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%%%%% 	SPANISH
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%%%%% 	Fuente
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%%%%% 	GRÁFICOS, CAJAS
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%%%%%%%%%%%%%%%
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%%%%% 	MACROS
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%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% 	COLORES
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\definecolor{amarillO}{RGB}{248,172,0} %amarilloICMAT
\definecolor{azulD}{RGB}{64,80,147} %azulDULCI
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%%*****************************************
%% Colores tema
%%*****************************************
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%%%%%%%%%%%%%%%%%%%%%%%%%%


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%--------------------------------------------------------------------------


%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Datos
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\title[Matemáticas con experimentos]{Matemáticas con experimentos}
\institute{Universidad Autónoma de Madrid}
\date{5 de julio de 2017}
\author[F. Chamizo, D. Raboso]{Fernando Chamizo (UAM-ICMAT) \and Dulcinea Raboso}


\begin{document}

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\titlepage
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\section{Tarjetas perforadas}
%%%%%%%%%%%%
% SLIDE 1  %
%%%%%%%%%%%%
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Tarjetas perforadas
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    \node[anchor=north east,align=right,ct2!70!black] at (10,0) {La baraja del\\ mago matemático};
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%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 2  %
%%%%%%%%%%%%
\begin{frame}{Sistema binario}
\vskip-10pt
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Es un sistema de numeración en el que los números se representan utilizando solamente dos cifras, {\color{rojo1}0} y {\color{rojo1}1}.
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\vskip5pt

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%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 3  %
%%%%%%%%%%%%
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\vskip12.5pt

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%%%%%%%%%%
\section{Topología}
%%%%%%%%%%%%
% SLIDE 4  %
%%%%%%%%%%%%
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Topología
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%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 5  %
%%%%%%%%%%%%
\begin{frame}{Deformaciones sin cortes}
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    \node (t2) at (2.4,-3.5) {\includegraphics[width=3.2cm]{./images/tira1v}};
    \draw[fill=white]
        ($(t2.south west)+(45:3pt)$) circle [radius=2pt]
        ($(t2.north east)+(-135:3pt)$) circle [radius=2pt];
    \draw[fill=rojo1]
        ($(t2.north west)+(-45:3pt)$) circle [radius=2pt]
        ($(t2.south east)+(135:3pt)$) circle [radius=2pt];
    \node[anchor=east] at (.8,-3.5) {\includegraphics[height=1cm]{./images/f3}};
    \node at (7.8,-3.5) {\includegraphics[width=2.3cm]{./images/moeb1v}};
}
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 6  %
%%%%%%%%%%%%
\begin{frame}{Banda de Möbius}

\begin{miblock}
La \textbf{banda de Möbius} es una superficie con una sola cara y un solo borde que además es un objeto no orientable, es decir, invierte el sentido de los objetos que viajan sobre ella.
\end{miblock}
\vskip12.5pt

\begin{tcbraster}[raster force size=false,enhanced]
    \begin{tcolorbox}[add to width=-1.8cm,sharp corners,,drop fuzzy shadow=ct2!70!black,colback=white,colframe=ct2!70!black,size=small]
    \includegraphics[width=3cm]{./images/diestrozurdo}
    \end{tcolorbox}
    \begin{tcolorbox}[add to width=1.8cm,halign=flush left,size=fbox,frame empty,interior empty]
    Si un habitante (bidimensional) de la banda de Möbius se cansara de ser diestro, no tendría más que dar un paseo por su mundo.
    \end{tcolorbox}
\end{tcbraster}

\begin{tikzpicture}[>=latex]
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-3.5);
    \node[inner sep=0pt] at (5,-1.5) {\includegraphics[width=10.5cm]{./images/mpaseo}};
    \draw[->,line width=1pt] (1.2,-2.05) to[bend right=15] +(30:.7cm);
    \draw[->,line width=1pt] (4.2,-.65) to[bend right=50] +(175:.85cm);
    \draw[->,line width=1pt] (5.6,-1.42) to[bend right=15] +(260:.4cm);
    \draw[->,line width=1pt] (7.72,-2.2) to[out=210,in=200] +(-5:.7cm);
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 7  %
%%%%%%%%%%%%
\begin{frame}{Extrañas propiedades}
\begin{tikzpicture}[inner sep=0pt]
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-7.3);
    \draw[rounded corners,draw=ct2,line width=1pt] (0,-.3) rectangle (10,-6.8);%
    \draw[dashed,line width=1pt,ct2] (5.3,-.3) -- (5.3,-6.8);
    \node[anchor=east] at (.8,-5.5) {\includegraphics[height=1cm]{./images/f3}};
    \node[anchor=west] at (4,-5.5) {\includegraphics[height=1cm]{./images/f3r}};
    \node at (2.4,-1.5) {\includegraphics[width=3.2cm]{./images/tira2v}};
    \node at (2.4,-3.5) {\includegraphics[width=3.2cm]{./images/tira2v}};
    \node at (2.4,-5.5) {\includegraphics[width=3.2cm]{./images/tira2v}};
    \node[anchor=east] at (.8,-3.5) {\includegraphics[height=1cm]{./images/f3}};
    \node at (7.8,-1.5) {\includegraphics[width=2.5cm]{./images/cil2v}};
    \node at (7.8,-3.5) {\includegraphics[width=2.3cm]{./images/moeb2v}};
    \node at (7.9,-5.5) {\includegraphics[width=2.8cm]{./images/banda2v}};
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 8  %
%%%%%%%%%%%%
\begin{frame}{Extrañas propiedades}

