Letter K

Ekta K. Kalra (Entry 1)

Here's a conceptual description of what it might look like:

• Background: The diagram features a dark space setting, with a deep black background sprinkled with distant stars to give it a cosmic feel.

• Light Source: A high-intensity beam of light, possibly from a futuristic spacecraft or cosmic entity, is visibly interacting with diffused particles in space. The light source could be a glowing point or a concentrated beam, shown in vivid white or blue hues, illuminating the surrounding environment.

• Charged Particles: The particles in space, represented as glowing specks or spheres, are charged and scattered throughout the scene. Some of these particles may be moving, their paths guided by the high-intensity light, which creates visible trails or curving lines to indicate motion.

• Electromagnetic Radiation: You can visualize electromagnetic radiation interacting with the charged particles as waves or pulses traveling from the light source, depicted in a ripple or grid-like pattern. The particles respond to these waves, altering their trajectories.

• Sound Interaction: Sound waves, although not visible in space, could be represented by subtle vibrations or oscillations around the particles. These could be shown as waves radiating outward from the diffused particles, affecting their motion or causing slight disturbances, perhaps in the form of smaller particles drifting or vibrating as the sound interacts with them.

Ekta K. Kalra (Entry 2)

When a diffused particle becomes entrapped within a stream of high-intensity, it may be unable to escape a confined space. This phenomenon could be influenced by additional factors, such as the presence of dense sound waves surrounding the light wave or the proximity of intense electromagnetic radiation. In certain cases, this unique interplay of forces can result in the indefinite confinement of diffused particles within an expanse of darkness.

Ekta K. Kalra (Entry 3)

Scenario Description:

A diffused particle orbits the Sun at extremely high speed—approaching the speed of light. At its nth revolution, it collides with another unique diffused particle that approaches perpendicularly.

Drawing Requirements:

    Depict a large-sized Sun at the center.

    A mid-sized diffused particle should follow a clear orbital path around the Sun, maintaining a realistic scale.

    A smaller, high-speed diffused particle should be shown approaching perpendicularly, extremely close to the orbiting particle, just before impact.

    The scene should be left open-ended, allowing the audience to interpret the possible outcome.

    Do not depict an explosion or burst of energy.

Niko Kapa

Video https://drive.google.com/file/d/11XbQQY2dvGngT59JuwC5xEDWAf57YJ68/view?usp=sharing

People that belong in the autistic spectrum can get obsessively fixated with patterned conceptions such as shapes and systems, presenting hypersensitivity to sensory input. Connecting body rhythms and visual art, pulse of human body becomes melody orchestrated through vibrations in digital space. Body rhythms are treated as physical disturbances from within, interacting with external stimuli. The continuous loop of transmogrified patterns is digitally manipulated by interacting with artist’s breath, heartbeat and neural activity, which function as ‘activators’, promoting the production of ‘digital morphogens’, pattern disturbances that generate interacting systems. Considering aniconism in Islamic culture as the predecessor of algorithmic design, the mechanical intersects with the natural creating iterative arrays corresponding to the concept of ‘self-organization’. Converted to electrical signals that interact with Arabic drone music frequencies, geometries expanding and contracting infinitely on a continuum generate an Arabic carpet of interconnected and intersecting elements, evolving and collapsing to themselves. 

An attitude of disobedience to rhythmical pattern, improvising according to a complex set of preestablished rules and conventions, instrumentalist’s modulations (‘maqam’) communicate personal perspective of the predefined. Switching from one ‘maqam’ to another, installation superimposes such sound patterns insinuating the emotional complexity neurodivergent people have to deal with in their daily routine.

Suzanne Kelly (Entry 1)

Title: Jupiter Drops

Jupiter Drops explores the nature of the gas giant, its convection system’s patterns, and my fascination with theories of change and time.  

The swirls and movements caused by the planet’s currents, 'weather' and convection echo the progressions I see when considering spiral neolithic art, which I imagine how might appear if kinetic over time, over space, perhaps through different dimensions.  

This is my first offshoot from my ongoing Celtic and neolithic-inspired work, and is a jump from exploring manmade patterns to considering the related massive, beautiful patterns organically formed by chemical and heat interactions.  I see similarities between the geometry of planetary changes and carved stone swirls.  It to me is something beyond the Fibonacci series or geometric progressions; there is something I feel which I try to capture about time and eternity.

