boxrule=1pt, coltitle=white]
La ilusión óptica más increíble de la que tengo noticia fue creada para un encuentro en honor de Martin Gardner (1914-2010), un famoso divulgador matemático.
Está basada en ideas del mago Jerry Andrus (1918--2007). 
\end{miblock}

\vspace{5pt}


\begin{miblock}[halign=flush left,colframe=ct00,boxrule=1pt, coltitle=white]
Consiste en una especie de dragón tridimensional que parece mover la cabeza cuando lo miramos con un solo ojo. El efecto también se captura en vídeo. Si tienes curiosidad, mira por ejemplo:
\begin{center}
 \url{https://www.gathering4gardner.org/3d-dragon-illusion}
\end{center}
\end{miblock}

\

{\footnotesize La perspectiva se comenzó a estudiar matemáticamente en el Renacimiento por su interés en la pintura.

}

\end{frame}


%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\section{Filtros}
\begin{frame}
\vskip2cm
\begin{tcolorbox}[enhanced,center,width=10.0cm,boxsep=10pt,halign=center,%
                  fontupper=\Large,colupper=ct0,%
                  colback=white!97!ct2,colframe=ct2!70!black,%
                  sharp corners=downhill,drop fuzzy shadow=ct2!70!black]
Filtros y más filtros
\end{tcolorbox}
\vfill
\end{frame}


%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}
\frametitle{Filtros y más filtros}

\vspace{-10pt}

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white, title=Imágenes con números]
El modelo matemático para una foto en blanco y negro es una matrix de números que indican el tono de gris de cada píxel. 
\end{miblock}

\vskip7pt

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
\begin{center}
   \includegraphics[height=20pt]{./images/ctones.png}
\end{center}


 \ \hspace{-14pt}
 \begin{tabular}{c}
   \includegraphics[height=70.177pt]{./images/grid1.png}
 \end{tabular}
 \hspace{-10.17pt}{\scriptsize$\to$}\hspace{-10.17pt}
 \begin{tabular}{c}
   \includegraphics[height=70.177pt]{./images/grid2.png}
 \end{tabular}
 \hspace{-10pt}
 \begin{tabular}{c}
   \includegraphics[height=70.177pt]{./images/grid3.png}
 \end{tabular}
 \hspace{-10.17pt}{\scriptsize$\to$}\hspace{-10.17pt}
 \begin{tabular}{c}
   \includegraphics[height=70.177pt]{./images/grid4.png}
 \end{tabular}
\end{miblock}

\vspace{5pt}

{\footnotesize Para los pitagóricos de hace 25 siglos y para los ordenadores actuales, casi todo son números. 

}


\end{frame}




%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}



\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
¿Qué ocurre si al transmitir la foto por un canal defectuoso algunos píxeles pasan a cero o a uno con cierta probabilidad?
\end{miblock}


\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
   \includegraphics[height=70pt]{./images/icmat_bw.jpg}
   \includegraphics[height=70pt, trim = {15.5cm 11cm 44.0cm 8cm}, clip]{./images/icmat_bw.jpg}
\end{miblock}

\pause

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
   \includegraphics[height=70pt]{./images/icmat_noisy.jpg}
   \includegraphics[height=70pt, trim = {15.5cm 11cm 44.0cm 8cm}, clip]{./images/icmat_noisy.jpg}
   
{\tiny $(\text{Probabilidad defecto}=15\%$)}
\end{miblock}

\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}



\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white, title=¿Tiene arreglo?]
Empleamos los \emph{filtros}
$
 \text{media}\Big(
 \hspace{-4pt}
  \begin{tabular}{c}
   \raisebox{-4.0pt}{\includegraphics[height=20pt]{./images/block3.png}}
  \end{tabular}
  \hspace{-4pt}
  \Big)
$
y
$
 \text{mediana}\Big(
 \hspace{-4pt}
  \begin{tabular}{c}
   \raisebox{-4.0pt}{\includegraphics[height=20pt]{./images/block3.png}}
  \end{tabular}
  \hspace{-4pt}
  \Big).
$
\end{miblock}

\pause

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
   \includegraphics[height=70pt]{./images/icmat_blur.jpg}
   \includegraphics[height=70pt, trim = {15.5cm 11cm 44.0cm 8cm}, clip]{./images/icmat_blur.jpg}
\end{miblock}

\pause

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
   \includegraphics[height=70pt]{./images/icmat_clean.jpg}
   \includegraphics[height=70pt, trim = {15.5cm 11cm 44.0cm 8cm}, clip]{./images/icmat_clean.jpg}
\end{miblock}

\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}


\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
Hay filtros con muchos propósitos. Por ejemplo, ¿Qué ocurre si aplicamos a la imagen de partida el filtro correspondiente a  multiplicar los valores de  los bloques $3\times 3$ de píxeles con la tabla 
\[
   L = \begin{tabular}{c}
   \includegraphics[height=50pt]{./images/block3_l.png}
  \end{tabular}
  \qquad \text{(Filtro de Laplace)}
\]

\vspace{-10pt}

y sumar los resultados?
\end{miblock}

\pause

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
   \includegraphics[height=70pt]{./images/icmat_L.jpg}
   \includegraphics[height=70pt, trim = {15.5cm 11cm 44.0cm 8cm}, clip]{./images/icmat_L.jpg}
\end{miblock}

{\footnotesize Obtenemos un detector de aristas, que recuerda a un dibujo  a lápiz.}

