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\def\imagetop#1{\vtop{\null\hbox{#1}}}

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\title{The 17 Worlds of Planar Ants}

\author{Dror Bar-Natan}

\institute{\url{http://drorbn.net/mc21}}

\date{MathCamp by Web, July 2021}

\begin{document} \beamertemplatenavigationsymbolsempty
\setbeamertemplate{headline}{\null\hfill\text{\footnotesize\purple Better vaxxed!}}

\begin{frame}\null\smash{\rlap{\hspace{-0.5in}\rotatebox[origin=c]{45}{\parbox{0.8in}{\begin{center}\large\red Doodle on this page!\end{center}}}}}
  %\titlepage
  \begin{center}
  \Large The 17 Worlds of Planar Ants
  \vfill
  \par\normalsize Dror Bar-Natan, \url{http://drorbn.net/mc21}
  \vfill
  \par\large MathCamp by Web, July 2021
  \end{center}

  \vfill\normalsize
  {\bf Abstract.} My goal is to get you hooked, captured and unreleased until you find all 17 in real life, around you.

  \vskip 2mm
  We all know that the plane can be filled in different periodic manners: floor tiles are often square but sometimes hexagonal, bricks are often laid in an interlaced pattern, fabrics often carry interesting patterns. A little less known is that there are precisely 17 symmetry patterns for tiling the plane; not one more, not one less. It is even less known how easy these 17 are to identify in the patterns around you, how fun it is, how common some are, and how rare some others seem to be.

  \vskip 2mm
  Gotta Catch 'Em All!
\end{frame}

\begin{frame}
Thanks for inviting me to MathCamp! Just to feel a little closer, here's a picture of the lecture room:
\[ \includegraphics[width=0.8\textwidth]{LectureRoom.jpg} \]
If you can, please turn your video on! (And mic, whenever needed).
\end{frame}

\begin{frame}
\parbox[t]{0.7\linewidth}{
{\bf\red Reading.} An excellent book on the subject is {\em The Symmetries
of Things} by J.~H.~Conway, H.~Burgiel, and C.~Goodman-Strauss, CRC
Press, 2008.

\vskip 2mm
Another nice text is {\em Classical Tessellations and Three-Manifolds}
by J.~M.~Montesinos,  Springer-Verlag, 1987.

\vskip 2mm
And another is {\em Tilings~\&~Patterns} by B.~Gr\"unbaum and G.~C.~Shephard, Dover, 2016.
}\hfill\imagetop{\includegraphics[width=0.25\linewidth]{../../2014-08/17Worlds/ConwayEtAl.jpg}}
\end{frame}

\begin{frame} \null
{\bf\red (Easy) Question 1.} In what ways can you make $\$2$ change, using coins denominated $\$\frac12$, $\$\frac23$, $\$\frac34$, $\$\frac45$, $\$\frac56$, etc.?

%{\def\f#1#2{\frac{#1}{\red\mathbf #2}}
%{\bf\red Answer.}
%$2 = \f12 + \f12 + \f12 + \f12 = \f23 + \f23 + \f23 = \f34 + \f34 + \f12 = \f56 + \f23 + \f12$,
%and that's it.
%}

\vskip 8mm
{\bf\red (Harder) Question 2.} Why am I asking?
\end{frame}

\def\PatternSlide#1#2{{%
\begin{frame}\begin{center}
\includegraphics[width=0.49\linewidth]{../../Projects/Tilings/#1.jpg}
\hfill
\includegraphics[width=0.49\linewidth]{../../Projects/Tilings/#1_.jpg}
\par #2
\end{center}\end{frame}
}}

