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Michael Winter 1 year ago
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@article{doi:10.1002/j.1538-7305.1961.tb03975.x,
author = {Wang, Hao},
title = {Proving Theorems by Pattern Recognition — II},
journal = {Bell System Technical Journal},
volume = {40},
number = {1},
pages = {1-41},
year = {1961}
}

@book{berger1966undecidability,
title={The Undecidability of the Domino Problem},
author={Berger, R.},
isbn={9780821812662},
series={Memoirs ; No 1/66},
url={https://books.google.com/books?id=mLfTCQAAQBAJ},
year={1966},
publisher={American Mathematical Society}
}

@Article{Robinson1971,
author="Robinson, Raphael M.",
title="Undecidability and nonperiodicity for tilings of the plane",
journal="Inventiones mathematicae",
year="1971",
month="Sep",
day="01",
volume="12",
number="3",
pages="177--209",
issn="1432-1297",
doi="10.1007/BF01418780",
url="https://doi.org/10.1007/BF01418780"
}

@book{Grunbaum:1986:TP:19304,
author = {Gr\"{u}nbaum, Branko and Shephard, G C},
title = {Tilings and Patterns},
year = {1986},
isbn = {0-716-71193-1},
publisher = {W. H. Freeman \& Co.},
address = {New York, NY, USA},
}

@article{Kari:1996:SAS:245761.245817,
author = {Kari, Jarkko},
title = {A Small Aperiodic Set of Wang Tiles},
journal = {Discrete Math.},
issue_date = {Nov. 15, 1996},
volume = {160},
number = {1-3},
month = nov,
year = {1996},
issn = {0012-365X},
pages = {259--264},
numpages = {6},
url = {http://dx.doi.org/10.1016/0012-365X(95)00120-L},
doi = {10.1016/0012-365X(95)00120-L},
acmid = {245817},
publisher = {Elsevier Science Publishers B. V.},
address = {Amsterdam, The Netherlands, The Netherlands},
}

@article{DBLP:journals/corr/JeandelR15,
author = {Emmanuel Jeandel and
Micha{\"{e}}l Rao},
title = {An aperiodic set of 11 Wang tiles},
journal = {CoRR},
volume = {abs/1506.06492},
year = {2015},
url = {http://arxiv.org/abs/1506.06492},
archivePrefix = {arXiv},
eprint = {1506.06492},
timestamp = {Mon, 13 Aug 2018 16:48:45 +0200},
biburl = {https://dblp.org/rec/bib/journals/corr/JeandelR15},
bibsource = {dblp computer science bibliography, https://dblp.org}
}

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\bibcite{Grunbaum:1986:TP:19304}{4}
\bibcite{Kari:1996:SAS:245761.245817}{5}
\bibcite{DBLP:journals/corr/JeandelR15}{6}

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\begin{thebibliography}{1}

\bibitem{doi:10.1002/j.1538-7305.1961.tb03975.x}
Hao Wang.
\newblock Proving theorems by pattern recognition — ii.
\newblock {\em Bell System Technical Journal}, 40(1):1--41, 1961.

\bibitem{berger1966undecidability}
R.~Berger.
\newblock {\em The Undecidability of the Domino Problem}.
\newblock Memoirs ; No 1/66. American Mathematical Society, 1966.

\bibitem{Robinson1971}
Raphael~M. Robinson.
\newblock Undecidability and nonperiodicity for tilings of the plane.
\newblock {\em Inventiones mathematicae}, 12(3):177--209, Sep 1971.

\bibitem{Grunbaum:1986:TP:19304}
Branko Gr\"{u}nbaum and G~C Shephard.
\newblock {\em Tilings and Patterns}.
\newblock W. H. Freeman \& Co., New York, NY, USA, 1986.

\bibitem{Kari:1996:SAS:245761.245817}
Jarkko Kari.
\newblock A small aperiodic set of wang tiles.
\newblock {\em Discrete Math.}, 160(1-3):259--264, November 1996.

\bibitem{DBLP:journals/corr/JeandelR15}
Emmanuel Jeandel and Micha{\"{e}}l Rao.
\newblock An aperiodic set of 11 wang tiles.
\newblock {\em CoRR}, abs/1506.06492, 2015.

