Letter R

Eaas Rahman

An illustration that explores the concept of interacting random walks, where multiple figures traverse independent yet interwoven paths, influencing one another’s trajectories in unpredictable ways.

The torii gates serve as symbolic thresholds, representing transitions between different states, akin to a random walker crossing from one region to another. The elegantly dressed woman stepping out of a limousine follows a seemingly deliberate path, whereas the winged marionette suspended from a twisted tree is completely at the mercy of external forces - mirroring constrained diffusion and non-autonomous movement.

A man walks toward an imposing gothic structure, while an incongruous wet floor sign presents an unexpected environmental obstruction, altering the possible movement paths of other figures. The overall composition captures a complex system of interacting agents, where movement is influenced not just by internal intent but also by external constraints, much like interacting particle systems, diffusive processes, and stochastic motion in mathematical modelling.

By intertwining structure, randomness, and external influences, this piece visually expresses how mathematical patterns emerge in dynamic systems. It embodies the beauty of probability, the unpredictability of movement, and the deeply interconnected nature of seemingly independent journeys.

Aditya Raj

This artwork visualizes anomalous diffusion, a process where particles move unpredictably, deviating from standard Brownian motion. It is inspired by Lévy flights, a type of random walk with occasional long jumps mixed with clusters of short steps. The glowing, thread-like pathways represent these movements, where tangled regions correspond to sub-diffusive behavior, in which particles get temporarily trapped, while sweeping curves depict super-diffusion, where rapid jumps occur.

Mathematically, anomalous diffusion appears in systems where classical models fail, such as polymer dynamics, biological foraging, and turbulent flows. Unlike Gaussian diffusion, where displacement grows as the square root of t, Lévy flights follow power-law distributions, leading to nontrivial scaling properties. The swirling structures in the artwork reflect self-similarity and fractal geometry, which are key features of these stochastic processes.

The color gradient illustrates the passage of time and shows how randomness generates intricate, evolving patterns. This fusion of art and mathematics reveals the hidden order in chaotic systems, making complex theoretical ideas visually accessible. By capturing both structure and spontaneity, this work highlights the deep mathematical beauty underlying many natural and physical phenomena.

Zahra Ranjbaripour

Title: The Application of Chemical Formulas in Creating Anomalous Mathematical Patterns in Art

 My artistic process employs a unique mathematical approach to color distribution, rooted in chemical formulas. Rather than selecting pigments randomly, I calculate the spread of each color on the canvas based on the atomic and molecular mass of elements within a chemical compound. This generates precise numerical patterns that guide the composition of my paintings.

 This method reveals an anomalous mathematical pattern by transforming scientific measurements into a visual system. Unlike traditional color-selection techniques, my approach adheres to a structured yet unconventional mathematical rule, where each element's contribution is determined by its proportional atomic mass. The result is an organic but non-random distribution of color, unveiling the hidden numerical relationships embedded in chemistry.

 By applying mathematical reasoning to artistic creation, my work dissolves the boundary between science and art. These paintings visually and harmoniously interpret chemical structures, translating formulas into dynamic and aesthetic compositions. As such, they exemplify anomalous mathematical patterns in art—demonstrating how chemistry and mathematics can merge to produce singular visual expressions. 

John Reese

I create digital art using MSPaint. Though a somewhat primitive program by some standards, I find that I learn a lot by pushing the boundaries of its capabilities. 

I initially started with the idea of drawing fractals, and I have a fair amount of work in that vain, some patterns of which are commonly known. Symmetry also plays a big role in my work, which allows me to do a bit of cutting and pasting. But scale also plays a big role, and it is upon zooming in, in many of my works, where one will find abundant detail invisible, or barely visible, at larger scales. The appearance of such detail, as one zooms in, seems to me well into the nature of diffusion, but there are other aspects of my work that do also. (Don't get too focused on the symmetry. ;-) ) All such details are my own creation, in the sense that I do not use AI when creating them, but often need to work at the pixel level (though the software does allow for 8x magnification, which makes this process a little easier).

Again, the level of detail here may be more than intended, but just a few more lines to say, my work often starts with simple geometric shapes, but I find simple shapes give rise to complex shapes, when layered and laid out in a grid. And within a grid, each small scale shape can serve as an independent easel of sorts, though the use of color, and symmetry within the larger scale grid can give a cohesiveness to the work as a whole. 