\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-7);
    \node[anchor=north,text width=10.4cm,align=flush left,draw=ct2,fill=white,rounded corners,line width=1pt] at (5,0)
        {Al cortar por la mitad un cilindro de papel, se obtienen dos cilindros con el mismo radio y la mitad de altura que el original.};
    \node[inner sep=0pt] at (2.5,-1.9) {\includegraphics[width=2.8cm]{./images/cortescil1}};
    \node[inner sep=0pt] at (2,-3.5) {\includegraphics[width=2.8cm]{./images/cortesmoeb1}};
    \node[inner sep=0pt] at (1.5,-5.5) {\includegraphics[width=2.8cm]{./images/cortesmoeb2}};
    \draw[->,line width=1.5pt,rojo1] (4.2,-2) -- +(2.3,0);
    \node at (8,-1.8) {\includegraphics[width=2.5cm,height=.6cm]{./images/cil}};
    \node at (8.5,-2.3) {\includegraphics[width=2.5cm,height=.6cm]{./images/cil}};
    \node[draw=ct2,fill=white,rounded corners,line width=1pt,inner sep=7pt,anchor=north west,text width=5.3cm,align=flush center] at (4,-3)
        {Si se hace lo mismo con la banda de Möbius, ¿qué sucede?};
    \node[inner sep=0pt] at (9,-6.2) {\includegraphics[width=2cm]{./images/tijeras2}};
    \node[draw=ct2,fill=white,rounded corners,line width=1pt,inner sep=7pt,anchor=north west,text width=5.3cm,align=flush center] at (4,-4.8)
        {¿y si el corte se hace a un tercio de distancia?};
    \node[inner sep=0pt] at (9,-6.2) {\includegraphics[width=2cm]{./images/tijeras1}};
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 9  %
%%%%%%%%%%%%
\begin{frame}{El problema del camarero}
\vspace*{-10pt}

\begin{tcbraster}[raster force size=false,enhanced]
    \begin{tcolorbox}[add to width=-2cm,sharp corners,drop fuzzy shadow=ct2!70!black,colback=white,colframe=ct2!70!black,size=small,halign=center]
    \includegraphics[width=1.5cm]{./images/vaso}
    \end{tcolorbox}
    \begin{tcolorbox}[add to width=2cm,halign=flush left,size=small,frame empty,interior empty]
    Si realizamos \textbf{dos} rotaciones de la mano mientras sostenemos en la palma un vaso de agua, terminamos en la posición inicial.
    \end{tcolorbox}
\end{tcbraster}

\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-3.25);
    \node[anchor=west,inner sep=0,text width=6cm,align=flush left,font=\small] at (-.35,-1.3)
        {La mano hace una rotación sobre su hombro, retorciendo el brazo, y luego otra rotación pasando por debajo, desenrollándolo.};
    \node[inner sep=0] at (8,-1.3) {\includegraphics[width=3.5cm]{./images/diract1}};
\only<1>{
    \coordinate (i) at (9.7,-1.6);
    \draw[rojo1,line width=2pt,opacity=.8] (i) -- +(45:.15) -- +(225:.15) (i) -- +(139:.15) -- +(315:.15);
    \draw[ct2!80,line width=2pt,>=latex,->] (7,-.05) to[bend right=20] (6.3,-.4);
    \draw[ct2!80,line width=2pt,>=latex,->] (6.3,-1.8) to[bend right=20] (7,-2.2);
}
\only<2>{
    \node[inner sep=0,opacity=.7] at (8,-1.3) {\includegraphics[width=3.5cm]{./images/diracb1}};
    \node[inner sep=0,font=\small] at (5,-2.9) {\structure{¡La trayectoria describe el borde de una banda de Möbius!}};
}
\end{tikzpicture}

\begin{miblock}
El problema está relacionado con el \emph{espín} de los llamados \emph{fermiones}, y ``explica'' por qué un electrón tiene que girar $720^\circ$ para recuperar su ``estado'' original.
\end{miblock}

\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 10 %
%%%%%%%%%%%%
\begin{frame}{Cinturón de Dirac}

