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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 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' > Wang tiles < / NuxtLink > . Like in the game , the rule is that edges of adjacent dominoes in a tiling must match .
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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 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' > Wang 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 as an infinite puzzle with a peculiar characteristic : given 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 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 : given 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 existence of aperiodic tilings would mean that such an algorithm <span class="italic">does not</span> exist. However, in 1964, 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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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 1964, 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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a few thoughts on how things fit together ...
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( free entrance )
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( free entrance )
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in collaboration with MAREIKE YIN - YEE LEE
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in collaboration with MAREIKE YIN - YEE LEE
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Exhibition Opening - 17 Nov 2023 | 19 Uhr
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Exhibition Opening - 17 Nov 2023 | 19 Uhr
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Exhibition Closing - 01 Dec 2023 | 19 Uhr
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Exhibition Closing - 01 Dec 2023 | 19 Uhr
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Gallery Hours - Wednesday to Friday | 13 - 20 Uhr & Saturday | 11 - 18 Uhr
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Gallery Hours - Wednesday to Friday | 13 - 20 Uhr & Saturday | 11 - 18 Uhr
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Lichthof Ost , HU Berlin Hauptgebäude , Campus Mitte , Unter den Linden 6 ( U - Bahn Unter den Linden oder Museuminsel )
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Lichthof Ost , HU Berlin Hauptgebäude , Campus Mitte , Unter den Linden 6 ( U - Bahn Unter den Linden oder Museuminsel )
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The exhibition will feature individual and collaborative works by Michael Winter and Mareike Yin - Yee Lee in a constellation designed specifically for the Lichthof Ost exhibition room of the Humboldt University . The original kinetic sculpture Winter created to visualize the aperiodic tilings of the history of the domino problem will be juxtaposed with recent works by Yin - Yee Lee as well as collaboratively created realizations of the tilings . The works on display by Yin - Yee Lee will highlight selections from her Hidden Lakes and Missing Pieces series in which enigmatic outlines of lakes and various shapes encourage observers to perceive similarities and differences in form , pattern , and repetition between the pieces and to mentally fill in blank space . The collaborative realizations of the tilings will include prints generated by Winter with the aid of a computer that incorporate images and color schemes by Yin - Yee Lee as well as a floor mosaic of drawings on mirrors . The exhibition plays on the macro versus the micro , transformation , and how topologies of various color combinations , relationships between shapes and gradients reflect in space in order to illuminate "a few thoughts on how things fit together..."
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The exhibition will feature individual and collaborative works by Michael Winter and Mareike Yin - Yee Lee in a constellation designed specifically for the Lichthof Ost exhibition room of the Humboldt University . The original kinetic sculpture Winter created to visualize the aperiodic tilings of the history of the domino problem will be juxtaposed with recent works by Yin - Yee Lee as well as collaboratively created realizations of the tilings . The works on display by Yin - Yee Lee will highlight selections from her Hidden Lakes and Missing Pieces series in which enigmatic outlines of lakes and various shapes encourage observers to perceive similarities and differences in form , pattern , and repetition between the pieces and to mentally fill in blank space . The collaborative realizations of the tilings will include prints generated by Winter with the aid of a computer that incorporate images and color schemes by Yin - Yee Lee as well as a floor mosaic of drawings on mirrors . The exhibition plays on the macro versus the micro , transformation , and how topologies of various color combinations , relationships between shapes and gradients reflect in space in order to illuminate "a few thoughts on how things fit together..."