\PatternSlide{S442/AgaKhan-5}{A decorated window at the Aga Khan Gallery, Toronto 2014}
\PatternSlide{442/FransRestaurant}{Floor tiles at Fran's Restaurant \& Bar, Toronto 2014}
\PatternSlide{442/TorontoPublicLibrary}{A chair at the Toronto Public Library, 2014}
\PatternSlide{442/TissueBox}{A box of tissues}
\PatternSlide{oo/KleinBottleSofa}{A living room sofa at the Karshon's, 2012}

\begin{frame}\begin{center}\resizebox{0.9\linewidth}{!}{\input{Gotta2.pdftex_t}}\end{center}\end{frame}
\begin{frame}\begin{center}\resizebox{0.9\linewidth}{!}{\input{Gotta1.pdftex_t}}\end{center}\end{frame}
\begin{frame}\begin{center}\includegraphics[width=0.92\linewidth]{R.png}\end{center}\end{frame}
\PatternSlide{632/Floor@CharidiseBaldwin2}{Floor tiles in a restaurant in Toronto's Baldwin Street, 2018}
\PatternSlide{442/TissueBox}{A box of tissues}
\begin{frame}\begin{center}\includegraphics[width=0.92\linewidth]{RR.png}\end{center}\end{frame}
\PatternSlide{S442/AgaKhan-5}{A decorated window at the Aga Khan Gallery, Toronto 2014}
\PatternSlide{S632/RomaFiumicino}{A food court at the Roma Fiumicino airport, 2017}
\begin{frame}\begin{center}\includegraphics[width=0.92\linewidth]{M.png}\end{center}\end{frame}
\PatternSlide{SS/Tissues}{A packet of tissues}
\PatternSlide{22S/RicksCafeKensington}{At Rick's Cafe in Toronto's Kensington Market, 2014}
\begin{frame}\begin{center}\includegraphics[width=0.92\linewidth]{G.png}\end{center}\end{frame}
\PatternSlide{oo/KleinBottleSofa}{A living room sofa at the Karshon's, 2012}
\PatternSlide{So/IndianaUniversityCarpet}{A carpet seen at Indiana University, 2016}
\PatternSlide{0/GroningenBikeParking}{A bike parking in Groningen, 2020}

\begin{frame} {\bf Homework.}

Go out and find them all! At home, around the corner, a mile away. Take pictures and upload them to \url{http://drorbn.net/mc21/upload} using the file format {\bf name}-{\bf type}-{\bf description}.jpg, where
\begin{itemize}
\item {\bf name} is your name or alias.
\item {\bf type} is the type of the tiling pattern, using the Conway conventions but with the ``$\ast$'' replaced with the English letter ``s''. In other words, {\bf type} is one of 2222, 333, 442, 632, s2222, s333, s442, s632, 4s2, 3s3, 2s22, 22s, ss, so, oo, 22o, or 0.
\item {\bf description} is a short description.
\end{itemize}
As an example, see the file {\tt Dror-4S2-StClairWSubway.jpg} there.

We will start our class tomorrow with a quick discussion of the patterns you will find --- but I can only promise to look at whatever will be uploaded at least two hours ahead of class.

{\bf Privacy note.} Whatever you upload I may post on my web site. So make sure the pictures you upload don't include anything personal.
\end{frame}

\begin{frame} {\bf Prize.}

US\$50 if you find a 333 in time for our second meeting!
\begin{itemize}
\item You must find it ``natural'' --- it can't be your own drawing, or within a book on symmetries, or in a museum that has an exhibit on symmetries (I think \href{https://momath.org/}{MoMath} has one).
\item In the unlikely event that more than one person will find a 333, I'll split the prize between all winners.
\item I've paid the prize twice before, but it's really tough. I've only seen a 333 ``in nature'' three times in more than 20 years of looking for it!
\end{itemize}

\end{frame}

\PatternSlide{333/JaimeInDorne}{\underline{\hspace{2in}} of the TV series \underline{\hspace{2in}} visiting \underline{\hspace{1in}}}

\begin{frame} See you tomorrow! \end{frame}

\begin{frame}\begin{center}\resizebox{0.9\linewidth}{!}{\input{Gotta1.pdftex_t}}\end{center}\end{frame}
\begin{frame}\begin{center}\resizebox{0.9\linewidth}{!}{\input{Gotta2.pdftex_t}}\end{center}\end{frame}
\PatternSlide{333/JaimeInDorne}{\underline{\hspace{2in}} of the TV series \underline{\hspace{2in}} visiting \underline{\hspace{1in}}}

\begin{frame} Let's look at what you found\ldots\hfill\url{http://drorbn.net/mc21/upload} \end{frame}

