In 2021 an unconventional pair of collaborators embarked on a bold experiment. For two decades Steven Rayan, a mathematician and mathematical physicist, and Jeff Presslaff, a freelance composer, pianist and trombonist, prepared to solution one big dilemma: Could they translate a mathematical physics investigation paper right into new music<\/a>? Also, would their musical development sound excellent?<\/p>\n In September Rayan and Presslaff released their brainchild, \u201cMath + Jazz: Seems from a Quantum Potential.\u201d<\/a> Specifically two several years to the date that Rayan, a researcher at the College of Saskatchewan, and Presslaff, who\u2019s dependent in Winnipeg, Canada, 1st linked more than e-mail, they collected a 15-piece \u201chyperbolic band\u201d of musicians to execute the five-segment concert at the College of Saskatchewan. Each individual segment corresponded to a part of Rayan\u2019s investigation report<\/a>.<\/p>\n Part musical effectiveness and component lecture, the live performance was played to \u201ca packed dwelling,\u201d Rayan suggests. The lecture portion dissected the paper\u2019s scientific ideas and illustrated how all those ideas ended up transmogrified into music<\/a>. Some of the illustrations were being literal: the slideshow featured hyperbolic art established by Elliot Kienzle.<\/p>\n Pulling off the live performance was no effortless feat. Due to the fact many of the musicians weren\u2019t community, the band hadn\u2019t rehearsed the new music collectively in man or woman right until the night time just before the live performance, Rayan notes.<\/p>\n The music was based mostly on Rayan\u2019s 2021 Science Advancements<\/em> article \u201cHyperbolic band theory,\u201d which he wrote with Joseph Maciejko of the University of Alberta. Their goal was to take a look at no matter whether band theory\u2014which scientists use to take into account the power ranges of supplies and the atoms that they\u2019re made of\u2014could be reformulated to demonstrate hyperbolic elements, which have irregular, warped preparations.<\/p>\n In band theory a material\u2019s power ranges are assumed of as becoming contained in sheetlike bands hovering over the supplies they belong to. These shadowy bands symbolize the material\u2019s quantum homes, and interactions among these bands have implications for the material\u2019s conduct.<\/p>\n Rayan and Maciejko succeeded in getting a band idea that works in the wonky planet of hyperbolic geometry, a unusual geometrical realm that breaks Euclid\u2019s \u201cparallel postulate.\u201d<\/a> Also named Euclid\u2019s fifth postulate, this rule tells us the next: Suppose you are offered a line. For any place that is not on that line, there will be only a single line that the two goes through that place and is parallel to the original line. In hyperbolic land, a bare minimum of two lines will go by way of the stage although also remaining parallel to the provided line.<\/p>\n The study \u201cis a entire new tactic to designing materials\u2014especially quantum materials\u2014by re-engineering their geometry from the inside out,\u201d Rayan states. The approach requires altering the material\u2019s band composition to build the ideal variations in the material\u2019s attributes. \u201cThey can get on strange, exotic geometries,\u201d he suggests.<\/p>\n This may possibly glimpse like, for instance, covering a curved surface by tiling it with octagons so that there aren\u2019t any gaps among the shapes, which are nonoverlapping. To human eyes, the edges of these octagons surface curved, and the styles look like they are distinct sizes, Rayan notes. But \u201cif you had a distinctive variety of eye that sees the planet in a hyperbolic way\u2014maybe insectlike compound eyes\u2014[the octagons] may all glimpse the identical to you,\u201d he suggests.<\/p>\n The function gained a great deal of focus from other scientists. \u201cI\u2019m really impressed by the connection in between material science and algebraic geometry which was unearthed by the authors of this paper,\u201d says Michael Groechenig, a mathematician at the College of Toronto, who wasn\u2019t included with the posting.<\/p>\n Rayan is energized to utilize his results to researching unconventional resources with the potential for \u201cdisruptive applications,\u201d this kind of as in quantum computing. \u201cIt\u2019s instead delightful to see anyone show an critical software of these approaches of such a concrete character,\u201d Groechenig claims. The paper is \u201can invitation for us pure math folks to leave our comfort and ease zone a minimal and to examine hitherto uncharted territory,\u201d he provides.<\/p>\n Producing a mathematical live performance<\/a> is its have kind of disruptive application of the research. \u201cI did not want [the music] to be impressionistic,\u201d Presslaff suggests. \u201cI needed it to be definitely genuine to mathematics…. I\u2019ve just witnessed far too numerous cross-disciplinarity projects that just strike me as superficial. The science aspect may be rigorous, and the art side is very not rigorous.”