\end{frame}




%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}


\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
El resultado se mejora con la operación más complicada:
\vspace{-10pt}
\[ 
  S =
  \frac 18
  \sqrt{
   \Big(
   \begin{tabular}{c}
   \raisebox{-3.32pt}{\includegraphics[height=20.12pt]{./images/block3_h.png}}
  \end{tabular}
   \Big)^2
   +
   \Big(
   \begin{tabular}{c}
   \raisebox{-3.32pt}{\includegraphics[height=20.12pt]{./images/block3_v.png}}
  \end{tabular}
   \Big)^2
   }
  \qquad \text{(Filtro de Sobel)}
\]

\vspace{-10pt}

y ajustando el contraste.
\end{miblock}

\pause

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
   \includegraphics[height=70pt]{./images/icmat_Ss.jpg}
   \includegraphics[height=70pt, trim = {15.5cm 11cm 44.0cm 8cm}, clip]{./images/icmat_Ss.jpg}
\end{miblock}


\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
   \includegraphics[height=70pt]{./images/icmat_S8.png}
   \includegraphics[height=70pt, trim = {15.5cm 11cm 44.0cm 8cm}, clip]{./images/icmat_S8.png}
\end{miblock}

\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}

\

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white, title= Base de gran parte del tratamiento de imágenes y sonidos]
\begin{center}
 {\bf\color{blue} Toda onda es superposición de tonos puros}
\end{center}
\end{miblock}

\vskip7pt

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
¿Y qué es un tono puro? La altura en función del tiempo de un punto que gira en una circunferencia:

 \begin{tabular}{c}
   \includegraphics[height=70pt]{./images/puret.png}
 \end{tabular}

\end{miblock}

\end{frame}




%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}


\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
\noindent Por ejemplo, la onda triangular

 \begin{tabular}{c}
   \includegraphics[height=50pt]{./images/tri1.png}
 \end{tabular}
 
\noindent es muy similar a la superposición

 \begin{tabular}{c}
   \includegraphics[height=50pt]{./images/tri2.png}
 \end{tabular}

\end{miblock}

\vskip7pt

\begin{miblock}[halign=flush left,colframe=ct0,boxrule=1pt, coltitle=white]
{\footnotesize 
Si las circunferencias generan los tonos puros y estos todas las gráficas, 
¿no podríamos hacer \emph{todos los dibujos} combinando círculos?

}

\end{miblock}

\

{\tiny \url{http://tiny.cc/k1jtzz}}


\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\section{Números}
\begin{frame}
\vskip2cm
\begin{tcolorbox}[enhanced,center,width=10.0cm,boxsep=10pt,halign=center,%
                  fontupper=\Large,colupper=ct0,%
                  colback=white!97!ct2,colframe=ct2!70!black,%
                  sharp corners=downhill,drop fuzzy shadow=ct2!70!black]
¡Vaya ladrillo!
\end{tcolorbox}
\vfill
\end{frame}



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% SLIDE    %
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\begin{frame}
\frametitle{¡Vaya ladrillo!}



\vspace{-4pt}

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white, title=En las primeras civilizaciones\dots]
\[
  \begin{tabular}{c}
   \includegraphics[height=64pt]{./images/pyth.png}
 \end{tabular}
 \qquad\Longleftrightarrow\qquad 
 a^2+b^2= c^2
\]
\end{miblock}

\vspace{7pt}

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white, title= Hace 2300 años\dots]
\[
 a = m^2-n^2, 
 \qquad 
 b= 2mn,
 \qquad
 c=m^2+n^2
\]
{\small
Escogiendo enteros positivos $n<m$ se obtienen las soluciones enteras.  }
\end{miblock}

\vspace{-15pt}

{\scriptsize
\[
 \begin{array}{lcrrr}
  1,\ 2
  &\to&
  3,&4,&5,
  \\
  2,\ 3
  &\to&
  5,&12,&13,
 \end{array}
 \qquad\ \qquad
 \begin{array}{lcrrr}
  20,\ 24
  &\to&
  176,&960,&976,
  \\
  20,\ 25
  &\to&
  225,&1000,&1025.
 \end{array}
\]

}



\end{frame}




%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}


\vspace{-2pt}

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white, title=Hace 300 años\dots]
\[
 \begin{tabular}{c}
   \includegraphics[height=50.123pt]{./images/cub0.png}
 \end{tabular}
 \qquad
 \begin{tabular}{c}
   \includegraphics[height=50.123pt]{./images/cub2.png}
 \end{tabular}
\]
{\small
Hay una fórmula que da las dimensiones de infinidad de ladrillos con aristas enteras y diagonales de las caras también enteras.}
\end{miblock}

\vspace{-4pt}

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white, title=Hoy\dots\hfill {\tiny(tarea para futuros matemáticos)}]
\[
 \begin{tabular}{c}
   \includegraphics[height=50.123pt]{./images/cub1.png}
 \end{tabular}
 \qquad
 \begin{tabular}{c}
   \includegraphics[height=50.123pt]{./images/cub3.png}
 \end{tabular}
\]
{\small
Se desconoce si existe alguno con la diagonal interior también entera.}
\end{miblock}

{\scriptsize
Si existe, dicha diagonal tiene más de 15 cifras.}

\end{frame}






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\vskip12.5pt
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\invisible{\draw[step=1cm,help lines] (-.35,0) grid (10.35,-8);}
\node at (5,-2.8) {Esta presentación está disponible en};
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\end{document}