\end{thebibliography}

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\begin{document}

\textit{\textbf{a history of the domino problem}} \\
a performance-installation

\begin{flushright}
michael winter \\ schloss solitude and cdmx; 2018 - 2019 \\
\end{flushright}

\bigskip

\begin{center}
\begin{tabular}{cc}

\centering
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\centering
\includegraphics[width=0.49\linewidth]{selects/discos.png}
\end{tabular}
\end{center}

\bigskip
\bigskip

\textbf{Description / note}

The domino problem, first posed by Hao Wang in 1961, is an epistemological question that asks whether there exists an algorithm to determine if an arbitrary finite set of tiles with colored edges can cover the plane such that adjacent edges match color. He conjectured that if a set of such tiles covers the plane, it can only do so periodically. However, in 1966, his student, Robert Berger, proved that the problem is undecidable (that is, there is no general algorithm) by showing the existence of a set of tiles that can only cover the plane aperiodically. This initial set contained more than 20000 tiles. Over the past 60 years, there has been a continual reduction in the size of provably aperiodic sets to the most recent discovery of a set of 11 tiles along with a proof that no smaller sets exist. It is a beautiful narrative / history of a particular epistemological problem that challenged a group of people not only to solve it, but to understand it to the extent possible.

\textit{a history of the domino problem} is a performance-installation that visualizes and sonifies aperiodic tilings in order to trace the history of the domino problem. The tilings are visualized using a cryptographic scheme in which two `shadow images', each which look completely random independently, are combined / overlayed at various orientations to reveal the tilings in a form that replaces the colored edges of the original constructions with binary codes. The shadow images are printed on photomasks typically used to manufacture computer chips: quartz wafers with a chrome coating etched at a pixel size of 20 microns. A high-precision, motorized multiaxis stage aligns the shadow images to reveal the tilings (along with 3 other images of poetic texts inspired by the history of the domino problem). The whole apparatus rests on a light source that illuminates the photomasks which are then magnified and projected. The visualizations are accompanied by musical compositions generated from the tilings that can be realized live by performers as intermittent augmentations within the installation or as singular pieces in concert.

The aim of the work is to create an artistic experience that demonstrates the aesthetic qualities of these found mathematical objects while also functioning as a sort of historical record.

This is ultimately a piece about how things fit together.

\bigskip

\textbf{Installation, performance setting, and technical requirements}

As an installation, the apparatus that aligns the image should be centered in a dark room such that observers can view the photomasks up close. Ideally, this should be set up with a teleprompter mirror at a 45 degree angle above the apparatus so that the viewer can see the photomasks without having to bend over. On the other side of the mirror, a camera is placed such that the resulting projected image aligns with what the viewer sees in the teleprompter mirror. The camera side must be darkened out with a cover in order for as much reflection to the viewer as possible.

In the installation, recordings of the musical pieces are played back; sometimes randomly and sometimes in sync with the respective tilings from which they were generated. The installation can be augmented (e.g. for an exhibition opening) by live performances of the musical pieces instead of the recordings. If so, direct access to the apparatus should be avoided in order for a situation where the observers can view the projected images and listen to the musical accompaniment in a tranquil and focused environment.

The computer code needed to run the installation along with all the code that generated the piece and schematics to rebuild the installation are available at \url{https://gitea.unboundedpress.org/mwinter/a_history_of_the_domino_problem}.

\bigskip

\textbf{Partial historical timeline of the domino problem}

pre-history:\\
\begin{tabular}{p{0.8in}p{1.25in}p{4.25in}}
17th century & Leibniz & pioneer of binary arithmetic and the idea of computing machines \\
ca 1928 & Hilbert & posed the original "Entsheidungsproblem" \\
ca 1931 & Goedel & first showed that their exists truths that are undecidable with a finite set of axioms \\
ca 1936 & Turing & invented the concept of the modern day computer and showed its limits yet was unfortunately persecuted for his sexuality despite being a key figure in the triumph of the allied nations against the nazi regime
\end{tabular}

conjecture and first proof:\\
\begin{tabular}{p{0.8in}p{1.25in}p{4.25in}}
ca 1961 & Wang & conjectured that aperiodic tilings of the plane did not exist \\
ca 1966 & Berger & showed that an aperiodic set 20000+ tiles exist (using his method this was quickly reduced to 104 then 92 by Berger and Knuth, respectively)
\end{tabular}