\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-7);
    \node[anchor=west,text width=3cm] at (-.2,-2.5)
        {Sujeta un cinturón a una barra\ldots};
    \node[anchor=east,text width=3.5cm,align=flush right] at (10.2,-4)
        {\ldots y retuércelo unas cuantas veces.};
    \node[anchor=north,inner sep=0] at (6,0) {\includegraphics[width=8cm]{./images/dirac3}};
    \node[inner sep=0,opacity=.5] at (3.85,-3.3) {\includegraphics[width=2.4cm]{./images/d3f}};
    \node[inner sep=0] at (3.85,-3) {\includegraphics[width=.5cm]{./images/d3m}};
    \node[text width=10cm,align=flush center,draw=ct2,fill=white,rounded corners,line width=.5pt] at (5,-6)
        {Si no permitimos girar la barra verticalmente,\\ \structure{¿se puede desenrollar?}};
\end{tikzpicture}
\end{frame}
%%%%%%%%%%%
\section{Curvas misteriosas}
%%%%%%%%%%%%
% SLIDE 11 %
%%%%%%%%%%%%
\begin{frame}
\vskip3cm
\begin{tcolorbox}[enhanced,center,width=8.5cm,boxsep=10pt,halign=center,%
                  fontupper=\Large,colupper=ct2!70!black,%
                  colback=white,colframe=ct2!70!black,%
                  sharp corners=downhill,leftrule=5pt,rightrule=5pt,drop fuzzy shadow=ct2!70!black]
Curvas misteriosas
\end{tcolorbox}
\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-.35,0) grid (10.35,-3);
    \node[anchor=north east,align=right,ct2!70!black] at (10,0) {Ruedas, reflejos\\ y cangrejos};
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 12 %
%%%%%%%%%%%%
\begin{frame}{El problema}

\begin{tcbraster}[raster width=9.5cm,raster halign=center,raster force size=false,enhanced,halign=flush center,size=small]
    \begin{tcolorbox}[add to width=1.6cm,frame empty,interior empty]
    Si hacemos girar, sin deslizar, una moneda sobre otra\\[5pt]
    \structure{¿cuántos giros completos dará?}\\[5pt]
    \begin{tikzpicture}
    \draw[step=1cm,help lines,opacity=0] (0,0) grid (5,-2.5);
        \node[inner sep=0,ct2!80] at (1,-1) {\scalebox{3}{\textbf{?}}};
        \node[inner sep=0,ct2] at (2.5,-1.5) {\scalebox{3}{\textbf{?}}};
        \node[inner sep=0,ct2!80] at (4,-2) {\scalebox{3}{\textbf{?}}};
        \node[inner sep=0,ct2!60] at (3.5,-.5) {\scalebox{2}{\textbf{?}}};
        \node[inner sep=0,ct2!60] at (1.5,-2) {\scalebox{2}{\textbf{?}}};
        \node[inner sep=0,ct2!30] at (4.5,-1) {\scalebox{1.5}{\textbf{?}}};
        \node[inner sep=0,ct2!30] at (2,-.5) {\scalebox{1.5}{\textbf{?}}};
        \node[inner sep=0,ct2!30] at (.5,-2.2) {\scalebox{1.5}{\textbf{?}}};
        \node[inner sep=0,ct2!30] at (3.1,-2.1) {\textbf{?}};
    \end{tikzpicture}
    \end{tcolorbox}
    \begin{tcolorbox}[add to width=-1.6cm,sharp corners,drop fuzzy shadow=ct2!70!black, colback=white,colframe=ct2!70!black]
    \begin{tikzpicture}
    \draw[step=1cm,help lines,opacity=0] (-1.25,-2) grid (1.25,2);
        \node[inner sep=0] at (0,0) {\includegraphics[height=3cm]{./images/giro8}};
        \draw[>=latex,->,ct2,line width=2pt] (.2,1.8) to[bend left=40] (1,1);
    \end{tikzpicture}
    \end{tcolorbox}
\end{tcbraster}

