“The understanding of complexity and the use of the creativity of nature, the continuation of the work of nature are the grand challenges for the scientists of the 21st century.”
— Ilya Prigogine, Is Future Given?
“The collision of hail or rain with hard surfaces, or the song of cicadas in a summer field. These sonic events are made out of thousands of isolated sounds; this multitude of sounds, seen as totality, is a new sonic event.”
— Giannis Xenakis, Formalized Music: Thought and Mathematics in Composition
Overview
The Poincaré Trajectories Digital Art Collection is an interdisciplinary research focusing on the theoretic implications of dynamical systems theory, complex systems theory in music composition, and on the synergy between visual art, music and science.
This NFT collection marries applied math, blockchain and music into a single, captivating experience.
Each NFT in this collection encapsulates the essence of chaos through mathematical models, accompanied by compositions resonating with the underlying frequencies.
Impact
The mathematical underpinnings add an intellectual layer to the experience.
Collectors are drawn to the uniqueness of each NFT, as no two are alike.
The Collection
01
Matteo likes dynamical systems. Chaos, stochastic processes, high-dimensional geometry. He plays gypsy jazz guitar and is co-founder of a fintech startup. Arnold’s cat map is a chaotic map from the torus into itself.
02
David from “common” music production has approached generative music, loves technology and multimedia and always tries to merge these three things in his works, under the name of “Flux Theory”.
03
Mattia is a student in computer engineering. He likes to test the real world and learn things that go beyond his course of study. Arnold’s cat map is a chaotic map from the torus into itself.
04
Stefano is an aerospace engineer. His work gravitates around fluid dynamics, but he is involved in everything that can be called geeky. Arnold’s cat map is a chaotic map from the torus into itself.
05
Andrey Nikolaevich Kolmogorov was a Soviet mathematician who contributed to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity. Arnold’s cat map is a chaotic map from the torus into itself. A few words extracted from speeches by Kolmogorov have been stretched and granulated in a semi-randomic way to create an audio texture.
06
Vladimir Igorevich Arnold was a Soviet and Russian mathematician. Best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he made contributions in dynamical systems theory, algebra, catastrophe theory, topology, algebraic geometry, symplectic geometry, symplectic topology, differential equations, classical mechanics, differential geometric approach to hydrodynamics, geometric analysis and singularity theory. He is widely regarded as one of greatest mathematicians of all time. Arnold’s cat map is a chaotic map from the torus into itself. A few words extracted from speeches by Arnold have been stretched and granulated in a semi-randomic way to create an audio texture as accompaniment to the sounds made with synths.
07
Jürgen Kurt Moser was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations. Arnold’s cat map is a chaotic map from the torus into itself. The audio part is made by a cover of a Moser Track named Cadenza, remade with Synths and Samples.
08
Benoit B. Mandelbrot was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as “the art of roughness” of physical phenomena and “the uncontrolled element in life”. His math- and geometry-centered research included contributions to such fields as statistical physics, meteorology, hydrology, geomorphology, anatomy, taxonomy, neurology, linguistics, information technology, computer graphics, economics, geology, medicine, physical cosmology, engineering, chaos theory, econophysics, metallurgy, and the social sciences. Arnold’s cat map is a chaotic map from the torus into itself. A few words extracted from speeches by Mandelbrot have been stretched and granulated in a semi-randomic way to create an audio texture as accompaniment to the sounds made with synths.
09
Giannis Klearchou Xenakis was a Romanian-born Greek-French avant-garde composer, music theorist, architect, performance director and engineer. Xenakis pioneered the use of mathematical models in music such as applications of set theory, stochastic processes and game theory and was also an important influence on the development of electronic and computer music. Among the numerous theoretical writings he authored, the book Formalized Music: Thought and Mathematics in Composition is regarded as one of his most important. Arnold’s cat map is a chaotic map from the torus into itself. The music is created through systems giving different notes, lengths, dynamics and speeds to various samplers of Strings Instruments in a stochastic way, imitating Xenakis style.