Paulina Rega

The image was created by manipulating the arrangement of iron filings with magnets. Although it is easy to imagine the resulting regular, almost geometric structures arranged on a surface, I try to master the effect of magnetic force (by overlapping and moving the magnets) in order to obtain and then fix the image in a way that does not obviously evoke the method of their creation. In the same way, on a day-to-day basis we do not notice the forces of nature affecting us in invisible, sometimes surprising ways. 

Patrick Reichherzer

Video https://drive.google.com/file/d/1mAGzuAEtl7wul9Son3ljYYaCDA9lypHX/view?usp=sharing

This visualisation portrays cosmic rays traversing the complex topology of astrophysical magnetic fields. Each magnetic field line is coloured according to field strength, illustrating spatial variations in magnetic intensity. Cosmic rays spiral along these lines, guided by underlying magnetic structures.

Three stochastic phenomena emerge:

First, magnetic field lines spatially diffuse—a standard process in numerous astrophysical contexts. Second, magnetic field strengths fluctuate through bounded random walks, constrained between zero and their maximal intensity, introducing anomalous diffusion. Lastly, cosmic rays primarily follow magnetic field lines but occasionally switch to other lines due to scattering off magnetic fluctuations. Initially, this scattering produces anomalous diffusion within heterogeneous plasma conditions. Over extended periods, cumulative randomness converges toward classical diffusion, consistent with the central limit theorem.

This artwork, grounded in physically accurate magnetohydrodynamic (MHD) simulations and explicitly not generated by artificial intelligence, encapsulates the subtle yet profound role of stochastic processes in high-energy astrophysical plasmas. It vividly illustrates how cosmic rays navigate turbulent magnetic fields—ubiquitous in magnetized plasmas that constitute over 99% of visible matter—capturing essential themes at the heart of the "Stochastic systems for anomalous diffusion" research programme.

Franck Renand

Title: FRACTOR

A highly vibrant digital artwork inspired by anomalous mathematical patterns, featuring a complex interplay of red, blue, and yellow hues. The image showcases a chaotic fractal landscape with intricate structures, a glowing Lorenz attractor, and unpredictable geometric distortions, reflecting the beauty of stochastic systems.

References:

Annual Report 2023-2024: Uncertainty Quantification and Stochastic Modelling of Materials, p.8.

https://www.newton.ac.uk/event/ssd/

https://doi.org/10.1017/S0004972720000258

https://doi.org/10.1016/j.chaos.2019.04.039

Matt Roberts and Apolline Louvet

How does stochastic geometry affect evolution? This picture illustrates the dynamics of a stochastic growth model arising from population genetics, spreading from the bottom of the image to the top and starting from a flat interface. Stochasticity in reproduction at the front edge leads to an increasingly irregular front interface, that advances several times faster than in the deterministic version of the model. This is because growth at the front edge is driven by random "spikes", that are relatively infrequent but can then grow sideways at a linear speed, pulling the front. You can see several examples of these spikes in the picture. Understanding their properties has direct implications for evolution in expanding populations. When reproduction is only local (descendants stay close to their parents), this spiking phenomenon is enough to lead to a sharp increase of the expansion speed, though not to switch from a linear to a super-linear expansion regime.

Isaac Robson

My work is a testament to how diffusion models can produce novel methods to art creation. Combining AI generation, digital collage, and classical & modern themes, I want my work to demonstrate the way computational process can be used like a numerical alchemy. Engineered gaussian profiles gamify ways that sequential systems become aesthetic form, much like Fibonaccian patterns and Harmonic Ratios. The work is usually developed through modes of feedback, reprocessing the result like formula as it stretches further into its latent potential. Abstract formula hybridises with human formats seamlessly.  

Radoslav Rochallyi (Entry 1)

My paintings show mathematical anomalies in visual form. I do this by looking at geometry, topology, and symbolic abstraction. The first painting shows a warping of a grid that looks like it's being pulled by gravity, which represents a kind of space that isn't Euclidian. The second piece references cellular automata and periodicity, using a black-and-white grid interrupted by a dotted spiral—a nod to irrational cycles and non-repeating patterns. The third piece combines math symbols (Lagrangians, variational calculus) with a portrait, showing the connection between human thinking and formal systems. The combination of mathematical formulas and artistic creativity challenges our perception of the relationship between human thinking and mathematical certainty. Each piece is both a visual metaphor and a way of thinking about the differences that happen when order meets chaos.