In my exploration through paintings patterns and colours change in different lighting, I hope this piece succeeds in looking differently - perhaps even seeming to move - at times.  

My work looks at the relationships between the geometry of our past, our current mathematical understanding of how our world and universe work, and nature, posing questions and expressing things I have no other vocabulary for.

Suzanne Kelly (Entry 2)

Title: Newgrange study

Newgrange Study explores the relationships between the geometry of our ancestors, our current mathematical understanding of how our world and universe work, and patterns found in nature, posing questions and expressing things I have no other vocabulary for.

What is this work about?  Science, patterns, ancient history, neolithic art, fossils, art, colour fascinate me.  I discovered the timeless, enigmatic beauty of Celtic coin design and became a collector.  I started incorporating Celtic coin symbols in artworks in the early 1990s, and after a visit to the neolithic site Newgrange, it occurred to me to suggest that I explore merging them visually.  Overall, spiral shapes came to the fore in my work – they speak of galaxies, ammonites, storm systems, art of primitive peoples worldwide, growth, time.  Our ancestors carved them into their monuments all over the world throughout time.  The spiral, relationships and interactions between colours, change, movement, landscapes, passing of time… these are what I’m attempting to explore in this work.  

This piece looks quite differently in different lighting; the values of the colours play against each other.  It is hard to capture in a single photograph, as I intended. 

Suzanne Kelly (Entry 3)

Title: Newgrange Spacetime

The painting Newgrange Spacetime considers the spiral form and how it radiates on a galactic level.  The spiral and the Fibonacci progression have left the stone carving and appear as constellations, nebulae, solar systems.  Are these items inches apart from each other on the canvas, or light years and dimensions apart from each other in space?  Are some that seem fainter moving towards the viewer or away from them?  Ideas of rotation, progression, growth, recession, explosion.  Faint spirals, circles and lines either trace where an object has been or where it is going. 

As part of my 40-year series Newgrange, this is a further study musing on geometry, archaeology, symbology and the cosmos.  I have taken the spirals off the stone monuments and compared them to galaxies.  I seek to look for relationships and ideas for which I don’t have a vocabulary other than the medium of paint.  When others are able to see the concepts I am thinking about, dialogue has occurred which raises ideas, and this is one of my goals in this ongoing work.

Nidal Khaddour

This artwork visualizes the mathematical concept of pattern and sequence through the lens of music and architecture.

The piano keys, a central motif, represent a discrete sequence of notes, each a distinct element within a structured order. Similarly, the building facades are composed of repeating geometric patterns, lines, rectangles, and squares, illustrating the principles of tessellation and symmetry. The varying heights and widths of the buildings suggest arithmetic progressions, while the color variations within them hint at combinatorial possibilities.

The overall composition, with its structured blocks and orderly arrangement, reflects the mathematical precision inherent in both music and urban design. The artwork invites viewers to see the underlying mathematical harmony in seemingly disparate elements, emphasizing the universal language of patterns and structures that connect art, music, and mathematics

Kulsoom Kamil Khan

My painting explores anomalous diffusion by capturing the interaction of watercolor with different mediums: salt, baking soda, oil, and lotion. Drops of black watercolor paint were made to run freely over the surface of the canvas interspersed with various substances that created unique and irregular patterns. The paint ran faster in some places, then slower, reacting in unpredictable ways in a heterogeneous environment. The baking soda created granular texture, the salt crystals resulted in fractal-like patterns, oil repelled the water paint creating distinct boundaries, and the lotion dollops depending on their size, either resulted in swirling patterns or acted as blocks for the paint to run around. Biases towards a certain path that the paint took were also noted, with specific areas of the canvas being darker than the rest, sometimes having no paint running through. This interplay of mediums with paint exhibits how different particle systems influence the movement of fluids, highlighting the complexity of anomalous diffusion and its departure from the standard Brownian motion. Thus, the emergent structures in this piece reinforce how environmental factors can disrupt standard diffusion, affecting the spread of substances and resulting in unexpected, unique and beautiful structures of distinct complexity.

