The image shows where prime numbers fall when the integers are arranged by their remainders modulo 37. Each angular direction represents a residue class, and each stone marks a position corresponding to a prime. In this circular layout, arithmetic progressions become visible through gaps, alignments, and the uneven distribution of primes. Each stone is procedurally generated by subdividing a box into Voronoi cells and selectively removing parts. The scene was rendered in TouchDesigner using a custom path tracer, allowing for rich shadows and indirect lighting.

A Bird in Flight is generated entirely by mathematical formulas. The image consists of 20,001 circles whose centers and radii are defined by trigonometric expressions, producing the structure of a bird with outstretched wings. Rather than tracing an existing image, the work was created by designing equations whose collective geometry forms a recognizable figure, illustrating how mathematics alone can generate complex visual art.  

Culturally Japanese temari balls consist of a core wrapped in layers of padding followed by thread, which serves as the base for embroidery.  Each ball in this assemblage of interwoven symmetry orbs uses woven stitching procedures to encode vertex and edge data from spherical Catalan solids. As an artisan and mathematician, I am acutely aware that parameter choices at the outset effect results of the interweaving of the spheres in the middle column in a complex manner. The visual symmetry of completed temari create space for the mind and soul to rest.

This computer simulation shows an ideal way of arranging a limited amount of material so it can best withstand uniform pressure from all sides. In the image, white regions are solid material and black circular areas are empty space. The pattern is based on the mathematical “coated sphere” construction from the theory of optimal materials. Our recent work proved that any best possible design must contain infinitely fine internal structure like here, although the holes do not have to be circular. Created as a digital simulation/rendering.

The Borromean rings, named after the Italian Borromeo family who used them in their coat of arms, were known even earlier as the Viking symbol Valknut or Odin’s Triangle.
The three rings are linked so that no two are directly connected, yet all three together cannot be separated. Removing one ring causes the entire structure to fall apart.
The image was created using the Surfer software; the final formula is shown on the left. It was obtained through the following construction process:

• Start with the formula for a standard torus:
  (x²+y²+z²+a²-b²)²-4a²(x²+y²) = 0.
• Scale one variable to deform the torus into an elliptical shape:
  (cx²+y²+z²+a²-b²)²-4a²(cx²+y²) = 0.
• Create three copies of the resulting formula and cyclically permute the variables in each one (x→y→z→x).

Imagine a torus (a doughnut shape) inflated until its inner surface touches itself, creating a singularity formed by two touching cusps. This is the basic shape of the depicted apple. We can cut it open by adding an inequality, for example z + 4y < 1, which removes all points on one side of an intersecting plane, or rather, ‘eats’ them.
The apple was created using the Surfer software.

Elliptic curves are plane curves defined by equations of the form y² = x³ + ax + b. Their homogeneous counterparts, y²z = x³ + axz² + bz³, describe a family of algebraic surfaces with parameters a and b. Shown here are three such surfaces (red, green, and yellow) with slightly different parameter values, together with a semi-transparent black sphere that adds a touch of mystery.
The image was created using the Surfer software.

A Möbius strip has only one edge. Sewing this edge to the boundary of a disk produces a closed, non-orientable surface, which must intersect itself in three-dimensional space. In 1901, Werner Boy surprised his teacher, David Hilbert, by showing that such a surface can be perfectly smooth. This image shows slices of two Boy surfaces (mirror images), generated from a formula by François Apéry (shown on the left) using the Surfer software.

Find out more about the Boy surface in the user gallery The Real Projective Plane.

This image was grown from an algorithm I designed. It is similar to the classic space-filling H-Tree in quality, but instead of using a recursively-defined fractal tree growth process, new branches appear in the order of the sizes of available spaces to grow. Thus, the largest empty spaces are filled-in first; so the image gradually becomes more uniformly-dense. The choice of when to stop growth is both aesthetic and practical. It was generated with pure vanilla JavaScript and rendered to an html canvas, then saved as an image file. Some shading effects were added to the branches as a post-process. 

This image was grown from an algorithm I designed. As the structure grows, new branches appear in the order of the sizes of available spaces to grow. The largest empty spaces are filled-in first; so the image gradually becomes more uniformly-dense. The generator that determines its complexity is a precisely-defined “branch motif” based on a cosine curve with adjustable parameters. It is chaotic: slight changes in the parameters can make a huge difference in the outcome. The choice of when to stop growth is both aesthetic and practical. It was generated with pure vanilla JavaScript and rendered to an html canvas, then saved as an image file.

This image was created with a custom fractal curve algorithm. There are many adjustable parameters involved. I use a custom genetic algorithm, which allows me to search the space to find self-avoiding curves with visual qualities that fit my aesthetics. There is no randomness: it is based on complex integer math to determine a fractal curve generator, which is then iterated to about 3-6 fractal levels of detail. The code is written in pure vanilla JavaScript, and rendered to an html canvas. Once I have saved the chosen fractal curve, I use an imaging tool to fill-in the spaces carved-out by the curve, and add shading effects.