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< span class = "font-bold" > Exhibition Opening - 17 Nov 2023 | 19 Uhr < / span >
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< span class = "font-bold" > Exhibition Opening - 17 Nov 2023 | 19 Uhr < / span >
Lichthof Ost , HU Berlin Hauptgebäude , Campus Mitte , Unter den Linden 6 ( U - Bahn Unter den Linden oder Museuminsel )
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Lichthof Ost , HU Berlin Hauptgebäude , Campus Mitte , Unter den Linden 6 ( U - Bahn Unter den Linden oder Museuminsel )
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< span class = "font-bold" > Exhibition Closing - 01 Dec 2023 | 19 Uhr < / span >
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< span class = "font-bold" > Exhibition Closing - 01 Dec 2023 | 19 Uhr < / span >
Lichthof Ost , HU Berlin Hauptgebäude , Campus Mitte , Unter den Linden 6 ( U - Bahn Unter den Linden oder Museuminsel )
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Lichthof Ost , HU Berlin Hauptgebäude , Campus Mitte , Unter den Linden 6 ( U - Bahn Unter den Linden oder Museuminsel )
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< span class = "font-bold" > Public lecture + Concert ( free entrance ) - 22 Nov 2023 | 19 : 30 Uhr ( doors open at 19 : 00 Uhr ) < / span >
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< span class = "font-bold" > Public lecture + Concert ( free entrance ) - 22 Nov 2023 | 19 : 30 Uhr ( doors open at 19 : 00 Uhr ) < / span >
with Prof . JARKKO KARI ( Turku University ) , moderated by Prof . Dr . GAËTAN BOROT ( HU Berlin )
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with Prof . JARKKO KARI ( Turku University ) , moderated by Prof . Dr . GAËTAN BOROT ( HU Berlin )
the abstract of Prof . JARKKO KARI ' s Lecture is provided below
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the abstract of Prof . JARKKO KARI ' s Lecture is provided below
performance by KALI ENSEMBLE
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performance by KALI ENSEMBLE
Fritz - Reuter - Saal , HU Berlin Universitätsgebäude ( am Hegelplatz ) , Dorotheenstraße 24 ( U - Bahn Unter den Linden oder Museuminsel )
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Fritz - Reuter - Saal , HU Berlin Universitätsgebäude ( am Hegelplatz ) , Dorotheenstraße 24 ( U - Bahn Unter den Linden oder Museuminsel )
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< span class = "font-bold" > Concert - 23 Nov 2023 | 20 : 30 Uhr ( doors open at 20 : 00 Uhr ) < / span >
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< span class = "font-bold" > Concert - 23 Nov 2023 | 20 : 30 Uhr ( doors open at 20 : 00 Uhr ) < / span >
performance by KALI ENSEMBLE
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performance by KALI ENSEMBLE
< NuxtLink to = 'https://www.km28.de/' > KM28 < / NuxtLink >
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< NuxtLink to = 'https://www.km28.de/' > KM28 < / NuxtLink >
Karl - Marx - Str . 28 , 12043 Berlin ( U - Bahn Karl - Marx - Platz )
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Karl - Marx - Str . 28 , 12043 Berlin ( U - Bahn Karl - Marx - Platz )
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About the Public lecture
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< span class = "font-bold" > From Wang Tiles to the Domino Problem : A Tale of Aperiodicity < / span >
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This presentation delves into the remarkable history of aperiodic tilings and the domino problem . Aperiodic tile sets refer to collections of tiles that can only tile the plane in a non - repeating , or non - periodic , manner . Such sets were not believed to exist until 1964 when R . Berger introduced the first aperiodic set consisting of an astonishing 20 , 426 Wang tiles . Over the years , ongoing research led to significant advancements , culminating in 2015 with the discovery of a mere 11 Wang tiles by E . Jeandel and M . Rao , alongside a computer - assisted proof of their minimality . Simultaneously , researchers found even smaller aperiodic sets composed of polygon - shaped tiles . Notably , Penrose 's kite and dart tiles emerged as early examples, and most recently, a groundbreaking discovery was made - a solitary aperiodic tile known as the "hat" that can tile the plane exclusively in a non-periodic manner. Aperiodic tile sets are intimately connected with the domino problem that asserts how certain tile sets can tile the plane without us ever being able to establish their tiling nature with absolute certainty. Moreover, aperiodic tilings hold a distinct visual aesthetic allure. In today' s musical presentation , their artistic appeal transcends the visual domain and extends into the realm of music .