Sheyda Kharaghani

Video https://drive.google.com/file/d/1ItPW4yQMDdTxfkBFt1NzZOTJFTUy_Fgu/view?usp=sharing

Title: "N and Anomalous Patterns: A Play of Chance and Symmetry"

This piece is inspired by the concept of anomalous mathematical patterns and is built using a single element—the letter N. At first glance, N is just a typographic symbol, but in this animation, it transforms into a dynamic and self-interacting pattern.

The mathematical ideas behind this work manifest in several ways:

- Symmetry and randomness: The movement and shape variations reflect the interplay between order and disorder, a key concept in the theory of stochastic systems and anomalous diffusion.

- Reflective and iterative processes: The evolving forms of N resemble random paths and interacting networks, similar to mathematical models used to describe the motion of particles in constrained spaces.

- Minimalism and complexity: Despite being constructed from a single letter, the resulting pattern is a multi-layered, evolving system, echoing the complexity found in mathematical and physical processes.

This piece is not just a visual representation of a random pattern; it is an exploration of how a simple element can evolve into a complex and dynamic system.

Masoud Khorasani

This work is 3d dark digital style and shows the physical body in the tesseract space made of cubes with parallel lines, which has order in the midst of disorder.

Ghazal Kian 

My artwork is inspired by the intricate orsi windows of Isfahan, a masterpiece of Persian architecture where mathematics and art unite. These traditional stained-glass windows, designed with precise geometric patterns, embody centuries of mathematical exploration in Islamic art.

In my painting, I recreated one of these windows using symmetry, tessellations, and golden ratio proportions, emphasizing the harmony between structure and aesthetics. Each segment follows mathematical principles, such as repetitive polygons and star patterns, which are fundamental in Islamic geometry. The interplay of colors mimics the way light filters through real orsi windows, casting dynamic reflections that shift with perspective.

Beyond aesthetics, my work highlights the functional brilliance of these designs. The geometric grids are not just decorative; they regulate sunlight, create natural ventilation, and maintain privacy—showcasing how mathematics can enhance both beauty and function in architecture.

By painting this pattern, I aimed to bridge traditional craftsmanship with modern appreciation, demonstrating how mathematics is not merely an abstract concept but a vital force shaping our world. Through this piece, I invite viewers to see numbers not just as calculations, but as a language of symmetry, balance, and timeless artistry.

Ayodeji Kingsley

Title: Spot-aneous Thoughts: Ladybug's Fractal Paradox

Dimensions - 35x 18x 3 inches

Medium- Metal Scrap

Year- 2024

This scrap metal sculpture, “Spot-aneous Thoughts," explores the deliberate disruption of the fractal's predictable self-similarity, visually representing how seemingly stable mathematical patterns can have hidden irregularities and similarities through the lens of a ladybug. The iconic seven spots of lady bugs, often perceived as a convex pentagon, are deliberately distorted, revealing hidden complexities.

The sculpture transforms the pentagon into a concave form, highlighting the inherent variations within seemingly stable patterns by shifting one vertex of pentagon inward. The spots could also be reimagined as a square enclosing a triangle, echoing the fractal-like Sierpiński pattern.

This visual transformation leads to a mathematical exploration. Using the square number equation (Sn = n^2), we derive n=±2. Inputting these values into the pentagonal number equation (Pn = n(3n-1)/2) yields 5 and 7, with the latter corresponding to the ladybug's spot count.

Further, a novel relationship is derived: Nth Pentagonal Number = 2(Nth Square number) - Nth Triangular number as against the widely established relationship : Pn = Sn = Tn-1 which is only valid for n greater or equal to 2. This, alongside the Pythagorean triple (3,4,5), highlights the interconnectedness of these number sequences, mirroring the ladybug's spot deviation.

Diane M Kjellgren (Entry 1)

This AI image effectively illustrates Anomalous Diffusion in five ways through its symbolic structure, pattern, and material composition.

🌪 **1. Spiral Geometry** The central spiral, reminiscent of a nautilus shell (Fibonacci Spiral), symbolizes non-linear pathways. Movement doesn’t follow a straight line or uniform speed. It follows curved, recurring paths, dependent on prior steps. 

🌀 **2. Fractal Complexity** The nested spiral structure with the flower petals and layered shell structures (spiral within a spiral) shows multiscale behavior. This is common in fractal media where movement varies from constrained at small scales to freer in different environments. Particles explore environments of varying resistance or permeability.