<\/p>\n Rayan agreed with that purpose. \u201cI experienced a dedication to not just producing music that was in some way loosely influenced by the math and the science but rather in some way retelling the mathematics term for term, equation for equation, in a musical kind,\u201d he states.<\/p>\n But embracing that problem also expected that each authorities go away their ease and comfort zones and learn concepts from every other\u2019s spots of know-how. Presslaff immersed himself in subject areas from linear algebra and topology that assisted illuminate the interior workings of the research paper. Rayan dove into \u201ctrying to recognize, as substantially as probable, the advanced musical tips [Presslaff] introduced to the desk.\u201d<\/p>\n The pair exchanged tips for around 18 months before Presslaff even began creating the songs. \u201cIt\u2019s amazing that I never met Jeff in individual until finally the day just before the performance on September 20,\u201d Rayan says. \u201cIt was all on Zoom due to the fact of the pandemic and for the reason that of length. It was a fascinating way of working\u2014that we could accomplish this even via purely virtual means.\u201d<\/p>\n It\u2019s tricky to pinpoint no matter if Rayan and Presslaff achieved their grand objective: to convert the primary tips from Rayan\u2019s paper specifically into jazz new music. Unlike in mathematics investigate, there is no \u201cproof\u201d that they attained their objectives. Still, the duo is delighted with their outcome. \u201cGetting it completed was just hardly ever a specific thing until even, say, six months before the overall performance,\u201d Rayan suggests.<\/p>\n Hyperbolic Band performer Shah Sadikov, a Baltimore-based violist, claims a emphasize of the live performance was when Presslaff utilized a double fugue, a musical strategy that is \u201cvery tough to apply,\u201d to signify the system of constructing an \u201cinfinite shape,\u201d Sadikov claims. Mathematically, that meant developing an object with \u201cno commencing, no finish,\u201d Sadikov claims. Musically, developing a double fugue includes making \u201cone strategy the foundation of the musical piece and then you choose the correct very same thought [and] you put it a little bit later on major of it,\u201d and so on, he says. \u201cYou build these levels of strategies. And then you can use counter suggestions to that, possibly getting the similar musical concept [and] placing it backwards or upwards,\u201d he notes.<\/p>\n For Rayan, a emphasize was listening to Presslaff\u2019s musical choose on \u201cthe so-termed particle-wave duality or the place-momentum duality in hyperbolic band concept.\u201d In that context, momentums can get on a lot more dimensions than positions can. \u201cWe wished to seize in the audio the jump from, say, two proportions to 4 proportions in the simplest of these materials, which are based on octagonal hyperbolic lattices,\u201d Rayan says.<\/p>\n \u201cHearing [Presslaff\u2019s] makes an attempt at introducing extra voices in the music that seize the extra levels of freedom, the sudden soar to two added dimensions, was a moving encounter for me,\u201d Rayan suggests. \u201cI cherished viewing the audience try to hear these extra voices just after [his] clarification of them.\u201d<\/p>\n The live performance bundled 1 other inventive component: Kienzle\u2019s hand-drawn illustrations. Now a graduate scholar at the University of California, Berkeley, Kienzle developed the mathematical artwork for a related research task<\/a> that he and Rayan labored on though he was an undergraduate university student at the University of Maryland, College Park. \u201cThis was an attempt to tell the tale via a visual lens,\u201d Rayan says. In the live performance individuals illustrations assisted enhance the musical and verbal explanations of the math and science.<\/p>\n Rayan sees reinterpreting this do the job by means of musical and artistic lenses as a way of bringing it whole circle. Considerably of the mathematical and scientific ideas showcased in his papers borrow concepts from the earth of artwork. For occasion, \u201chyperbolic tilings are extremely reminiscent of\u201d Dutch graphic artist M. C. Escher\u2019s legendary woodcuts, he notes. Rayan designs to continue checking out new ways of fusing mathematical and creative perspectives to \u201cgive again to art\u201d though also sprouting new insights for his exploration.<\/p>\n<\/div>\nHyperbolic Band Principle<\/h2>\n
![\"Purple,](\"https:\/\/static.scientificamerican.com\/sciam\/assets\/Image\/2023\/1-s2_0-S0001870822004819-gr006.jpg?w=900\")
<\/a>Translating to New music<\/h2>\n
Double Fugues and Infinite Designs<\/h2>\n