010, 011, 012
The Tinkerbell map is a discrete-time dynamical system. The origin of the name is uncertain; however, the graphical picture of the system (as shown to the right) shows a similarity to the movement of Tinker Bell over Cinderella Castle, as shown at the beginning of all films produced by Disney. Audio made with custom presets of Moog sub 37 and Access T Virus. Each preset has various random parameters on the modulation generators (LFOs, envelopes etc).
013, 014, 015
The Kaplan–Yorke map is a discrete-time dynamical system. It is an example of a dynamical system that exhibits chaotic behavior. Audio made with custom presets of Moog sub 37 and Access T Virus. Each preset has various random parameters on the modulation generators (LFOs, envelopes etc).
016, 017, 018
The Duffing map is a discrete-time dynamical system. It is an example of a dynamical system that exhibits chaotic behavior. It is a discrete version of the Duffing equation.
019, 020, 021
Chaotic map of a stochastically perturbed pendulum. Sounds made with samples recorded over the years, randomly mixed and stretched / “granulated”.
022
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden ‘qualitative’ or topological change in its behavior. Bifurcations occur in both continuous systems (described by ordinary, delay or partial differential equations) and discrete systems (described by maps).
The name “bifurcation” was first introduced by Henri Poincare’ in 1885 in the first paper in mathematics showing such a behavior.
The music inspiration is given by the sudden change of one frame to another, which creates an “explosion” effect to the track.
023
Bifurcation, particular. Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Audio: granular synthesis that emulates movement.
024, 025, 026, 027
Perturbed pendulum. Integrable systems may be seen as very different in qualitative character from more generic dynamical systems, which are more typically chaotic systems. The distinction between integrable and nonintegrable dynamical systems has the qualitative implication of regular motion vs. chaotic motion and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated into an exact form. Audio: granular synthesis which follows the animation.
028
Perturbed pendulum. In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state. Audio: granular synthesis which follows the animation.
029, 030, 031
Self-similarity in perturbed pendulum. In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Audio: granular synthesis which follows the animation.
032, 033, 034
Standard map bifurcation, particular. The standard map, also known as the Chirikov–Taylor map or as the Chirikov standard map, is an area-preserving chaotic map constructed by a Poincaré’s surface of section of the kicked rotator. Audio made with custom presets of Moog sub 37 and Access T Virus. Each preset has various random parameters on the modulation generators (LFOs, envelopes etc), and the music tries to follow the “fall into the void” effect given by the animation.
035
Standard map, zoom-out. In mathematics, an iterated function is a function from X to X (that is, a function from some set X to itself) which is obtained by composing another function f : X to X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again in the function as input, and this process is repeated. Music: simple oscillators that grow like a “swarm” as the zoom of the plot decreases.
036, 037, 038
Pendulum bifurcations. In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Audio made with custom presets of Acces T Virus. Each presets has various random parameters on the modulations generators (LFOs, envelopes etc) and and stretched samples.
039
Stability Islands, standard map. In celestial mechanics, orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other. Audio made with custom presets of Moog sub 37 and Acces T Virus. Each presets has various random parameters on the modulations generators (LFOs, envelopes etc).
040, 041
Standard Map, loops. Statistical mechanics has been applied in non-equilibrium statistical mechanics to the issues of microscopically modeling the speed of irreversible processes that are driven by imbalances. Audio made with custom presets of Moog sub 37 and Acces T Virus. Each presets has various random parameters on the modulations generators (LFOs, envelopes etc).
042, 043, 044, 045
From nearly-integrable to chaotic trajectories, standard map.
Audio made with custom presets of Moog sub 37 and Acces T Virus, that have been recorded, processed and stretched.
046, 047
Liouville’s theorem is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system. Audio: granular synthesis of various samples and improvisation sessions of semi-generative synths, recorded, processed and stretched.
048
Dynamical model of cancer growth, which includes the interactions between tumour cells, healthy tissue cells, and activated immune system cells, clearly leading to chaotic behavior. Audio: Granular Synthesis.
049, 050, 051
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers. Audio: granular synthesis of various samples and improvisation sessions of semi-generative synths, recorded, processed and stretched.