first wave of reduction:\\
\begin{tabular}{p{0.8in}p{1.25in}p{4.25in}}
ca 1971 & Robinson \& Lauchli & 56 and 40 tiles, respectively; using a similar technique of tiling arbitrarily large squares discovered independently
\end{tabular}

second wave of reduction:\\
\begin{tabular}{p{0.8in}p{1.25in}p{4.25in}}
ca 1986 & Penrose \& Amman & 32 and 16 tiles, respectively; using a method that translates different, non-squared aperiodic tiles into Wang tiles
\end{tabular}

third wave of reduction:\\
\begin{tabular}{p{0.8in}p{1.25in}p{4.25in}}
ca 1996 & Kari \& Culik & 13 tiles using an new construction with aperiodic integer sequences
\end{tabular}

final reduction\\
\begin{tabular}{p{0.8in}p{1.25in}p{4.25in}}
ca 2015 & Jaendel \& Rao & 11 tiles with an incredible computer-assisted proof that no smaller aperiodic sets exist
\end{tabular}

\bigskip

\textbf{Selected Bibliography}

\nocite{*}
\bibliographystyle{unsrt}
\bibliography{hdp}

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Berger
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Robinson
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Penrose
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Ammann
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Kari
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Jaendal-Rao

\vspace*{\fill}

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+ 21
- 0
latex/documentation/texput.log View File

@@ -0,0 +1,21 @@
This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Arch Linux) (preloaded format=pdflatex 2019.9.6) 26 NOV 2019 11:07
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
**hdp.tex

! Emergency stop.
<*> hdp.tex
*** (job aborted, file error in nonstop mode)

Here is how much of TeX's memory you used:
3 strings out of 492623
109 string characters out of 6135670
56958 words of memory out of 5000000
4018 multiletter control sequences out of 15000+600000
3640 words of font info for 14 fonts, out of 8000000 for 9000
1141 hyphenation exceptions out of 8191
0i,0n,0p,1b,6s stack positions out of 5000i,500n,10000p,200000b,80000s
! ==> Fatal error occurred, no output PDF file produced!

+ 3
- 4
supercollider/main.scd View File

@@ -25,7 +25,6 @@ s.record(~dir +/+ ".." +/+ "recs" +/+ "berger_knuth.wav", duration: (30 * 60));
s.record(~dir +/+ ".." +/+ "recs" +/+ "robinson.wav", duration: (30 * 60));


//Potential TODO: add (de)crescendo markings and update synthdef to have the fades
~penroseTiling = ~penrose.value(120, 5, ~seed);
~penroseCreateSynths.value;
~penroseMusic = ~penroseMusicify.value(~penroseTiling, 0, 0, 99, 99, 6, 3, 3.5, ~seed);
@@ -43,7 +42,7 @@ s.record(~dir +/+ ".." +/+ "recs" +/+ "penrose.wav", duration: (30 * 60));
~ammannSound.play;
~ammannTranscribe.value(~ammannMusic);
~visualize.value(~ammann.value(200, 200), 0, 0, name: "ammann")
s.record(~dir +/+ ".." +/+ "recs" +/+ "ammann.wav", duration: 300);
s.record(~dir +/+ ".." +/+ "recs" +/+ "ammann.wav", duration: (30 * 60));


~kariTiling = ~kari_culik.value(500, 500, 0, 0, true);
@@ -53,7 +52,7 @@ s.record(~dir +/+ ".." +/+ "recs" +/+ "ammann.wav", duration: 300);
~kariSound.play;
~kariTranscribe.value(~kariMusic);
~visualize.value(~kari_culik.value(200, 200, 0, 5, true), 0, 0, scale: 1, name: "kari");
s.record(~dir +/+ ".." +/+ "recs" +/+ "kari_culik.wav", duration: 300);
s.record(~dir +/+ ".." +/+ "recs" +/+ "kari_culik.wav", duration: (30 * 60));


~jaendelTiling = ~jaendel.value(14, 0, 0, 0);
@@ -63,7 +62,7 @@ s.record(~dir +/+ ".." +/+ "recs" +/+ "kari_culik.wav", duration: 300);
~jaendelSound.play;
~jaendelTranscribe.value(~jaendelMusic);
~visualize.value(~jaendelTiling, 0, 0, name: "jaendel");
s.record(~dir +/+ ".." +/+ "recs" +/+ "jaendel_rao.wav", duration: 300);
s.record(~dir +/+ ".." +/+ "recs" +/+ "jaendel_rao.wav", duration: 30 * 60);