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- Jarkko Kari
About the Public lecture
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< span class = "font-bold" > From Wang Tiles to the Domino Problem : A Tale of Aperiodicity < / span >
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This presentation delves into the remarkable history of aperiodic tilings and the domino problem . Aperiodic tile sets refer to collections of tiles that can only tile the plane in a non - repeating , or non - periodic , manner . Such sets were not believed to exist until 1964 when R . Berger introduced the first aperiodic set consisting of an astonishing 20 , 426 Wang tiles . Over the years , ongoing research led to significant advancements , culminating in 2015 with the discovery of a mere 11 Wang tiles by E . Jeandel and M . Rao , alongside a computer - assisted proof of their minimality . Simultaneously , researchers found even smaller aperiodic sets composed of polygon - shaped tiles . Notably , Penrose 's kite and dart tiles emerged as early examples, and most recently, a groundbreaking discovery was made - a solitary aperiodic tile known as the "hat" that can tile the plane exclusively in a non-periodic manner. Aperiodic tile sets are intimately connected with the domino problem that asserts how certain tile sets can tile the plane without us ever being able to establish their tiling nature with absolute certainty. Moreover, aperiodic tilings hold a distinct visual aesthetic allure. In today' s musical presentation , their artistic appeal transcends the visual domain and extends into the realm of music .
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- Jarkko Kari
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< NuxtLink class = "text-3xl font-bold" to = 'https://unboundedpress.org/' > Michael Winter - composer | sound artist < / NuxtLink >
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< NuxtLink class = "text-3xl font-bold" to = 'https://unboundedpress.org/' > Michael Winter - composer | sound artist < / NuxtLink >
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My practice as a composer and sound artist is diverse , ranging from music created by digital and acoustic instruments to installations and kinetic sculptures . Each piece typically explores one simple process and often reflects various related interests of mine such as phenomenology , mathematics , epistemology , algorithmic information theory , and the history of science . To me , everything we experience is computable . Given this digital philosophy , I acknowledge even my most open works as algorithmic ; and , while not always apparent on the surface of any given piece , the considerations of computability and epistemology are integral to my practice . I often reconcile epistemological limits with artistic practicality by considering and addressing the limits of computation from an artistic and experiential vantage point and by collaborating with other artists , mathematicians , and scientists in order to integrate objects , ideas , and texts from various domains as structural elements in my pieces . My work also aims to subvert discriminatory conventions and hierarchies by exploring alternative forms of presentation and interaction , often with minimal resources and low information .
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My practice as a composer and sound artist is diverse , ranging from music created by digital and acoustic instruments to installations and kinetic sculptures . Each piece typically explores one simple process and often reflects various related interests of mine such as phenomenology , mathematics , epistemology , algorithmic information theory , and the history of science . To me , everything we experience is computable . Given this digital philosophy , I acknowledge even my most open works as algorithmic ; and , while not always apparent on the surface of any given piece , the considerations of computability and epistemology are integral to my practice . I often reconcile epistemological limits with artistic practicality by considering and addressing the limits of computation from an artistic and experiential vantage point and by collaborating with other artists , mathematicians , and scientists in order to integrate objects , ideas , and texts from various domains as structural elements in my pieces . My work also aims to subvert discriminatory conventions and hierarchies by exploring alternative forms of presentation and interaction , often with minimal resources and low information .
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My work has been presented at venues and festivals throughout the world such as REDCAT , in Los Angeles ; the Ostrava Festival of New Music in the Czech Republic ; Tsonami Arte Sonoro Festival in Valparaiso , Chile ; the Huddersfield New Music Festival in the United Kingdom ; and Umbral Sesiones at the Museo de Arte Contemporáneo in Oaxaca , Mexico . Recordings of my music have been released by XI Records , Another Timbre , New World Records , Edition Wandelweiser , Bahn Mi Verlag , Tsonami Records , and Pogus Productions . In 2008 , I co - founded the wulf . , a Los Angeles - based organization dedicated to experimental performance and art . From 2018 to 2019 , I was a fellow / artist - in - residence at the Akademie Schloss Solitude in Stuttgart , Germany . I currently reside in Berlin .
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My work has been presented at venues and festivals throughout the world such as REDCAT , in Los Angeles ; the Ostrava Festival of New Music in the Czech Republic ; Tsonami Arte Sonoro Festival in Valparaiso , Chile ; the Huddersfield New Music Festival in the United Kingdom ; and Umbral Sesiones at the Museo de Arte Contemporáneo in Oaxaca , Mexico . Recordings of my music have been released by XI Records , Another Timbre , New World Records , Edition Wandelweiser , Bahn Mi Verlag , Tsonami Records , and Pogus Productions . In 2008 , I co - founded the wulf . , a Los Angeles - based organization dedicated to experimental performance and art . From 2018 to 2019 , I was a fellow / artist - in - residence at the Akademie Schloss Solitude in Stuttgart , Germany . I currently reside in Berlin .
