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< a href = "#contributors" class = "px-3" > contributors < / a >
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< span class = "italic" > a history of the domino problem < / span > is a performance - installation that traces the history of an epistemological problem in mathematics about how things that one could never imagine fitting together , actually come together and unify in unexpected ways . The work comprises a set of musical compositions and a kinetic sculpture that sonify and visualize rare tilings ( more commonly known as mosaics ) constructed from dominoes . The dominoes in these tilings are similar yet slightly different than those used in the popular game of the same name . As opposed to rectangles , they are squares with various color combinations along the edges ( which can alternatively also be represented by numbers or patterns ) called < NuxtLink to = 'https://en.wikipedia.org/wiki/Wang_tile' > w ang tiles < / NuxtLink > . Like in the game , the rule is that edges of adjacent dominoes in a tiling must match .
< span class = "italic" > a history of the domino problem < / span > is a performance - installation that traces the history of an epistemological problem in mathematics about how things that one could never imagine fitting together , actually come together and unify in unexpected ways . The work comprises a set of musical compositions and a kinetic sculpture that sonify and visualize rare tilings ( more commonly known as mosaics ) constructed from dominoes . The dominoes in these tilings are similar yet slightly different than those used in the popular game of the same name . As opposed to rectangles divided into two regions with numbers between 1 and 6 , they are squares where each of the 4 edges is assigned a number ( typically represented by a corresponding color or alternatively , pattern ) called < NuxtLink to = 'https://en.wikipedia.org/wiki/Wang_tile' > W ang tiles < / NuxtLink > . Like in the game , the rule is that edges of adjacent dominoes in a tiling must match .
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The tilings sonified and visualized in < span class = "italic" > a history of the domino problem < / span > are rare because there is no systematic way to find them . This is due to the fact that they are < NuxtLink to = 'https://en.wikipedia.org/wiki/Aperiodic_tiling' > < span class = "italic" > aperiodic < / span > < / NuxtLink > . One can think of an aperiodic tiling like an infinite puzzle with a peculiar characteristic . G unlimited copies of dominoes with a finite set of color / pattern combinations for the edges , there is a solution that will result in a tiling that expands infinitely . However , in that solution , any periodic / repeating structure in the tiling will eventually be interrupted . This phenomenon is one of the most intriguing aspects of the work . As the music and the visuals are derived from the tilings , the resulting textures are always shifting ever so slightly .
The tilings sonified and visualized in < span class = "italic" > a history of the domino problem < / span > are rare because there is no systematic way to find them . This is due to the fact that they are < NuxtLink to = 'https://en.wikipedia.org/wiki/Aperiodic_tiling' > < span class = "italic" > aperiodic < / span > < / NuxtLink > . One can think of an aperiodic tiling as an infinite puzzle with a peculiar characteristic : g unlimited copies of dominoes with a finite set of color / pattern combinations for the edges , on can form a tiling that expands infinitely . However , in that solution , any repeating structure in the tiling will eventually be interrupted . This phenomenon is one of the most intriguing aspects of the work . As the music and the visuals are derived from the tilings , the resulting textures are always shifting ever so slightly .
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The original Domino Problem asked if there exists an algorithm / computer program that , when given as input a finite set of dominoes with varying color combinations for the edges , can output a binary answer , ` yes' or ` no ' , whether or not copies of that set can form an infinite tiling . The problem was first posed by Hao Wang in 1961 , who conjectured that the answer is positive and that such an algorithm does exist . The reason why the Domino Problem is inextricably linked to whether or not aperiodic tilings exist is the following . The existence of aperiodic tilings would mean that such an algorithm < span class = "italic" > does not < / span > exist . However , in 1966 , his student , Robert Berger , proved him wrong by discovering an infinite , aperiodic tiling constructed with copies of a set of 20 , 426 dominoes . With the original problem solved , mathematicians then took on the challenge of finding the smallest set of dominoes that would construct an infinite aperiodic tiling . Over the past 60 years , this number has been continually reduced with the contributions of many different mathematicians until the most recent discovery of a set of 11 dominoes along with a proof that no smaller sets exist . It is a remarkable 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 .
