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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. Given 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.
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<p>
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The Domino Problem and its corresponding history are somewhat vexing and difficult to describe. The original 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 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. The problem was first posed by Hao Wang in 1961. He actually conjectured that aperiodic tilings do not 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 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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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.
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<NuxtLink class="text-3xl font-bold" to='https://unboundedpress.org/'>Michael Winter - composer | sound artist</NuxtLink>
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