Letter L

Viduranga Landers (Entry 1)

"Collatz Fractal Formation" captures the dynamic genesis of mathematical patterns emerging from chaos. This time-lapse video documents the progressive generation of fractals based on my generalized Collatz conjecture algorithm, revealing not just the final form but the mesmerizing process of mathematical self-organization.

The video shows how individual sequences—following the rules n/Y when divisible by Y, or ((n * (X + 1) + (X - r))/Y) otherwise—gradually converge into coherent structures. Starting from a single point, each iteration adds thousands of paths that branch according to precise mathematical relationships defined by parameters Z (base number), Y (divisor), m (power), and division angle.

What makes this process particularly fascinating is witnessing order emerge from apparent randomness. Early in the generation, the pattern appears chaotic, but as more sequences are processed, clear geometric structures materialize. This visual transformation mirrors the underlying mathematical principle: deterministic rules producing seemingly unpredictable yet ultimately ordered outcomes.

The time dimension adds another layer to understanding these anomalous patterns. Fractal structures bloom like mathematical flowers, revealing how simple recursive rules can generate extraordinary complexity—a perfect visualization of emergence in mathematical systems and a window into the boundary between disorder and pattern.

Viduranga Landers (Entry 2)

These artworks explore anomalous mathematical patterns through visualizations of fractals generated from a generalized Collatz conjecture. While the classic Collatz conjecture follows the 3n+1 rule, my work extends this to a generalized form using several more parameters based on the work "A generalization of the Collatz problem and conjecture" by M. Bruschi (2008).

The generalized algorithm follows two steps: n/Y when n is divisible by Y, or ((n * (X + 1) + (X - r))/Y) otherwise, where r = n mod X and X = Z^m. By adjusting Z (base number), Y (divisor), m (power), and angleDiv (π/n rotation), entirely new fractal geometries emerge.

Each iteration creates a line segment rotated according to the sequence value's relationship to Y. As thousands of starting values are processed simultaneously, intricate patterns reveal the hidden structure beneath these seemingly chaotic sequences.

What makes these patterns "anomalous" is their unpredictable regularity—mathematical chaos contained within elegant structures. While individual sequences appear random, the visual aggregate displays stunning self-similarity and order across scales.

This work visualizes the boundary between order and chaos in number theory. The emergent aesthetics aren't arbitrary artistic choices but direct manifestations of pure mathematical relationships, revealing how anomalous patterns in mathematics can generate remarkable visual coherence.

Valerie Large

This piece of relief sculpture was an experiment to see what would happen when joining a number of irregular but similar shapes which would perhaps suggest microbes or particles of some sort and joining them together by randomly selecting them and seeing how they could fit together. Each piece was slightly different in size and shape although roughly the same (as natural forms often are). The randomly selected shapes were fitted (sewn) together, starting at a central point and working outwards. The slight relief effect of joining the shapes was again unforced and the individual shapes interact with one another and become distorted by their connection.

However the decision to end the piece was an artistic one, it was finished when it looked, but up to this point it relied to a large extent on randomness.

Ao Lei (Entry 1)

Video https://drive.google.com/file/d/1cPdwLhXC3GcZWSeJ7dJ8JrJoEuEMXhHG/view?usp=sharing

Stereographic Sine-Gordon transforms the mathematical poetry of quantum field theory into visual experience. By projecting the two-dimensional Sine-Gordon equation—where particles exist as wave disturbances—into curved three-dimensional space, the work reveals hidden symmetries typically accessible only through equations. As the system evolves, ephemeral "kinks" and "breathers" emerge and dissolve, their fluid choreography manifesting the elegant topology beneath quantum excitations. The installation occupies a liminal space between rigorous physics and pure abstraction, inviting viewers to contemplate the aesthetic dimension of mathematical structures that shape our understanding of reality.

Ao Lei (Entry 2)

Trinion explores an imaginary mathematical structure extending beyond complex numbers through procedural mesh generation. The real-time computational system renders impossible geometries based on trinion algebra—a speculative framework I've developed. As the algorithm progresses, intricate crystalline structures emerge and transform, revealing different facets of this theoretical construct. These ethereal architectures possess unique topological properties that could only exist within mathematical fiction, inviting contemplation of how abstract constructs shape our understanding of reality

Deanna L Leonard (Entry 1)

Title: Mathematical Networks in Motion: The Fusion of Random Networks and Fluid Dynamics in Art

These images transform the mathematical elegance of random networks and fluid dynamics into futuristic, sculptural forms. The intricate, web-like structures enveloping the human face and head resemble random networks, where interconnected nodes form organic, evolving patterns. These formations are akin to Voronoi diagrams or Delaunay triangulations, which describe natural partitioning in cellular structures, neural pathways, and even urban planning. The dynamic, flowing nature of these structures also reflects fluid dynamics, specifically how surfaces adapt and morph under external forces, akin to the Navier-Stokes equations governing fluid motion.