\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 13 %
%%%%%%%%%%%%
\begin{frame}
\vskip12.5pt
\begin{tikzpicture}
\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}
            \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);
\invisible<13>{\node[inner sep=0] at (0,.7) {\Acht};}
%%%%
\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}{
\only<\time>{%
    \def\ang{\p+\g}
    %\draw[rojo1] (0,0)--(\p:1);
    %\draw (0,0)--(\ang:1);
%%%%MÓVIL
    \begin{scope}[shift={(\ang:2)},rotate=2*\g]
        \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);
        \node[inner sep=0,rotate=2*\g] at (0,.7) {\Acht};
        %%%%
        %\draw[rojo1] (0,0)--(\p:-1);
        %\draw (0,0)--(\p-\g:-1);
    \end{scope}
}}
\only<13>{
    \def\g{-330}
    \def\ang{\p+\g}
%%%%MÓVIL
    \begin{scope}[shift={(\ang:2)},rotate=2*\g]
        \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);
        \node[inner sep=0,rotate=2*\g] at (0,.7) {\Acht};
%%%%
        \filldraw[draw=rojo1,fill=rojo1!10] (0,0)-- (\p:-1) arc (\p:\p-30:-1) -- cycle;
        \filldraw[draw=rojo1,fill=ct1!10] (0,0)-- (\p-30:-1) arc (\p-30:\p-60:-1) -- cycle;
        \node[rojo1] at (\p-15:-.5) {$\alpha$};
        \node[rojo1] at (\p-45:-.5) {$\alpha$};
        \filldraw[rojo1] (\p:-1) circle (1pt);
    \end{scope}
    \filldraw[draw=rojo1,fill=rojo1!10] (0,0)-- (\p:1) arc (\p:\p+30:1) -- cycle;
    \node[rojo1] at (\p+15:.5) {$\alpha$};
    \draw[dashed,thick] (0,0)--++(0,1.5)--+(0,-2.8) (\p+30:2)--++(0,1.25)--+(0,-3);
    \filldraw[rojo1] (\p:1) circle (1pt);
}
\end{scope}
\only<13>{
    \node[inner sep=0pt] at (-2,-3) {Respecto a la vertical};
    \node[rounded corners,fill=white,draw=ct2,line width=1pt] at (2.5,-3)
        {$\begin{aligned}\alpha &= \text{giro del centro}\\ \alpha+\alpha &= \text{giro del punto}\end{aligned}$};
}
\only<1>{
    \node[text width=10.4cm,align=flush left,rounded corners,fill=white,draw=ct2,line width=.5pt] at (0,-2.5)
        {En la práctica, las monedas resbalan. Con CDs es más fácil verlo, usando el agujero central para ir desplazando el CD que gira.};}
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 14 %
%%%%%%%%%%%%
\begin{frame}
\vskip12.5pt
\begin{tikzpicture}
\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}
            \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);
%%%%
\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
    \begin{scope}[shift={(\ang:2)},rotate=2*\g]
        \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[rojo1] (\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}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 15 %
%%%%%%%%%%%%
\begin{frame}{Curvas misteriosas}

\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 curva que describe la trayectoria de cada punto del borde se llama \textbf{cardioide}.
        Ves algo parecido cuando pones un anillo o incluso una taza bajo una fuente de luz oblicua.};
    \begin{scope}[shift={(2.5,-3.5)}]
        \node[inner sep=0,draw=ct1,line width=2pt] {\includegraphics[width=3cm]{./images/ring2}};
    \end{scope}
    \begin{scope}[shift={(7.5,-3.5)}]
        \node[inner sep=0,draw=ct1,line width=2pt] {\includegraphics[width=3cm]{./images/taza}};
    \end{scope}
    \node[text width=10.5cm,align=flush left,inner sep=0pt] at (5,-6)
        {El efecto requiere que la fuente sea puntual. Es claro con una bombilla, una linterna o un foco halógeno pero no con un fluorescente. Es muy fácil verlo bajo estas condiciones.};
\end{tikzpicture}

\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 16 %
%%%%%%%%%%%%
\begin{frame}{Curvas misteriosas}

\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)
        {Si acercamos la fuente de luz hasta el borde del anillo, el reflejo se encoge hasta formar una cardioide.
        La segunda curva, una vez duplicada, se llama \textbf{nefroide} y es lo que se obtendría si el CD que gira fuera un miniCD de radio la mitad.};
    \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))});
    \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)});
    \end{scope}
    \node[inner sep=0pt] at (2.5,-6) {\structure{Luz en el borde}};
    \node[inner sep=0pt] at (7.5,-6) {\structure{Luz lejana}};
\end{tikzpicture}

\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 17 %
%%%%%%%%%%%%
\begin{frame}{¿Por qué se ven estas figuras?}

\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)
        {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) {\structure{Luz en el borde}};
    \node[inner sep=0pt] at (7.5,-6) {\structure{Luz lejana}};
\end{tikzpicture}

\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 18 %
%%%%%%%%%%%%
\begin{frame}{Más curvas misteriosas\ldots}

\begin{tikzpicture}
\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 hoja 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 que no aparecen en la realidad.};
\end{tikzpicture}
\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 19 %
%%%%%%%%%%%%
\begin{frame}{Más curvas misteriosas\ldots}

\begin{tikzpicture}
\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}
\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 20 %
%%%%%%%%%%%%
\begin{frame}{El efecto moiré}

\begin{miblock}
{\color{rojo1}\ding{55}} \ 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}
{\color{verde2}\ding{51}} \ También se utiliza en sentido positivo para efectuar mediciones ópticas con cierta precisión y alguna vez ha sido propuesto para dotar de medidas de seguridad a las tarjetas bancarias.
\end{miblock}

\vskip10pt

Las matemáticas que hay debajo tienen que ver con el análisis de Fourier que afirma que es posible escribir funciones como superposición de ondas. El efecto de moiré viene de una interferencia en ellas.