Radoslav Rochallyi (Entry 2)

My paintings show mathematical anomalies in visual form. I do this by looking at geometry, topology, and symbolic abstraction. The first painting shows a warping of a grid that looks like it's being pulled by gravity, which represents a kind of space that isn't Euclidian. The second piece references cellular automata and periodicity, using a black-and-white grid interrupted by a dotted spiral‚ a nod to irrational cycles and non-repeating patterns. The third piece combines math symbols (Lagrangians, variational calculus) with a portrait, showing the connection between human thinking and formal systems. The combination of mathematical formulas and artistic creativity challenges our perception of the relationship between human thinking and mathematical certainty. Each piece is both a visual metaphor and a way of thinking about the differences that happen when order meets chaos.

Joel Rodriguez

This painting is titled "Among Nature" and it is a painting that represents the idea of anomalous diffusion. The painting is about a house in the middle of a landscape that is divided into two parts, with the other part becoming a single island in the distance. It is surrounded by a vast ocean, multiple flower fields, blue snowy mountains, a lone bicycle, a pair of shoes near the steps, and the morning sky. Additionally, you can also see a sandy, beaten path that extends directly from the front door into the distance. However, the main focus of the piece is the smoke coming from the house's chimney that forms an example of anomalous diffusion. The smoke is represented with mostly "cold" colors (red-green, blue-green, red-violet) and takes a mostly random form, going upwards and spreading out diagonally towards the sky. The smoke has this form because it represents the probability of the colored smoke going for a "random walk" or going off into random directions instead of being fixed into one direction or "process." It provides an aesthetic to the concept of "anomalous diffusion," and its many examples that can be seen in our daily lives.

Laura Elidedt Rodriguez

Video https://drive.google.com/file/d/1ofu81zhFxDEMNbsKBWYG-VHlI2G5DvVD/view?usp=sharing

"Not All Who Wander Are Lost" is a three-screen audiovisual installation exploring anomalous diffusion as a way of existence. It juxtaposes the movement of Mimosa pudica and Physarum polycephalum, two organisms that challenge traditional diffusion models through memory, adaptation, and constraint-based navigation.

One screen displays Mimosa pudica reacting to touch, demonstrating subdiffusive behavior—its response slows over repeated stimuli as the plant "remembers" past disturbances. Another screen shows Physarum polycephalum, which expands superdiffusively, leaving behind a self-secreted trail to avoid retracing its path. The third screen translates diagrams into sound, using spectrograms derived from research on stochastic systems for anomalous diffusion. The sound guides the digital intervention of the original recordings, making a loop of digital modification. 

Lloyd Ericson Castro Rodriguez (Entry 1)

At first glance, this photo captures a lightning discharge against a stormy background, appearing chaotic. Anomalous diffusion, however, reveals the complex, branching path of the lightning. The trajectory of the lightning suggests a superdiffusive process, possibly driven by long-range correlations within the atmospheric electric field—unlike typical Brownian motion, where particle displacement scales linearly with time.

Although visually striking, the image signifies complex stochastic systems where conventional models fall short. Tools used to describe anomalous diffusion in various phenomena—from plasma physics to biological transport—such as fractional Brownian motion or continuous-time random walks, may help model the erratic yet structured path of the lightning. The faint glow surrounding the lightning's path hints at the underlying statistical fluctuations that drive this non-Gaussian behavior.

Thus, this photograph invites reflection on how seemingly random events can reveal underlying mathematical patterns through the lens of stochastic systems, highlighting the widespread occurrence of anomalous diffusion in nature.

Lloyd Ericson Castro Rodriguez (Entry 2)

The sophisticated realm of stochastic systems governing how particles travel in uncommon ways is illustrated through the chaotic yet orderly filaments of a plasma lamp. Here, periodic long jumps deviate from the usual expectations of how distance should scale with time, as the motion of ionized particles exhibits a Lévy flight pattern, contrasting with the expected trajectories of classical Brownian motion. 

The plasma arcs generate intricate, fractal-like branching structures—characteristic of fractional diffusion processes observed in turbulent plasmas, disordered materials, and even biological transport systems. The variations in the electric field properties give rise to non-Gaussian distributions of particle steps, leading to superdiffusion—a fundamental feature of anomalous transport. 

This fascinating complexity reflects mathematical models of stochastic differential equations incorporating power-law step distributions, where randomness and determinism intertwine to produce self-organizing patterns. By capturing this dynamic interplay, the image serves as a striking testament to how physical systems reveal mathematical anomalies, emphasizing the deep connection between art, physics, and the unpredictable nature of diffusive phenomena.

Rohini

A paper windmill, designed with the Fibonacci patterns, utilises visual aesthetics and functionality. The Fibonacci sequence—where each number is the sum of the two preceding ones (1, 1, 2, 3, 5, 8,…) is known for its natural and intricate occurrence in nature to architecture. From seeds of flowers,

spirals of shells, and galaxies, to the divides of rivers and trees. reflecting harmony within space and proportion.