🌸 **3. Heterogeneous Medium** The mix of flowers, shells, coral-like structures and minerals showcases a heterogeneous medium, with dense and constricted areas (like the petals and shells) slowing diffusion (subdiffusion) and more open and permeable areas allowing free movement (super diffusion). 

💧 **4. Energy Landscapes** The image suggests energy wells, where particles may get temporarily trapped in local minima before escaping, visually represented by nested and enclosed elements. 

🧬 **5. Natural Transport Systems** The composition evokes real-world diffusion systems, such as cellular transport and nutrient movement in soils or coral reefs, where anomalous diffusion is observed. 

Diane M Kjellgren (Entry 2)

This stunning image of a lush, spiraling garden path winding through a hill represents anomalous diffusion in five ways.

🌿 **1. Spiral Pathways = Non-Linear Trajectories** The garden's winding paths forming spirals and loops represent the behavior of particles in anomalous diffusion. Anomalous diffusion follows non-linear, curved, or recursive paths, requiring a traveler (or particle) to navigate complex routes. 

🧭 **2. Human in the Maze = A Diffusing Particle** The person at the center symbolizes a particle moving through subdiffusive or superdiffusive evironments. The garden’s structure influences the person’s movement, which acts as an environmental constraint.

🌸 **3. Spatial Heterogeneity = Variable Transport Zones** This garden has different plant types, textures, and densities. Dense and thick areas symbolize low-mobility regions (subdiffusion). Open areas represent high mobility paths (superdiffusion). 

🌀 **4. Fractal Geometry = Multiscale Diffusion** The garden’s recursive spiral design and repeating structures hint at multiscale dynamics. In anomalous diffusion, movement behavior varies with scale, reflecting the complexity of porous rock, cells, or ecosystems.

🌄 **5. Natural Landscape = Real-World Diffusion Media** The garden in a forested, hilly area illustrates anomalous diffusion in nature, such as how soil and root systems distribute nutrients and seeds while animals navigate the terrain.

Diane M Kjellgren (Entry 3)

This AI image is a surreal depiction of a nautilus shell-like structure with a highly intricate, lace-like surface, and it can be metaphorically connected to anomalous diffusion in several ways:

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🐚 How This Image Represents It Metaphorically

1. Complex Geometry and Porosity

The intricate, net-like pattern of the shell can symbolize a heterogeneous medium, full of barriers, traps, or voids. Particles could get delayed or rerouted—mirroring subdiffusive behavior.

2. Fractal-Like Structure

Fractal geometries can be observed in the shell’s repeating patterns and fine branching, which are common in systems exhibiting anomalous transport, such as porous rocks, spongy tissues, or even brain matter.

3. Spiral Form (Fibonacci/Logarithmic Spiral)

The spiral suggests nonlinear progression—just as in anomalous diffusion, where movement doesn’t follow a straight or uniform path over time. Spirals also show scale invariance, often observed in anomalous diffusion models.

4. Organic Yet Abstract

The structure looks natural, yet non-standard—much like how anomalous diffusion arises in real-world complex systems (e.g., biological membranes, crowded cell environments) that defy ideal models.

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🧠 Bonus Thought: Nature’s Chaos and Order

The shell’s structure evokes a balance between chaos and order. Anomalous diffusion bridges the gap between randomness and constraint in real systems. 

Kenji Kojima

Video https://drive.google.com/file/d/1CQSDovpKlQJ7sRet8vWbbQbG3XQWyanq/view?usp=sharing

Video https://drive.google.com/file/d/19W3htrvV7Kz9osXyfIcEsSaAZABa_r1E/view?usp=sharing

Video https://drive.google.com/file/d/1rElU74dQCgg5qMsptKX2aW-SESR07c8J/view?usp=sharing

This project encrypts and decrypts the colors of Leonardo da Vinci's portraits. You will surely find beauty in the mixture of coded and complex colors that you will encounter in the process. Artist Kenji Kojima believes that the world around us is full of chaotic information. Our sensory organs act as filters to extract certain elements, such as sight and sound, from this chaos and construct our perception of the world. In this way, we create reality using the "keys" of our sensory organs, like cracking a code. Currently, all media is recorded in binary format. This project uses an encryption technique called a one-time pad to encrypt color information by applying a bitwise XOR between the original color data and a random number. The data is separated into the code and the key. Each color is composed of red, green, and blue information. The chaotic color mosaic is then decoded using the random number corresponding to the key, reconstructing the original image. Kojima believes that the visual information gained from this chaos is intertwined with the auditory information. He converted the encryption keys into 12-tone musical scales. https://kenjikojima.com/DaVinci/