\end{frame}
%%%%%%%%%%%
\section{Serie armónica}
%%%%%%%%%%%%
% SLIDE 21 %
%%%%%%%%%%%%
\begin{frame}
\vskip3cm
\begin{tcolorbox}[enhanced,center,width=8.5cm,boxsep=10pt,halign=center,%
                  fontupper=\Large,colupper=ct2!70!black,%
                  colback=white,colframe=ct2!70!black,%
                  sharp corners=downhill,leftrule=5pt,rightrule=5pt,drop fuzzy shadow=ct2!70!black]
Serie armónica
\end{tcolorbox}
\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-.35,0) grid (10.35,-3);
    \node[anchor=north east,align=right,ct2!70!black] at (10,0) {¿Sostener media escalera\\ sin una pared?};
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 22 %
%%%%%%%%%%%%
\begin{frame}{El problema}

\begin{tcbraster}[raster width=9.5cm,raster halign=center,raster force size=false,enhanced,halign=flush center,size=small]
    \begin{tcolorbox}[add to width=1.6cm,frame empty,interior empty]
    ¿Cuánto se puede separar de la vertical una pila de libros?\\[5pt]
    {\Large \structure{¡Sin caerse!}}\\[5pt]
    \begin{tikzpicture}
    \draw[step=1cm,help lines,opacity=0] (0,0) grid (5,-2.5);
        \node[inner sep=0,ct2!80] at (1,-1) {\scalebox{3}{\textbf{?}}};
        \node[inner sep=0,ct2] at (2.5,-1.5) {\scalebox{3}{\textbf{?}}};
        \node[inner sep=0,ct2!80] at (4,-2) {\scalebox{3}{\textbf{?}}};
        \node[inner sep=0,ct2!60] at (3.5,-.5) {\scalebox{2}{\textbf{?}}};
        \node[inner sep=0,ct2!60] at (1.5,-2) {\scalebox{2}{\textbf{?}}};
        \node[inner sep=0,ct2!30] at (4.5,-1) {\scalebox{1.5}{\textbf{?}}};
        \node[inner sep=0,ct2!30] at (2,-.5) {\scalebox{1.5}{\textbf{?}}};
        \node[inner sep=0,ct2!30] at (.5,-2.2) {\scalebox{1.5}{\textbf{?}}};
        \node[inner sep=0,ct2!30] at (3.1,-2.1) {\textbf{?}};
    \end{tikzpicture}
    \end{tcolorbox}
    \begin{tcolorbox}[add to width=-1.6cm,sharp corners,drop fuzzy shadow=ct2!70!black, colback=white,colframe=ct2!70!black]
    \begin{tikzpicture}
        \draw[step=1cm,help lines,opacity=0] (-1.25,-2) grid (1.25,2);
        \node[inner sep=0] at (0,0) {\includegraphics[height=3cm]{./images/libros}};
        \draw[>=latex,->,ct1,line width=2pt] (-.8,1.7) -- (.8,1.7);
    \end{tikzpicture}
    \end{tcolorbox}
\end{tcbraster}