\def\SubmissionSlide#1{{\begin{frame}\begin{center}\includegraphics[height=0.8\textheight]{Submissions/#1}\par{\tt #1}\end{center}\end{frame}}}
\SubmissionSlide{Dror-4S2-StClairWSubway.jpg}
\SubmissionSlide{Janelle-2222-BinderPaper.jpg}
\SubmissionSlide{Janelle-2222-PencilCase.jpg}
\SubmissionSlide{Janelle-22o-MathtownFloorAtTau.jpg}
\SubmissionSlide{Janelle-632-TissueBox.jpg}
\SubmissionSlide{Janelle-oo-floor.jpg}
\SubmissionSlide{Maggie-333-BrokenSierpinsky.jpg}
\SubmissionSlide{Maggie-442-BowlPattern.jpg}
\SubmissionSlide{Maggie-632-JLo'sPot.jpg}
\SubmissionSlide{Maggie-GG-JLo'sBag.jpg}

\begin{frame}
  What if we lived on Venus?
  \hfill\imagetop{\includegraphics[width=0.4\linewidth]{../../2014-08/17Worlds/Venus_412.jpg}}
\end{frame}

\begin{frame}\begin{center}\includegraphics[width=0.4\linewidth]{../../2014-08/17Worlds/DiscWorld1.jpg}\end{center}\end{frame}
\begin{frame}\begin{center}
  \includegraphics[width=0.33\linewidth]{../../2014-08/17Worlds/MillerEarth.jpg}%
  \includegraphics[width=0.33\linewidth]{../../2014-08/17Worlds/MillerEarth.jpg}%
  \includegraphics[width=0.33\linewidth]{../../2014-08/17Worlds/MillerEarth.jpg}
\end{center}\end{frame}
\begin{frame}\begin{center}\includegraphics[width=0.6\linewidth]{../../2014-08/17Worlds/AnnularEarth.jpg}\end{center}\end{frame}
\begin{frame}\begin{center}\includegraphics[width=0.4\linewidth]{../../2014-08/17Worlds/SphericalEarth.jpg}\end{center}\end{frame}

\begin{frame}\begin{center}
  \includegraphics[width=0.6\linewidth]{../../2014-08/17Worlds/RenaissanceHotel.jpg}
  \par The Jerusalem Renaissance Hotel
\end{center}\end{frame}
\begin{frame}\includegraphics[width=0.5\linewidth]{../../2014-08/17Worlds/RenaissanceEnlarged.jpg}\end{frame}
\begin{frame}\begin{center}
  \includegraphics[width=0.8\linewidth]{Doughnuts.jpg}
  \par Doughnuts stolen from the MathCamp Kitchen
\end{center}\end{frame}
\begin{frame}\begin{center}
  \includegraphics[width=0.6\linewidth]{InnerTube_640.jpg}
  \par A truck's tire inner tube and Itai
\end{center}\end{frame}


\begin{frame}\begin{center}
  \includegraphics[width=0.6\linewidth]{../../Projects/Tilings/SS/LaTortilleriaFloor.jpg}
  \par Floor of La Tortilleria, Toronto 2018
\end{center}\end{frame}
\begin{frame}\includegraphics[width=0.5\linewidth]{../../Projects/Tilings/SS/LaTortilleriaFloor_.jpg}\end{frame}
\begin{frame}\begin{center}
  \includegraphics[width=0.4\linewidth]{../../2014-08/17Worlds/TransparentAnnularEarth.jpg}
  \par Earth on a cylinder
\end{center}\end{frame}

\begin{frame}\begin{center}
  \includegraphics[width=0.6\linewidth]{../../Projects/Tilings/So/IndianaUniversityCarpet.jpg}
  \par A carpet seen at Indiana University, 2016
\end{center}\end{frame}
\begin{frame}\includegraphics[width=0.5\linewidth]{../../Projects/Tilings/So/IndianaUniversityCarpet_.jpg}\end{frame}
\begin{frame}\begin{center}
  \includegraphics[width=0.6\linewidth]{../../2014-08/17Worlds/EscherMobius_800.jpg}
  \par A M\"obius band by M.~C.~Escher
\end{center}\end{frame}