//~~~~~~~~~crypto visualizer code

+ 25
- 11
supercollider/stepper_control.scd View File

@@ -2,27 +2,27 @@
var imageDist, micronsPerStep, automation, imgPositions, curPos, tarPos,
netAddress, serialPort, serialListener,
moveTo, jogControl, jogHorizontal, jogVertical,
imgSelect, imgCalibrate, automate;
imgSelect, imgCalibrate, automate, lastSelect;

// init global vars
imageDist = 100; // in steps
imageDist = 300; // in microns
micronsPerStep = 0.0977;
automation = false;
imgPositions = 9.collect({nil});
curPos = Point.new(0, 0);
tarPos = Point.new(0, 0);
netAddress = NetAddr.new("127.0.0.1", 7777);
serialPort = SerialPort("/dev/ttyACM0", baudrate: 115200, crtscts: true);
~serialPort = SerialPort("/dev/ttyACM0", baudrate: 115200, crtscts: true);

// recieve motor feedback
serialListener = Routine({
~serialListener = Routine({
var byte, str, res, valArray,
stepper, limitSwitchNeg, limitSwitchPos, safeMode, limitPos;

safeMode = false;

loop{
byte = serialPort.read;
byte = ~serialPort.read;
if(byte==13, {
if(str[1].asString == "[", {
valArray = str.asString.interpret.postln;
@@ -38,8 +38,16 @@ serialListener = Routine({
}
});
if(automation, {
if((curPos.x - tarPos.x).abs < 100, {tarPos.x = imageDist.rand2});
if((curPos.y - tarPos.y).abs < 100, {tarPos.y = imageDist.rand2});
var centerPos = nil, dist = 0;
if(lastSelect != 0, {
centerPos = imgPositions[lastSelect].deepCopy;
dist = 300;
}, {
centerPos = imgPositions[4].deepCopy;
dist = imageDist / micronsPerStep;
});
if((curPos.x - tarPos.x).abs < 100, {tarPos.x = centerPos.x + dist.rand2});
if((curPos.y - tarPos.y).abs < 100, {tarPos.y = centerPos.y + dist.rand2});
moveTo.value(tarPos);
});
}, {
@@ -58,8 +66,8 @@ serialListener = Routine({

// send new coordinates to the arduino / motors
moveTo = {arg point;
serialPort.putAll(point.x.asInteger.asString ++ " " ++ point.y.asInteger.asString);
serialPort.put(10);
~serialPort.putAll(point.x.asInteger.asString ++ " " ++ point.y.asInteger.asString);
~serialPort.put(10);
};

jogControl = {arg axis;
@@ -90,7 +98,7 @@ jogHorizontal = jogControl.value('/jog_horizontal');
jogVertical = jogControl.value('/jog_vertical');

imgSelect = {
var lastSelect = nil;
//var lastSelect = nil;
OSCFunc({arg msg;
var imgIndex;
if(msg[1] > 0, {
@@ -102,10 +110,13 @@ imgSelect = {
netAddress.sendMsg("/STATE/SET", "{automate: 0}");
lastSelect = imgIndex;
}, {
lastSelect = 0;
/*
imgIndex = msg[1].neg - 1;
if(imgIndex == lastSelect, {
if(imgPositions[imgIndex] != nil, {tarPos = imgPositions[imgIndex].deepCopy; moveTo.value(tarPos)});
netAddress.sendMsg("/STATE/SET", "{img_" ++ (imgIndex + 1).asInteger.asString ++ "_select: " ++ (imgIndex + 1) ++ "}")});
*/
});
}, '/img_select', netAddress)
}.value;
@@ -140,7 +151,10 @@ automate = OSCFunc({arg msg;
9.do({arg i; netAddress.sendMsg("/STATE/SET", "{img_" ++ (i + 1).asString ++ "_select: " ++ (i + 1).neg ++ "}")});
}, '/automate', netAddress);
)

~serialPort.close
~serialPort = SerialPort.new("/dev/ttyACM0", baudrate: 115200, crtscts: true);
~serialListener.reset
~serialListener.play(AppClock);

(
// TODO:

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