In the context of a paper windmill, the blades can be arranged following the Fibonacci spiral, where the curvature and blades sequences form a pattern to follow the ratios. The design maximizes the efficiency of the windmill, as the placement of the spiral helps in the distribution of air flow. With the aid of the sequence to design the blade, rotation speed is increased with elimination of lag.

Geometric patterns known as fractals can repeat at various scales to produce patterns that are indefinitely complex. Many fractals cannot be distinguished analytically. Though it is still structurally derived from a 1-dimensional line, an infinite fractal curve can be thought of as looping through space in a different way than an ordinary line.

In essence, the windmill embodies a blend of art and science.

Maya Rosenbaum 

For this competition, I wanted to explore the intersection of mathematics, human perception, and the unpredictable nature of patterns. The concept of “anomalous mathematical patterns” immediately suggested an exploration of the tension between order and chaos, structure and fluidity.

Beyond mathematical theory, my artworks explore themes of identity, isolation, and the human tendency to break away from predefined structures while still being part of mathematical systems.

I’d like to challenge viewers’ perceptions of the nature of patterns—both in science and in life.

Paul Rozenboim

Video https://drive.google.com/file/d/1w2cradxbej7AENnANe7B8we_uU2YN8vx/view?usp=sharing

Universa Morphogenesis is a speculative video work exploring digital abiogenesis through mathematical patterns that govern complex natural systems. Inspired by reaction-diffusion models, chaos theory, fractals, and the Fibonacci sequence, the project imagines the emergence of a parallel world shaped by altered yet familiar laws of growth and form.

At its core, the work simulates anomalous diffusion resulting in organic structures that form, spread, and self-organize in unexpected ways. The visual sequences are rooted in mathematical processes where local interactions generate global complexity and create landscapes and ‘living matter’ that behave according to altered laws of motion, growth, and emergence.

The film avoids narration or characters, instead unfolding entirely through visual logic. Morphogenesis becomes a language: flora bloom through Fibonacci spirals, ecosystems emerge through recursive fractal branching, and chaotic feedback loops drive continuous transformation.

By blending motion design and digital simulation, Universa Morphogenesis bridges art and mathematics to speculate on how life might arise under different universal conditions—governed not by biology, but by math itself. This work invites viewers to reimagine nature through the lens of pattern, anomaly, and abstraction.

Ruta

This piece captures the skeletal remains of a decomposed leaf, exposing a delicate vascular network that follows the same mathematical principles found across nature. Though shaped by organic processes, the patterns within this leaf reflect stochastic systems at work—evidence that mathematics is embedded in the world around us.

The fragmented vein structures mirror anomalous diffusion, where movement is irregular and constrained by natural barriers. The decay process forms random networks, modeling subdiffusive transport, much like how particles navigate disordered environments. The gaps and breakages resemble percolation clusters, illustrating the transition between connectivity and fragmentation. The fine, interwoven pathways echo Lévy flights, seen in everything from animal foraging to fluid dynamics.

This leaf was chosen because it is a perfect, natural manifestation of these mathematical ideas—a reminder that stochastic systems don’t just exist in equations but shape the patterns of life itself. The artwork highlights how randomness, constraints, and structure emerge organically, making the invisible mathematics of diffusion visible and tangible. Light and shadow interact with its delicate form, creating a dynamic composition that bridges the worlds of science, mathematics, and art.

Inessa Ruzhitsky

This digital art explores the intersection of natural symbolism and mathematical structure through a vertical composition centered around a symbolic tree. The tree marks the boundary between day and night. On the left side, the sun appears as a golden Fibonacci spiral, and its light illuminates a meadow where flowers grow in phyllotactic patterns governed by the golden angle (~137.5°). A snail with a logarithmic spiral shell moves among the roots, which twist downward in branching curves resembling anomalous diffusion and fractal growth. Within the roots, miniature creatures live alongside hidden treasures and winding staircases, suggesting complexity beneath surface order.

On the right side, the tree’s branches fade into night. The sky gradually transforms into a calm sea, whose waves form nested spirals. A pearl lies inside a shell at the ocean floor, and treasure chests rest nearby—objects shaped and placed in accordance with the golden ratio and natural symmetry. Throughout the work, elements such as the petals, shells, waves, and sun reflect the deep connection between biological forms and mathematical aesthetics. The composition invites the viewer to see nature not as random, but as a layered structure shaped by mathematical laws and graceful irregularity.