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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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\newcommand{\Q}{\mathbb{Q}}
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%%%%% 	COLORES
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\definecolor{amarillO}{RGB}{248,172,0} %amarilloICMAT
\definecolor{azulD}{RGB}{64,80,147} %azulDULCI
\definecolor{azulS}{RGB}{51,51,178} %azulstructure
\definecolor{rojo1}{rgb}{0.7,0,0}
\definecolor{verdeC2}{RGB}{102,153,0}
%%*****************************************
%% Colores tema
%%*****************************************
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%%%%%%%%%%%%%%%%%%%%%%%%%%


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


%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Datos
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\title[La magia de los números]{La magia de los números}
\institute{I.E.S. Avenida de los Toreros}
\date{6 de abril 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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\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 2  %
%%%%%%%%%%%%
\begin{frame}{Sistema binario}
\vskip-10pt
\begin{miblock}
Es un sistema de numeración en el que los números se representan utilizando solamente dos cifras, {\color{rojo1}0} y {\color{rojo1}1}.
\end{miblock}
\vskip5pt

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    \node[white,fill=white,font=\small] at (.9,-2.4) {$6$};
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\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 3  %
%%%%%%%%%%%%
\begin{frame}
\vskip12.5pt

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%%%%%%%%%%
\section{Banda de Möbius}
%%%%%%%%%%%%
% SLIDE 4  %
%%%%%%%%%%%%
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\vskip3cm
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Banda de Möbius
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\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 5  %
%%%%%%%%%%%%
\begin{frame}{Un poco de topología}
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        ($(t2.north west)+(-45:3pt)$) 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}{Extrañas propiedades}

\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 5  %
%%%%%%%%%%%%
\begin{frame}{Un poco de topología}
\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 7  %
%%%%%%%%%%%%
\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=10.5cm,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 8  %
%%%%%%%%%%%%
\begin{frame}{El problema del camarero}
\vspace*{-10pt}

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\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}
%%%%%%%%%%%
\section{Serie armónica}
%%%%%%%%%%%%
% 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]
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 12 %
%%%%%%%%%%%%
\begin{frame}{}
\vskip12.5pt

\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-7);
    \node[font=\large] at (5,-1) {\structure{¿Cuánto se puede separar de la vertical una pila de libros?}};
    \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}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 13 %
%%%%%%%%%%%%
\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}
%%%%%%%%%%%
\section{Puzzle misterioso}
%%%%%%%%%%%%
% SLIDE 14 %
%%%%%%%%%%%%
\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]
¿Qué ven mis ojos?
\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) {\ldots o qué no ven.};
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 15 %
%%%%%%%%%%%%
\begin{frame}{El puzzle}

Se trata de una variante del puzzle creado por Pat Patterson ``\emph{The Vanishing Leprechaun}''. Podemos hacer desaparecer con nuestras propias manos una figura dibujada en un papel y que vuelva a aparecer cuando queramos.

\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,.2) grid (10.35,-5);
\only<2>{
    \node[inner sep=0pt,rotate=-90] at (3.5,-.7) {\includegraphics[width=1.5cm]{./images/tijeras1}};
}
\only<1-2>{
    \begin{scope}
    \clip (-.2,-1) rectangle (10.2,-3.3);
    \node[anchor=north,inner sep=0pt] at (5,-1) {\includegraphics[width=10.4cm]{./images/todo}};
    \end{scope}
    \draw[ct2,line width=1pt] (-.2,-1) rectangle (10.2,-3.3);
}
\only<1>{
    \node[text width=10cm,align=flush center] at (5,-4) {\structure{¿Cuántas figuras se ven en la imagen?}};
}
\only<2>{
    \draw[rojo1,dashed,line width=1pt] (-.2,-2.125) -- +(10.2,0) (3.835,-1) -- +(0,-1.125);
    \node[inner sep=0pt,rotate=-90] at (3.5,-.7) {\includegraphics[width=1.5cm]{./images/tijeras2}};
    \node[text width=10cm,align=flush center] at (5,-4.25) {Si hacemos unos cortes ``estratégicos'' e intercambiamos las partes superiores\ldots};
}
\only<3>{
    \begin{scope}
    \clip (-.2,-1) rectangle (10.2,-3.3);
    \node[anchor=north,inner sep=0pt] at (5,-1) {\includegraphics[width=10.4cm]{./images/b}};
    \node[anchor=north west,inner sep=0pt] (i) at (6.135,-1.02) {\includegraphics[width=10.4cm]{./images/i}};
    \node[anchor=north east,inner sep=0pt] at (6.165,-1.02) {\includegraphics[width=10.4cm]{./images/d}};
    \end{scope}
    \draw[ct2,line width=1pt] (-.2,-1) rectangle (10.2,-3.3);
    \draw[rojo1,dashed,line width=1pt] (-.2,-2.125) -- +(10.2,0) (i.north west) -- +(0,-1.125);
    \node[text width=10cm,align=flush center] at (5,-4) {\structure{¿Cuántas figuras se ven ahora?}};
}
\end{tikzpicture}
\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 15 %
%%%%%%%%%%%%
\begin{frame}{El puzzle}