\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 23 %
%%%%%%%%%%%%
\begin{frame}{Posición límite de equilibrio}
\vskip-5pt
\begin{tikzpicture}
\def\L{8}
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-7.4);
\only<1>{
    \node[anchor=west,inner sep=0pt] at (0,-1) {Con 2 libros:};
    \node[rounded corners,fill=white,draw=ct2,line width=1pt,text width=10cm,align=flush center] at (5,-3.5)
        {¿Por qué anchura máxima apostaríamos si disponemos de tantos libros como deseemos?};
    \node[text width=10cm,align=flush left,inner sep=0pt] at (5,-5.5)
        {Parece claro que el último libro no puede estar más allá de la base de sustentación, ¿pero es verdad?};
}
\begin{scope}[xshift=9.5cm,yshift=-.65cm,scale=.5]
    \draw[<->] (0,.5) -- +(-\L,0) node[above,pos=.5,font=\footnotesize] {$1$};
    \draw[ct2,top color=ct1!30,bottom color=ct1!,line width=1pt] (0,0) rectangle  +(-\L,-1) node[anchor=south west,font=\small] {$1$};
    \draw[ct2,top color=ct1!30,bottom color=ct1!,line width=1pt] (-\L/2,-1) rectangle +(-\L,-1) node[anchor=south west,font=\small] {$2$};
    \draw[<->] (-\L/2,-1.5) -- +(\L/2,0) node[below,pos=.5,font=\footnotesize] {$\dfrac12$};
\only<2>{
    \draw[<->,rojo1] (-\L-\L/2,-2.5) -- +(\L*3/4,0) node[below,pos=.5,font=\footnotesize] {$\displaystyle{c_2=\frac{1+1/2}{2}=\frac34}$};
    \draw[<->] (-\L*3/4,-2.5) -- +(\L/4,0) node[below,pos=.5,font=\footnotesize] {$\dfrac14$};
    \draw[dashed,gray] (-\L*3/4,-3) -- (-\L*3/4,.5);
}
\only<3->{
    \draw[ct2,top color=ct1!30,bottom color=ct1!,line width=1pt] (-\L*3/4,-2) rectangle +(-\L,-1) node[anchor=south west,font=\small] {$3$};
    \draw[<->] (-\L*3/4,-2.5) -- +(\L/4,0) node[below,pos=.5,font=\footnotesize] {$\dfrac14$};
}
\only<4>{
    \draw[<->,rojo1] (-\L-\L*3/4,-3.5) -- +(\L*5/6,0) node[below,pos=.5,font=\footnotesize] {$\displaystyle{c_3=\frac{2+1/2}{3}=\frac56}$};
    \draw[<->] (-\L*11/12,-3.5) -- +(\L/6,0) node[below,pos=.5,font=\footnotesize] {$\dfrac16$};
    \draw[dashed,gray] (-\L*11/12,-4) -- (-\L*11/12,.5);
}
\only<5->{
    \draw[ct2,top color=ct1!30,bottom color=ct1!,line width=1pt] (-\L*11/12,-3) rectangle +(-\L,-1) node[anchor=south west,font=\small] {$4$};
    \draw[<->] (-\L*11/12,-3.5) -- +(\L/6,0) node[below,pos=.5,font=\footnotesize] {$\dfrac16$};
}
\only<6>{
    \draw[<->,rojo1] (-\L-\L*11/12,-4.5) -- +(\L*7/8,0) node[below,pos=.5,font=\footnotesize] {$\displaystyle{c_4=\frac{3+1/2}{4}=\frac78}$};
    \draw[<->] (-\L*25/24,-4.5) -- +(\L/8,0) node[below,pos=.5,font=\footnotesize] {$\dfrac18$};
    \draw[dashed,gray] (-\L*25/24,-5) -- (-\L*25/24,.5);
}
\only<7->{
    \draw[ct2,top color=ct1!30,bottom color=ct1!,line width=1pt] (-\L*25/24,-4) rectangle +(-\L,-1) node[anchor=south west,font=\small] {$5$};
    \draw[<->] (-\L*25/24,-4.5) -- +(\L/8,0) node[below,pos=.5,font=\footnotesize] {$\dfrac18$};
}
\only<8->{
    \filldraw[ct2] (-13,-6.5) circle [radius=2pt] (-12.5,-6) circle [radius=2pt] (-12,-5.5) circle [radius=2pt];
    \draw[ct2,top color=ct1!30,bottom color=ct1!,line width=1pt] (-\L*363/280,-7) rectangle +(-\L,-1) node[anchor=south west,font=\small] {$n$};
    \draw[<->] (-\L*761/560,-8.5) -- +(\L/16,0) node[below,pos=.5,font=\footnotesize] {$\dfrac{1}{2n}$};
}
\only<8>{
    \draw[<->,rojo1] (-\L-\L*363/280,-8.5) -- +(\L*15/16,0) node[below,pos=.5,font=\footnotesize] {$\displaystyle{c_n=\frac{n-1+1/2}{n}}$};
}
\only<9->{
    \draw[ct2,top color=ct1!30,bottom color=ct1!,line width=1pt] (-\L*761/560,-8) rectangle +(-\L,-1) node[anchor=south west,font=\small] {$n+1$};
}
\only<9-10>{
    \draw[<->,azulS,thick] (-\L*761/560,-10.5) -- (0,-10.5) node[below,pos=.5,font=\small] {separación de la vertical $\displaystyle{=\sum_{k=1}^n\frac{1}{2k}}$};
}
\only<10>{
    \draw[<->,azulS,thick] (-\L*25/24,-6.5) -- (0,-6.5) node[below,pos=.5,font=\small] {$\displaystyle{\sum_{k=1}^4\frac{1}{2k}=1.0416\ldots>1}$};
}
\end{scope}
\only<11>{
    \node[rounded corners,draw=ct2,line width=.5pt,font=\footnotesize] at (5,-6.6)
    {$\displaystyle{\sum_{k=1}^\infty\frac{1}{2k}=\Big(\frac{1}{2}\Big)+\Big(\frac{1}{4}\Big)+\underset{2 \text{ num. } \ge 1/8}{\Big(\frac{1}{6}+\frac{1}{8}\Big)}+\underset{4 \text{ num. } \ge 1/16}{\Big(\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+\frac{1}{16}\Big)}+\dots}$};
}
\end{tikzpicture}
\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 24 %
%%%%%%%%%%%%
\begin{frame}{Ver para creer\ldots}