Ekaterina Kovshova

"Celestial riddle" is an abstract artwork, representing a study of anomalous mathematical patterns emerging on the edge of chaos and order. The work is inspired by the idea that the universe creates its own laws of harmony and structures, which manifest themselves in the most unexpected forms.

The fusion of deep blues and golden lines symbolizes the interaction of boundless cosmic depths and light bursting through the darkness. The light swirls and contours resemble natural fractals and mathematical patterns that appear both random and ordered.

The painting was created in the technique of a spontaneous flow of Chinese watercolor and controlled lines, where each element seems to pulsate with life and energy. In this art the idea is that in every disorder you can find a structure, and in every structure you can find a part of disorder.

"Celestial riddle" also resonates with the contest themes, such as:

Non-standard patterns and structures - Unusual patterns and structures, manifested in nature and abstract forms.

Self-organized complexity - A spontaneous emergence of complex systems of simple elements, transmitted through smooth lines and dynamic patterns.

This work invites the viewer to look into an endless dance of form and energy, born from countless intersections of randomness and regularity.

Kashyap Krishna (Entry 1)

Titled “1728/864/?”, this is a 3D representation of the knight’s path on a 5x5 chess board created by randomising the selection of the path, face of the cube, the colour of the yarn and the orientation of the chess board.

Kashyap Krishna (Entry 2)

Cables are all around us. They run along the highway, under our streets, in the walls of our homes, and even extend into the device you might be reading this on. The concept couldn't be simpler - a protective sheathing covers a metallic wire carrying an electric current. They come in different sizes and the cables at a power station are massive and get thinner as they get closer to lighting your bed side lamp. But they are also similar in a peculiar way. The axial cross-section of a high-voltage cable is similar to an electric vehicle charging cable is similar to your USB power cable is similar to ad infinitum. This work titled "The Grid" presents our electric grid as a fractal playing on the natural self-similarity that arises in the construction of cables.

Marek Krzyszton 

The artwork "Astrophysicist" uniquely combines art with mathematics, exploring an abstract vision of a scientist lost in cosmic structures. The cracks on its surface resemble fractals—self-similar patterns often found in mathematical descriptions of nature. The connection between the figure's eye and the glowing point in the background may symbolize the path of research, where mathematics is the key to understanding the universe. At the same time, the composition of the image is based on a harmonious balance of forms, suggesting the use of proportions and geometry—essential elements of mathematics in art.

Santosh Kumar Jha 

1. Mathematical Aspects: These photographs depict anomalous diffusion caused by the interaction of agitated foam, clear water, light refractions and/or reflections. Some of these also depicts super-diffusion, in which chaotic wave motion enhances particle movement, resulting in rapid and uneven mixing; as well as in contrast, also shows sub-diffusion - in which, water clarity and structured light patterns imply limited movement, most likely due to sediment interactions. The boundary between foam and clear water reveals divergent diffusion rates, resulting in an unstable, dynamic transition zone. The irregular spread of ripples and light refraction patterns demonstrate non-uniform transport, which is a critical feature of anomalous diffusion. This natural interaction of movement and resistance successfully depicts how molecular and macroscopic flows differ from standard diffusion models in real-world fluid dynamics. 

2. Aesthetic Aspects: This set of photographs portrays the artistic harmony of water in motion, combining turbulence and peace. The interaction of foamy waves and clean ripples provides a dynamic contrast, while sunlight refracts across the water to make fascinating patterns. The organic fluidity and shimmering reflections convey a feeling of natural rhythm and tranquil beauty.

3. Disclosure: I confirm that subjects were treated with respect and dignity; and so, no humans, wildlife, or the environment were damaged while taking photographs.

Ekaterina Kuzmina

Title: Time 

Is time a mathematical parameter or a philosophical category? Which of the definitions do we use more often in everyday life? I address the viewer with such characteristics as fluidity, openness and duration.