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< NuxtLink class = "text-3xl font-bold" to = 'http://www.mareikelee.com/' > MAREIKE YIN - YEE LEE - visual artist < / NuxtLink >
< NuxtLink class = "text-3xl font-bold" to = 'http://www.mareikelee.com/' > MAREIKE YIN - YEE LEE - visual artist < / NuxtLink >
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Mareike Yin ‑ Yee Lee ’ s multidisciplinary practice encompasses drawing , video , sculpture , found and made objects , printmaking , and artist books . Current works include installations , recordings and live performances produced in collaboration with musicians and composers with an emphasis on the relation between sight and sound . How we approach , perceive and respond to these form the basis of her recent works ‘ manifestations . Her immersive , site - specific installations explore the complex and tenuous nature of communication and how we experience space , drawing on gesture , sound , and memory to elicit responses that cannot be put into words . She redefines the architecture and temporality of the spaces in which she works . Lee ’ s work plays with the spaces between , across , and beyond , embracing the undefinable and subtle gradations , forging a language of colour , tone and space that seeks to articulate microcosms of daily life and sustained contemplation . Lee studied at Universita ̈ t der Ku ̈ nste , Berlin , Germany ; University of Toronto , Toronto , Canada ; and Nova Scotia College of Art and Design , Nova Scotia , Canada , where she was awarded the Joseph Beuys Scholarship and the Canada Millennium Award of Excellence . Recent projects include exhibitions and performances at Kunsthaus Kule Berlin ( 2020 ) , Kunstmuseum Kloster Unser Lieben Frauen Magdeburg ( 2019 ) , Galerie Kunstpunkt Berlin ( 2018 ) , Kunstbezirk Stuttgart , Kunst ( zeug ) haus Rapperswil - Jona Switzerland ( 2017 ) , Kunsthaus Interlaken ( 2017 ) , Neuer Kunstverein , Aschaffenburg ( 2016 ) , and KW Institute for Contemporary Art , Berlin ( 2016 ) .
Mareike Yin ‑ Yee Lee ’ s multidisciplinary practice encompasses drawing , video , sculpture , found and made objects , printmaking , and artist books . Current works include installations , recordings and live performances produced in collaboration with musicians and composers with an emphasis on the relation between sight and sound . How we approach , perceive and respond to these form the basis of her recent works ‘ manifestations . Her immersive , site - specific installations explore the complex and tenuous nature of communication and how we experience space , drawing on gesture , sound , and memory to elicit responses that cannot be put into words . She redefines the architecture and temporality of the spaces in which she works . Lee ’ s work plays with the spaces between , across , and beyond , embracing the undefinable and subtle gradations , forging a language of colour , tone and space that seeks to articulate microcosms of daily life and sustained contemplation . Lee studied at Universita ̈ t der Ku ̈ nste , Berlin , Germany ; University of Toronto , Toronto , Canada ; and Nova Scotia College of Art and Design , Nova Scotia , Canada , where she was awarded the Joseph Beuys Scholarship and the Canada Millennium Award of Excellence . Recent projects include exhibitions and performances at Kunsthaus Kule Berlin ( 2020 ) , Kunstmuseum Kloster Unser Lieben Frauen Magdeburg ( 2019 ) , Galerie Kunstpunkt Berlin ( 2018 ) , Kunstbezirk Stuttgart , Kunst ( zeug ) haus Rapperswil - Jona Switzerland ( 2017 ) , Kunsthaus Interlaken ( 2017 ) , Neuer Kunstverein , Aschaffenburg ( 2016 ) , and KW Institute for Contemporary Art , Berlin ( 2016 ) .
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< NuxtLink class = "text-3xl font-bold" to = 'https://www.facebook.com/KaliEnsemble' > KALI - performing ensemble < / NuxtLink >
< NuxtLink class = "text-3xl font-bold" to = 'https://www.facebook.com/KaliEnsemble' > KALI - performing ensemble < / NuxtLink >
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Kali is a new music ensemble based in the Hague . They primarily work with composers with whom they can collaborate and experiment over long periods . They aim to develop an artistic practice unique to their relationship with their collaborators . Over the past years , they have formed close and active relationships with several composers based in The Hague and abroad realizing many large - scale projects with great attention to detail .