\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,-.5) grid (10.35,-7.5);
    \begin{scope}
        \clip[rounded corners] (.25,-1.8) rectangle (4.75,-5.2);
        \node at (2.5,-3.5) {\includegraphics[width=4.5cm]{./images/stack1}};
    \end{scope}
    \draw[ct2,rounded corners,line width=1pt] (.25,-1.8) rectangle (4.75,-5.2);
    \begin{scope}
        \clip[rounded corners] (5.25,-1.8) rectangle (9.75,-5.2);
        \node[rotate=-4] at (7.5,-3.6) {\includegraphics[width=5.2cm]{./images/stack2}};
    \end{scope}
    \draw[ct2,rounded corners,line width=1pt] (5.25,-1.8) rectangle (9.75,-5.2);
    \node[font=\Large] at (5,-6) {\structure{¡Sin caerse!}};
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
\section{Velocidad límite}
%%%%%%%%%%%%
% SLIDE 25 %
%%%%%%%%%%%%
\begin{frame}
\vskip3cm
\begin{tcolorbox}[enhanced,center,width=8.5cm,boxsep=10pt,halign=center,%
                  fontupper=\Large,colupper=ct2!70!black,%
                  colback=white,colframe=ct2!70!black,%
                  sharp corners=downhill,leftrule=5pt,rightrule=5pt,drop fuzzy shadow=ct2!70!black]
Velocidad límite
\end{tcolorbox}
\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-.35,0) grid (10.35,-3);
    \node[anchor=north east,align=right,ct2!70!black] at (10,0) {Echa el freno};
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 26 %
%%%%%%%%%%%%
\begin{frame}{Un chiste malo}

\begin{tcolorbox}[enhanced,size=small,center,width=5.5cm,halign=center,sharp corners,drop lifted shadow=ct2!70!black, colback=white,colframe=ct2!70!black,boxsep=2pt]
\begin{tikzpicture}
    \node[inner sep=0] at (0,0) {\includegraphics[width=5cm]{./images/carretera}};
\draw[step=1cm,help lines,opacity=0] (-2.3,-1) grid (2.3,1);
    \node[inner sep=0,align=center] at (-.75,.9) {\scalebox{.85}{Reduzca}};
    \node[inner sep=0,align=center] at (-.75,.55) {\scalebox{.85}{a 100}};
    \node[inner sep=0,align=center] at (.65,1.05) {\scalebox{.6}{Reduzca}};
    \node[inner sep=0,align=center] at (.65,.8) {\scalebox{.6}{a 80}};
    \node[inner sep=0,align=center] at (1.85,1.1) {\scalebox{.4}{Reduzca}};
    \node[inner sep=0,align=center] at (1.85,.92) {\scalebox{.4}{a 60}};
\end{tikzpicture}
\end{tcolorbox}

\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-4);
\only<1-2>{
    \node[anchor=west,text width=6.5cm,align=flush left,inner sep=0pt] at (-.3,-2)
        {Un conductor va por la carretera y ve un cartel que avisa ``Reduzca a 100'', después otro que dice ``Reduzca a 80'' y así sucesivamente, hasta llegar a un gran cartel que indica:};
}
\only<2>{
    \begin{scope}[shift={(8.5,-2)}]
        \node[inner sep=0] at (0,0) {\includegraphics[width=3cm]{./images/cartel}};
        \node[inner sep=0,align=center] at (.2,.65) {Bienvenido\\ a Reduzca};
    \end{scope}
}
\only<3>{
    \node[text width=10.5cm,align=flush left,inner sep=0pt] at (5,-.6)
        {Hay 5 tramos de 20 km. El primero se recorre a 100 km/h, el segundo a 80 km/h, hasta el último que se recorre a 20 km/h.};
    \coordinate (p0) at (1,-2);\coordinate (p5) at (9,-2);
    \foreach \x in {1,2,3,4}
        \coordinate (p\x) at ($(p0)!\x/5!(p5)$);
    \draw[thick,azulS] (p0)--(p5);
    \foreach \i/\f/\v in {0/1/100,1/2/80,2/3/60,3/4/40,4/5/20}{
        \node[below,font=\footnotesize] at ($(p\i)!.5!(p\f)$) {\v km/h};
        \draw[very thick,azulS] ([yshift=-.1cm]p\i) node[above=5pt]{\v} -- ([yshift=.1cm]p\i);}
    \draw[very thick,azulS] ([yshift=-.1cm]p5) node[above=5pt]{0} -- ([yshift=.1cm]p5);
    \node[text width=10cm,align=center,inner sep=0pt] at (5,-3.5)
        {$\Delta t=\dfrac{\Delta x}{v}\quad \Rightarrow \quad T=\dfrac{20}{100}+\dfrac{20}{80}+\dfrac{20}{60}+\dfrac{20}{40}+\dfrac{20}{20}=2 \text{h.}\ 17 \text{m.}$};
}
\end{tikzpicture}