\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-7);
    \node[anchor=north west,inner sep=0] at (-.35,0) {Coloreando las partes superiores:};
    \begin{scope}
    \clip (-.2,-1) rectangle (10.2,-3.3);
    \node[anchor=north,inner sep=0pt] at (5,-1) {\includegraphics[width=10.4cm]{./images/todoc}};
    \end{scope}
    \draw[ct2,line width=1pt] (-.2,-1) rectangle (10.2,-3.3);
    \draw[rojo1,dashed,line width=1pt] (-.2,-2.125) -- +(10.2,0) (3.835,-1) -- +(0,-1.125);

    \begin{scope}
    \clip (-.2,-4) rectangle (10.2,-6.3);
    \node[anchor=north,inner sep=0pt] at (5,-4) {\includegraphics[width=10.4cm]{./images/b}};
    \node[anchor=north west,inner sep=0pt] (i) at (6.135,-4.02) {\includegraphics[width=10.4cm]{./images/ic}};
    \node[anchor=north east,inner sep=0pt] at (6.165,-4.02) {\includegraphics[width=10.4cm]{./images/dc}};
    \end{scope}
    \draw[ct2,line width=1pt] (-.2,-4) rectangle (10.2,-6.3);
    \draw[rojo1,dashed,line width=1pt] (-.2,-5.125) -- +(10.2,0) (i.north west) -- +(0,-1.125);
\end{tikzpicture}
\end{frame}
%%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 16 %
%%%%%%%%%%%%
\begin{frame}{Variante con ¿24? lápices}

\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-0.35,0) grid (10.35,-7);
\begin{scope}
    \clip (.68,-.15) rectangle (9.328,-2.25);
    \node[anchor=north,inner sep=0pt] at (5,0) {\includegraphics[height=2.4cm]{./images/omissing}};
\end{scope}
\begin{scope}
    \clip (.68,-2.45) rectangle (9.328,-4.55);
    \node[inner sep=0pt] at (5,-3.5) {\includegraphics[height=2.4cm]{./images/pmissing}};
\end{scope}
\begin{scope}
    \clip (.68,-4.73) rectangle (9.328,-6.83);
    \node[anchor=south,inner sep=0pt] at (5,-7) {\includegraphics[height=2.4cm]{./images/pmissing11}};
\end{scope}
\end{tikzpicture}

\end{frame}
%%%%%%%%%%%
%%%%%%%%%%
\setbeamertemplate{headline}{
\leavevmode
\begin{beamercolorbox}[wd=\the\paperwidth,ht=0ex,dp=0ex]{section in head/foot}
  \begin{tikzpicture}
    \useasboundingbox[fill=ct1](0,0) rectangle (\the\paperwidth,0.5cm);
    \fill[fondo!5!white] (0,0) rectangle (\the\paperwidth,-0.1);
    \node[anchor=north west,font=\sffamily\bfseries\tiny] at (0.5,0.45){};%
  \end{tikzpicture}
\end{beamercolorbox}
\vskip-17pt}
%%%%%%%%%%%%
% SLIDE 17 %
%%%%%%%%%%%%
\begin{frame}
\vskip12.5pt
\begin{tikzpicture}
\invisible{\draw[step=1cm,help lines] (-.35,0) grid (10.35,-8);}
    \node[anchor=west,text width=10cm,align=left] at (0,-2.5)
        {Podéis encontrar la presentación en:};
    \draw[azulS] (-.35,-3.5) -- (10.35,-3.5);
    \node[azulS!80!black,font=\large] at (5,-4) {\url{https://www.uam.es/fernando.chamizo}};
    \draw[azulS] (-.35,-4.5) -- (10.35,-4.5);
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 18 %
%%%%%%%%%%%%
\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]
Bonus track
\end{tcolorbox}
\begin{tikzpicture}
\draw[step=1cm,help lines,opacity=0] (-.35,0) grid (10.35,-3);
\end{tikzpicture}
\end{frame}
%%%%%%%%%%
%%%%%%%%%%%%
% SLIDE 9  %
%%%%%%%%%%%%
\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}
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\end{document}