The original Domino Problem asked if there exists an algorithm / computer program that , when given as input a finite set of dominoes with varying color combinations for the edges , can output a binary answer , ` yes' or ` no ', whether or not copies of that set can form an infinite tiling. The problem was first posed by Hao Wang in 1961, who conjectured that the answer is positive and that such an algorithm does exist. The existence of aperiodic tilings would mean that such an algorithm <span class="italic">does not</span> exist. However, in 1966, his student, Robert Berger, proved him wrong by discovering an infinite, aperiodic tiling constructed with copies of a set of 20,426 dominoes. The resolution of Wang' s original question led to new questions and mathematicians took on the challenge of finding the smallest set of dominoes that would construct an infinite aperiodic tiling . Over the past 60 years , this number has been continually reduced with the contributions of many different mathematicians until the most recent discovery of a set of 11 dominoes along with a proof that no smaller sets exist . It is a remarkable 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 .
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a few thoughts on how things fit together ...
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in collaboration with MAREIKE YIN - YEE LEE
in collaboration with MAREIKE YIN - YEE LEE
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Nov . - Exact dates and exhibition opening hours TBA soon !
Exhibition Opening - 17 Nov 2023 | 19 Uhr
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Exhibition Closing - 31 Nov 2023 | 19 Uhr
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Exact Gallery hours to be announced soon !
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Lichthof Ost , HU Berlin , Campus Mitte , Unter den Linden 6 , U Bahn Unter den Linden oder Museuminsel
Lichthof Ost , HU Berlin , Campus Mitte , Unter den Linden 6 , U Bahn Unter den Linden oder Museuminsel
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Exhibition Opening - TBA
Exhibition Opening - 17 Nov 2023 | 19 Uhr
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Lichthof Ost , HU Berlin , Campus Mitte , Unter den Linden 6 , U Bahn Unter den Linden oder Museuminsel
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Exhibition Closing - TBA
Exhibition Closing - 31 Nov 2023 | 19 Uhr
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Lichthof Ost , HU Berlin , Campus Mitte , Unter den Linden 6 , U Bahn Unter den Linden oder Museuminsel
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Lecture - Concert
Lecture - Concert
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performance by KALI ENSEMBLE
performance by KALI ENSEMBLE
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22 Nov 2023 | 19 Uhr
22 Nov 2023 | 19 : 30 Uhr
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Senatssal, HU Berlin , Campus Mitte , Unter den Linden 6 , U Bahn Unter den Linden oder Museuminsel
Reuter- Saal , HU Berlinm Universitätsgebäude am Hegelplatz , Dorotheentsraße 24 , U Bahn Unter den Linden oder Museuminsel
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Concert
Concert
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KM28
KM28
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Karl - Marx - Str . 28 , 12043 Berlin
Karl - Marx - Str . 28 , 12043 Berlin , U Bahn Karl - Marx - Platz
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a few selected articles :
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Hao Wang ( 1961 ) , Proving theorems by pattern recognition — II , Bell System Technical Journal , Volume : 40 , Issue : 1.
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Robert Berger ( 1966 ) , The undecidability of the domino problem , American Mathematical Society , Volume 1 , 1966.
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Jarkko Kari ( 1996 ) , A small aperiodic set of Wang tiles , Discrete Mathematics , Volume 160.
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Emmanuel Jeandel and Michael Rao , An aperiodic set of 11 Wang tiles , Advances in Combinatorics , Volume 1.
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a definitive book on tilings and patterns :
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Branko Grunbaum and G . C . Shephard , Tilings and Patterns , Dover Books ( originally published 1986 )
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a few useful links :
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< NuxtLink to = 'https://grahamshawcross.com/2012/10/12/aperiodic-tiling/' > https : / / g r a h a m s h a w c r o s s . c o m / 2 0 1 2 / 1 0 / 1 2 / a p e r i o d i c - t i l i n g / < / N u x t L i n k >
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< NuxtLink to = 'https://grahamshawcross.com/2012/10/12/wang-tiles-and-aperiodic-tiling/' > https : / / g r a h a m s h a w c r o s s . c o m / 2 0 1 2 / 1 0 / 1 2 / w a n g - t i l e s - a n d - a p e r i o d i c - t i l i n g / < / N u x t L i n k >
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< NuxtLink to = 'https://en.wikipedia.org/wiki/Wang_tile' > https : / / e n . w i k i p e d i a . o r g / w i k i / W a n g _ t i l e < / N u x t L i n k >
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< NuxtLink to = 'https://en.wikipedia.org/wiki/Aperiodic_tiling' > https : / / e n . w i k i p e d i a . o r g / w i k i / A p e r i o d i c _ t i l i n g < / N u x t L i n k >
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