Kali is a new music ensemble based in the Hague . They primarily work with composers with whom they can collaborate and experiment over long periods . They aim to develop an artistic practice unique to their relationship with their collaborators . Over the past years , they have formed close and active relationships with several composers based in The Hague and abroad realizing many large - scale projects with great attention to detail .
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< NuxtLink class = "text-3xl font-bold" to = 'https://users.utu.fi/jkari/' > Jarkko Kari - mathematician | invited guest < / NuxtLink >
< NuxtLink class = "text-3xl font-bold" to = 'https://users.utu.fi/jkari/' > Jarkko Kari - mathematician | invited guest < / NuxtLink >
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Jarkko Kari received his MSc and PhD degrees in mathematics from the University of Turku in Finland in 1986 and 1990 , respectively . He then worked for the Academy of Finland , and for Iterated Systems Inc . and the University of Iowa in the USA . Since year 2000 he has been a professor of mathematics at the University of Turku . His research interests include automata theory and the theory of computation , with emphasis on cellular automata , tilings and symbolic dynamics . Jarkko Kari has supervised twelve PhD theses , published over one hundred peer reviewed research articles and edited twenty conference proceedings and special issues on these topics . He serves in the editorial boards of eight scientific journals , and is currently a co - editor - in - chief of the journal Natural Computing . Jarkko Kari is a member of the Finnish Academy of Science and Letters since 2014.
Jarkko Kari received his MSc and PhD degrees in mathematics from the University of Turku in Finland in 1986 and 1990 , respectively . He then worked for the Academy of Finland , and for Iterated Systems Inc . and the University of Iowa in the USA . Since year 2000 he has been a professor of mathematics at the University of Turku . His research interests include automata theory and the theory of computation , with emphasis on cellular automata , tilings and symbolic dynamics . Jarkko Kari has supervised twelve PhD theses , published over one hundred peer reviewed research articles and edited twenty conference proceedings and special issues on these topics . He serves in the editorial boards of eight scientific journals , and is currently a co - editor - in - chief of the journal Natural Computing . Jarkko Kari is a member of the Finnish Academy of Science and Letters since 2014.
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< NuxtLink class = "text-3xl font-bold" to = 'https://www.mathematik.hu-berlin.de/de/forschung/forschungsgebiete/mathematische-physik/borot-mp-homepage' > Gaëtan Borot - mathematician | organizer | moderator < / NuxtLink >
< NuxtLink class = "text-3xl font-bold" to = 'https://www.mathematik.hu-berlin.de/de/forschung/forschungsgebiete/mathematische-physik/borot-mp-homepage' > Gaëtan Borot - mathematician | organizer | moderator < / NuxtLink >
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Gaëtan Borot was trained at École Normale Supérieure ( Paris ) in theoretical physicist and progressively moved to pure mathematics . He received his PhD from Universite d ' Orsay / CEA Saclay in 2011. After a postdoctorate in Geneva and a visiting scholarship at MIT , he worked as a Group Leader at the Max Planck Institute for Mathematics in Bonn . Since 2020 , he holds a bridge professorship between the Institute of Mathematics and the Institute of Physics of the Humboldt University of Berlin . He has worked on enumerative geometry , combinatorics , random matrix theory and mathematical aspects of quantum field theory , and likes to investigate the unexpected relations between seemingly different problems . He is also interested in scientific outreach .
Gaëtan Borot was trained at École Normale Supérieure ( Paris ) in theoretical physicist and progressively moved to pure mathematics . He received his PhD from Universite d ' Orsay / CEA Saclay in 2011. After a postdoctorate in Geneva and a visiting scholarship at MIT , he worked as a Group Leader at the Max Planck Institute for Mathematics in Bonn . Since 2020 , he holds a bridge professorship between the Institute of Mathematics and the Institute of Physics of the Humboldt University of Berlin . He has worked on enumerative geometry , combinatorics , random matrix theory and mathematical aspects of quantum field theory , and likes to investigate the unexpected relations between seemingly different problems . He is also interested in scientific outreach .
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a few selected articles :
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.
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.
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.
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.
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 :
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 )
Branko Grunbaum and G . C . Shephard , Tilings and Patterns , Dover Books ( originally published 1986 )
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a few useful links :
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 >
< 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 >
< 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 >
< 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 >
< 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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