The inspiration behind these pieces stems from the hidden mathematics in nature—how soap bubbles organize into networks, how air flows over organic surfaces, and how neural connections continuously reshape within the brain. By blending AI-generated forms with the laws of fluid motion and emergent connectivity, these artworks showcase how the unseen mathematical principles governing nature can be transformed into wearable, kinetic structures. They are not just futuristic fashion concepts; they are visual representations of how chaos and order coexist in the physical world. The images ask us to reconsider the boundaries between art, science, and nature, reminding us that behind every structure, there is a formula shaping its existence.

Deanna L Leonard (Entry 2)

Title: Interacting Random Walks and Fluid Dynamics: A Visual Representation of Movement

The attached images showcase the mathematical principles of interacting random walks and fluid dynamics, capturing the essence of fluid motion and chaotic dispersion. The intricate swirling patterns and dynamic structures reflect random walks, a concept in probability theory where particles or entities move unpredictably while still following a set of mathematical constraints. The way the liquid flows and disperses suggests stochastic movement, akin to how molecules in a gas spread over time, or how electrical currents diffuse through conductive surfaces.

At the same time, these images embody fluid dynamics, specifically the Navier-Stokes equations, which describe how fluids flow and respond to external forces. The way the liquid forms faces and dissipates into space mimics real-world turbulence, eddies, and laminar flows. The golden and metallic tones further suggest energy transfer, much like heat diffusion or wave propagation in physics.

The inspiration behind these artworks lies in the idea of transformation—how motion shapes identity, and how randomness can create harmony. The forms symbolize the impermanence of structure, the beauty of fluid motion, and the unseen mathematical principles that guide the natural world. These images invite viewers to embrace movement, fluidity, and the hidden patterns within chaos.

Deanna L Leonard (Entry 3)

Title: Equations of Water and Light

She drifts between wave and wind,

a whisper of motion, a breath unpinned.

Her face dissolves where rivers bend,

where fractals dance and chaos blends.

A symphony spun from fluid flow,

the Navier-Stokes let her body go.

Particles scatter, then reunite,

a golden ember in soft twilight.

The sky holds whispers of diffused desire,

reflections flicker, then expire.

A random walk through endless streams,

chaotic beauty wrapped in dreams.

The ocean’s mirror, still and deep,

carries echoes time must keep.

Anomalous tides, reflections bright,

math and nature fused in light.

From rippling silk to liquid skin,

she is motion, she is wind.

Equations sculpted, wild yet free,

a fleeting glimpse of infinity.

Mathematics, art, and poetry intertwine as different languages of the same truth—patterns, motion, and beauty found in nature and existence. Math defines the rhythm of the universe, art gives it form, and poetry captures its essence in words. Together, they reveal how structure and fluidity coexist, how chaos shapes harmony. This submission is a celebration of that unity—where equations become art, and numbers whisper poetry in waves of light and motion.

Kelvin Leong

Title: The Rhythm of Random from the World

This artwork presents dice-shaped figures walking across a stylised version of Earth. Each die displays a different mathematical symbol on its face—including addition, division, inequality signs, the square root of a negative number, and even a newly imagined symbol. These symbols represent the wide range of ideas in mathematics, from basic operations to abstract and unknown concepts.

The dice, traditionally symbols of chance and unpredictability, are shown in motion, as if following a common rhythm or invisible path. Their journey suggests that even random elements can move in harmony, guided by patterns we may not immediately see. Curved lines and musical notes around the globe reinforce this idea, hinting that randomness itself can contain a hidden structure—much like music composed from unexpected notes.

The inclusion of the square root of a negative number symbolises the world of imaginary and complex numbers, which are essential in understanding many scientific and natural phenomena. The undefined symbol adds a layer of mystery, suggesting that mathematics is still evolving, with new discoveries yet to come.

This artwork is a poetic reflection on how chaos, imagination, and structure coexist within both nature and mathematics.

Georgi Levchev (Entry 1)

In this digital collage, I present the idea of geometric arrangements in gallery spaces. The physical layout of an art gallery can be seen as a geometric problem. The goal is to use the space efficiently while enhancing visitors' aesthetic and educational experience.

The background depicts an ample art space built from a solid 3D geometric interior. In the middle, a woman's bust is divided vertically as a focal point, with the left side in warm yellow tones and the right in cool blue and red tones. This symbolizes the basic concepts of geometry, a branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and solids.

A complex collage of geometric shapes, lines, and patterns complements the background. A grid structure at the bottom resembles a landscape with pyramids and polyhedra, an expression of geometric solutions for maximum use of space, calculating the most efficient routes for the visitor through them. The Geometric principles ensure clear lines of sight so that somebody can view the artworks from different points in the same room.

The Color Palette used is vibrant, with contrasting warm and cool colours. The lighting is dramatic, with a bright light source in the lower centre, symbolizing symmetry and balance.

In this symmetrical composition in a modern and abstract style and with a mix of realistic and surreal elements, the intersection of art and mathematics offers fertile ground for innovation, discovery and interdisciplinary enrichment. As we continue to explore and expand these connections, we can expect a deeper understanding of the mathematical patterns that structure our world and the artistic expressions that give it meaning.