\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 27 %
%%%%%%%%%%%%
\begin{frame}{La realidad supera a la ficción\ldots}

\tcbox[enhanced,size=title,center,halign=center,sharp corners,drop lifted shadow=ct2!70!black,colback=white,colframe=ct2!70!black,%
left=0pt,right=0pt,top=0pt,bottom=0pt,boxsep=0pt,%
flip title,colbacktitle=ct2!70!black,fonttitle={\footnotesize},halign title=right,title={Faro Sabinar (Almería)}]
{\includegraphics[width=6.5cm]{./images/reduzca}}

\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 28 %
%%%%%%%%%%%%
\begin{frame}{Logaritmo neperiano}

\begin{miblock}
Los logaritmos neperianos son muy importantes y, aunque parezca mentira, J.~Neper los inventó en 1614 mediante una variante de este chiste.
\end{miblock}
\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-4);
    \node[text width=5.2cm,align=flush left,inner sep=0pt] at (2.5,-1.5)
        {En el caso de Neper, la distancia inicial era 1 y había infinitas señales, con lo cual el coche nunca llegaba.};
    \coordinate (p0) at (5.8,-1.5);\coordinate (p2) at (9.8,-1.5);
    \coordinate (p1) at ($(p0)!3/4!(p2)$);
    \draw[thick,azulS] (p0)--(p2);
    \draw[very thick,azulS] ([yshift=-.1cm]p0) node (a) [above=5pt]{1} -- ([yshift=.1cm]p0);
    \draw[very thick,azulS] ([yshift=-.1cm]p1) node[above=5pt]{$x$} -- ([yshift=.1cm]p1);
    \draw[very thick,azulS] ([yshift=-.1cm]p2) node[above=5pt]{0} -- ([yshift=.1cm]p2);
    \draw[->,thick,rojo1] (a.east) -- +(1,0);
    \node[rounded corners,fill=white,draw=ct2,line width=.5pt] at (5,-3.2)
        {$-\ln(x)=$ tiempo en llegar a distancia $x$};
\end{tikzpicture}

{\small Los logaritmos permiten pasar multiplicaciones a sumas, divisiones a restas y raíces a divisiones.}
\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 29 %
%%%%%%%%%%%%
\begin{frame}{Velocidad límite}

\begin{miblock}
Hay situaciones en las que la velocidad se comporta como el espacio recorrido en el chiste: se incrementa cada vez menos y hay una \textbf{velocidad límite}.
\end{miblock}
\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-5);
    \node[anchor=west,text width=7cm,align=center,inner sep=0pt] at (-.35,-1.5)
        {\structure{¿A qué velocidad cae la lluvia?}\\[10pt]
        \structure{¿Por qué los paracaidistas llevan gafas?}};
    \begin{scope}[shift={(8,-1)}]
        \foreach \x in {-.7,0,.7}{
        \foreach \y/\s in {-.8/1,-1.6/.9,-2.5/.8,-3.5/.7,-4.8/.6,-5.5/.6}{
            \fill[top color=blue!40!cyan!60, bottom color=blue!50,scale=\s] (\x+rand/5,\y+rnd/3)  to[in=-50,out=-120,loop] (\x,\y);}}
        \node [cloud, cloud puffs=9, draw=blue!50,top color=blue!40!cyan!60, bottom color=blue!50, minimum width=2.5cm, minimum height=1.5cm] at (0,0) {};
    \end{scope}
    \draw[decorate,decoration=brace] (9,-1.5)--(9,-2.5) node[pos=.5,right,font=\footnotesize] {$v$ crece};
    \draw[decorate,decoration=brace] (9,-2.6)--(9,-4.5) node[pos=.5,right,font=\footnotesize,align=left] {$v<v_{\ell}$\\[5pt]$v\sim $ cte};
\only<2>{
    \node[text width=10.5cm,align=flush left,inner sep=0pt] at (5,-4)
        {Dependiendo de las condiciones, se tiene:
    \begin{itemize}
    \item Gotas de lluvia gruesa: $v_{\ell}=36\ \text{Km/h}$
    \item Salto en paracaídas: $v_{\ell}=200\ \text{Km/h}$
    \end{itemize}};
}
\end{tikzpicture}

\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 30 %
%%%%%%%%%%%%
\begin{frame}{Velocidad límite}

Un ejemplo menos conocido es el de un imán que cae por un tubo metálico que no se pegue al imán.

\begin{tcbraster}[raster force size=false,enhanced]
    \begin{tcolorbox}[add to width=-1cm,size=small,halign=center,frame empty,interior empty]
    \begin{tikzpicture}
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    Un campo magnético que varía mueve los electrones de metal que a su vez, crean otro campo magnético.\\ Según la ley de Lenz, estos campos son opuestos.
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Rápidamente la velocidad del imán se acerca a la velocidad límite, que es pequeña si el imán es potente.

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