052
Fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Audio: granular synthesis that follows the growth of the animation.
053
Strange Attractor is used to describe an attractor (a region or shape to which points are ‘pulled’ as the result of a certain process) that displays sensitive dependence on initial conditions. Audio made with custom presets of Acces T Virus, “rotary effect”.
054, 055
An ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. Audio: granular synthesis of various samples and improvisation sessions of semi-generative synths, recorded, processed and stretched.
056, 057, 058
Langevin dynamics is an approach to the mathematical modeling of the dynamics of molecular systems. The approach is characterized by the use of simplified models while accounting for omitted degrees of freedom by the use of stochastic differential equations. Audio:custom presets with random elements of Moog sub 37 and Access virus TI, granular synthesis, samples recorded from custom max MSP synths.
059
The Log Periodic Power Law Singularity model provides a flexible framework to detect bubbles and predict regime changes of a financial asset. A bubble is defined as a faster-than-exponential increase in asset price, that reflects positive feedback loop of higher return anticipations competing with negative feedback spirals of crash expectations. It models a bubble price as a power law with a finite-time singularity decorated by oscillations with a frequency increasing with time. Audio generated by the plot information, transformed into MIDI notes and sent to synths.
060
Stochastic Celestial Mechanics.
Determinism is an idealization: real dynamical systems cannot be isolated from their environments and thus always experience stochastic influence. This work focuses on the implications of this in celestial mechanics. Inspired by the recently introduced Stochastic Two-Body problem, we here introduce the Stochastic Three-Body problem, in which the motion of the point mass is both influenced by the gravitational attraction of the primaries and by a stochastic perturbation, arising from random fluctuations of mass distributions in the solar system. Restricting our analysis on the planar, circular case, after deriving the equations of motion, we discuss the effect of the stochastic perturbation on the symplectic structure of the original dynamical system, and the relations with deterministic nearly-integrable dynamical systems in general. We show the implications of the stochastic perturbation, particularly via Ito’s lemma, for diffusion processes and the existence of weak first integrals. Audio: through a generative system built with Max MSP and Mira for Ipad that modify the frequency and the amplitude of a sound according to the position of a finger in the x y, the resulting spectrum of x(t) has tracked and “redrawn” as is does with a drawing with tracing paper. By repeating the process several times, and merging the takes, the result is a single spectrum obtained in a semi-random way.
061, 062, 063
White noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. Audio: Access Virus T recordings and noise that increases with zoom.
064, 065, 066
Brownian noise, is the type of signal noise produced by Brownian motion, hence its alternative name of random walk noise. It is an example of coloured noise. Audio: Access Virus T recordings and noise that increases with zoom.
067
A Levy flight is a random walk in which the step-lengths have a stable distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions. The use of the term has extended to also include cases where the random walk takes place on a discrete grid rather than on a continuous space. Audio: the entire x y z values have been exported, placed in text files and read at animation speed. Each values of the three dimensions over time has applied to various synthesis and sampler parameters, such as note, MIDI control changes, and effects, following the movement of the Levy Flight.
068
Nyan Cat is a YouTube video uploaded in April 2011, which became an internet meme. Arnold’s cat map is a chaotic map from the torus into itself. Audio: David’s cat samples, Nyan cat’s song redone with synths, granular synthesis.
069
Pikachu is widely considered to be the most popular and well-known Pokémon species. Arnold’s cat map is a chaotic map from the torus into itself. Audio: recreated pikachu’s voice in studio, various stretched samples, granular synthesis.
070
Marco Rossi is one of the main protagonists in Metal Slug. Arnold’s cat map is a chaotic map from the torus into itself. Audio: FXs samples and stretch / granular synthesis.
071 to 081
CryptoPunks is a non-fungible token (NFT) collection on the Ethereum blockchain. Arnold’s cat map is a chaotic map from the torus into itself. Audio: music made improvising on the animation in Loop, with random elements on synths, samples, granular synthesis etc.
Future Directions
Continuing to expand the collection with new mathematical concepts, together with the exploration of opportunities for real-world exhibitions to bridge the gap.