\begin{frame}\begin{center}
  \includegraphics[width=0.6\linewidth]{../../Projects/Tilings/oo/KleinBottleSofa.jpg}
  \par A living room sofa at the Karshon’s, 2012
\end{center}\end{frame}
\begin{frame}\includegraphics[width=0.5\linewidth]{../../Projects/Tilings/oo/KleinBottleSofa_.jpg}\end{frame}
\begin{frame}\begin{center}
  \includegraphics[width=0.33\linewidth]{../../2014-08/17Worlds/KleinBottle.jpg}
  A Klein bottle from \url{https://www.kleinbottle.com/}
\end{center}\end{frame}

\begin{frame}\begin{center}
  \includegraphics[width=0.6\linewidth]{BedCoverAndAnnie.jpg}
  \par A bed cover and Annie, 2000
\end{center}\end{frame}
\begin{frame}\includegraphics[width=0.5\linewidth]{BedCoverAndAnnie_.jpg}\end{frame}
\begin{frame}\begin{center}
  \includegraphics[width=0.5\linewidth]{Pillow.jpg}
  \par a $90^\circ90^\circ90^\circ90^\circ$ pillow
\end{center}\end{frame}

\begin{frame}\begin{center}
  \includegraphics[width=0.6\linewidth]{AngelaWuInLondonCropped.jpg}
  \par This one earned Angela Wu C\$50 a few years ago
\end{center}\end{frame}
\begin{frame}\includegraphics[width=0.5\linewidth]{../../Projects/Tilings/333/AngelaWuInLondon_.jpg}\end{frame}
\begin{frame}\begin{center}
  \includegraphics[width=0.5\linewidth]{Hamantashan.jpg}
  \par a $60^\circ60^\circ60^\circ$ Hamantashan
\end{center}\end{frame}

\begin{frame}\null
Similarly, 442 $\to$ a $90^\circ 45^\circ 45^\circ$ pillow or cookie, and 632 $\to$ a $90^\circ 60^\circ 30^\circ$ pillow or cookie.
\end{frame}

\begin{frame}\begin{center}
  \includegraphics[width=0.6\linewidth]{../../Projects/Tilings/S2222/BridgeheadCoffeeOttawa.jpg}
  \par Wallpaper at Bridgehead Coffee in Ottawa
\end{center}\end{frame}
\begin{frame}\includegraphics[width=0.5\linewidth]{../../Projects/Tilings/S2222/BridgeheadCoffeeOttawa_.jpg}\end{frame}
\begin{frame}\begin{center}
  \includegraphics[width=0.5\linewidth]{2222Polygon_240.jpg}
  \par a $90^\circ90^\circ90^\circ90^\circ$ rectangle
\end{center}\end{frame}

\begin{frame}\null
Similarly, 442 $\to$ a $90^\circ 45^\circ 45^\circ$ triangle, 333 $\to$ a $60^\circ 60^\circ 60^\circ$ triangle, and 632 $\to$ a $90^\circ 60^\circ 30^\circ$ triangle.
\par\begin{center}\includegraphics[width=0.4\linewidth]{Polygons.jpg}\end{center}
\end{frame}

\begin{frame}\begin{center}
  \includegraphics[width=0.6\linewidth]{PentagonalTiles.jpg}
  \par The powder room at the Kuperberg-Zieve's
\end{center}\end{frame}
\begin{frame}\includegraphics[width=0.5\linewidth]{PentagonalTiles_.jpg}\end{frame}
\begin{frame}\begin{center}
  \includegraphics[width=0.5\linewidth]{PlasticBag.jpg}
  \par A plastic bag
\end{center}\end{frame}

\begin{frame} {\bf Homework.}
\par What about \Di, \Dj, \Dk, and \Dp?
\end{frame}

\begin{frame}\begin{center}
{\red The 230 Worlds of Spacial Monkeys}
\par\includegraphics[width=0.4\linewidth]{../../2014-08/17Worlds/3D-1.jpg}\quad\includegraphics[width=0.4\linewidth]{../../2014-08/17Worlds/3D-2.jpg}
\par The 219 Worlds of Monkeys that Can't Tell Left From Right
\vskip 1mm
\par\footnotesize (Numbers and pictures from \arXiv{math/9911185} by Conway, Friedrichs, Huson, and Thurston; see also \url{http://webmineral.com/crystall.shtml})
\end{center}\end{frame}

\begin{frame} Thank You! \end{frame}

\end{document}