Georgi Levchev (Entry 2)

In this digital collage, I present the idea of geometric interactions in the art world, and I want to show the interaction between mathematics/geometry and art in our world.

I trace the connection between Man, the World of the Artist, Art Spaces, and Nature. I use geometry to maximize the use of space without overcrowding, where the objective is to use the artwork space effectively.

Behind every process or action stands a person/artist whose creations are a mixture of inspiration and a unique artistic style. However, to realize a painting, we always resort to mathematical calculations and geometric structuring of the canvas. However, seemingly different fields, art and mathematics, have intertwined paths throughout history, with each influencing and enriching the other.

The canvas was divided into two pyramidal fields, with the upper part symbolizing the artist's world, his creative temple filled with the light of inspiration, and the lower part presents the real world - a mixture of earth textures resembling rock formations, covered with a grid and a spiral with the golden ratio. The geometric shapes included in the composition are arranged in strict balance, and are the connection between the physical and creative aspects of the artist.

I use a color palette filled with blues, greens, ambers, and purples to create a surreal, ethereal atmosphere. The lighting is soft and diffuse, enhancing the work's dreamlike quality. The style is reminiscent of digital surrealism, combining realistic portraits with abstract geometric elements.

Leyla

İn my artwork, titled “Circle”, a abstract tree is depicted with geometric shapes. The small elements that create these shapes are taken from the crystalline structure of solid oxygen. My goal in depicting a tree with the crystalline structure of solid oxygen is to create a connection between form and meaning. I tried to avoid realistic imagery.

The artwork is created by painting each solid oxygen element in different colors. This means that there is no space, that is, no background, in the artwork: all spaces have become elements. The general structural scheme of the composition is the similar as the structural principles in carpets and rugs (frame, block, violation of symmetry).

İn order to create different forms with the same element and create meaning between these forms, I have created color games. One of these is the formation of blocks by grouping colors. I have also assembled the tree prototype from rhombus-shaped blocks through a mathematical game. That is, it was achieved by having 2 rectangular blocks, each with 6 rhombuses are displaced according to a certain regularity, creating a new form. This game also created asymmetry.

In a philosophical solution of the work, I have tried to create a whole from parts. In my opinion, if a tree is a part, then the forest is a whole. If the Earth is a part, then the solar system is a whole. But the solar system is also a part in the infinite universe. So, does completeness exist?

Nguyễn Khánh Linh

The image presents a visually striking and futuristic digital representation of a stochastic system affecting anomalous diffusion. It is structured on a dark, grid-like background that conveys a sense of order and mathematical precision, resembling a computational or scientific simulation environment. Across this structured space, multiple luminous, smooth lines flow in parallel, symbolizing deterministic pathways or trajectories often associated with classical diffusion models.

However, at the center of the image, these pathways encounter a significant disturbance—a complex, spherical structure composed of dispersed particles and glowing elements. This disruption appears to interfere with the orderly flow, causing the previously straight lines to bend, scatter, or redirect unpredictably. The sphere's fragmented appearance, with tiny glowing particles dispersing outward, visually represents randomness, turbulence, or probabilistic behavior, key characteristics of stochastic systems.

In physical and mathematical contexts, anomalous diffusion refers to deviations from classical Brownian motion, often caused by heterogeneities, memory effects, or stochastic fluctuations in the system. The image captures this idea by contrasting the structured, deterministic motion of particles with the unexpected, probabilistic deviation caused by the central disturbance.

The artistic use of glowing white and blue hues against a dark backdrop enhances the futuristic, high-tech aesthetic, reinforcing the notion of complex computational modeling or scientific exploration. The choice of a three-dimensional perspective, with the paths curving dynamically around the disturbance, further emphasizes the interaction between order and randomness in a visually engaging manner.

Gabrielle Llewellin

I am no mathematician but am drawn to conceptual nuanced and ambivalent discourses surrounding notions of space: the notion of Probability theory as based upon the concept of a sample space of mutually exclusive possibilities is what ignited fascination with this set brief. In personal research terms, a lens is placed on the notion of Imaging of the Tacit Dimension: Probability theory for me, inhabits this terrain.

Jean Constant in his paper Random Processes and Visual perception states:

A random process or stochastic process is a collection of random variables defined on an underlying probability space.

Thus, the challenge: to locate equivalence in visual terms of an underlying probability sample space of mutually exclusive possibilities.

I record/document visually what I perceive to be the in-between space/place of Being, imbued as it is with fallacious energies/frequencies/vibrations that challenge perception and apprehension. Thus, the image submitted is put forward as visual equivalent articulating a Probability (sample) Space, mapping random (physical and perceptual/visual) walks in woodland with river, waterfalls and weir to expose consciousness and perception to such

Possibilities.

What was emergent in the sub sequential experimental/random process and methodology of imaging, was additionally a chance visual anomalous diffusion that heralded the other.

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