In ordinary Euclidean space, regular dodecahedra always have the same shape, but in curved hyperbolic 3-space their angles change with their size. Here, the dodecahedra are scaled so that their faces meet at exactly 90°, allowing four of them to fit around every edge and tile space seamlessly — the {5,3,4} honeycomb. Over 52,000 polyhedra fill the scene, with a cube at each vertex echoing Escher’s “Cubic Space Division.” The geometry was generated with custom Python code and rendered in Blender 5.1 (Cycles render engine).

gum box shows an image of the Mandelbox, a 3d fractal system, that does not formally apply the principles of the Mandelbrot set in a mathematical sense, but rather in a geometric way. Put simply, the Mandelbox replaces a circular transformation in the two-dimensional complex plane with a kind of spherical transformation in three dimensions. As in the Mandelbrot set, a constant c is then added. This makes it possible to create shapes such as those shown in gum box.

The image is rendered with the open source software Mandelbulber, an experimental 3d fractal renderer. The fractal is a Mandelbox with negative scale and rotated folding planes.

This digital piece, created using a drawing tablet, stylus, and illustration software, explores the juxtaposition of two geometric principles — periodicity and aperiodicity — within a single image. Because any Hat tiling contains an underlying hexagonal lattice, this symmetry can be used as a compositional device. I developed a flower-like motif from each tile’s geometry, so that the Hat boundaries are inscribed in the flower distribution, while the flowers themselves arrange into a periodic hexagonal field, holding both orders at once. The work asks whether periodicity and aperiodicity are ever truly separate, or two faces of the same pattern.

An analog botanical composition of dried Hydrangea macrophylla, hand-arranged on paper and photographed. A hydrangea flowerhead is dome-like; florets occupy a curved surface without regular tiling. Flattened into a bounded disc, their geometry becomes visible: large, dark florets transition into smaller, paler ones with increasing density, while sepals vary in size, shape and number (3–5). The disc is treated as an organic packing problem, exploring how gradients of scale and colour draw local variation into global order. Seen as a whole, it recalls depictions of hyperbolic geometry and creates cohesion without repetition or a fixed formula.

Domain coloring allows for the visualization of complex functions with color-lookups via a pre-image. The white grid lines correspond to function values with integer real or imaginary part. All other complex numbers are mapped to different colors by their angle to the positive real axis, using four color spectra, one for each quadrant. By placing zeros and poles in a highly symmetric configuration, we deform the coordinate axes shown in black into heart shapes. The image was created digitally by a pixel-wise evaluation of the input function.

Minimal surfaces are surfaces which have the same curvature features as physical soap films. The construction of minimal surfaces with given features is a classical subject of differential geometry. In 1844, E. G. Björling showed that for each sufficiently benign space curve a narrow minimal surface strip can be found which contains the curve. Furthermore, it can even be specified how the strip shall twist around the curve.
The surface shown here is generated with the basic curve as a Helix along which the strip is twisted at constant speed.

This image shows a checkerboard projected onto a sphere. Each square contains a p-ball, a geometric shape that is either convex or concave depending on the value of the parameter p. In each quadrant, a convex p-ball is paired diagonally with a concave p-ball: their interior and exterior areas are exchanged, as are their colors. At the center, in three dimensions, is the familiar Euclidean ball.

The checkerboard was generated with a Python program using Matplotlib and NumPy. The area of each p-ball is computed using the gamma function, while the spherical effect is obtained by adjusting the parameters of a G’MIC filter.

The central triangle, composed of cubes in perspective, is inspired by the Reutersvärd (or Penrose) triangle. It was first drawn on an isometric grid, then inverted with respect to its circumscribed circle using GeoGebra software. The coloring was subsequently applied using GIMP.

Note: Inversion transforms a line segment into a circular arc.

This artwork is a tribute to both Maurits Cornelis Escher and Sir Roger Penrose. The “kite” and “dart” tiles of the Penrose P2 tiling—the first aperiodic tiling using only two tile shapes—have been replaced by two bird shapes that fill the entire plane; this was made possible by the ability to deform the edges of the tiles.
The focus was placed on aesthetics to reveal the beauty of mathematics.
The artwork was created using vector graphics in Illustrator. Each bird has been given a slight relief effect, which I call the “cookie effect.”

The Kolakoski sequence is a self-generating integer sequence consisting solely of 1’s and 2’s, appearing either separately or in consecutive pairs. Each element is predicted by an element appearing earlier in the sequence and each element predicts elements appearing later, shown here as a tree-like structure. Every branch point corresponds to a 2. If a 2 predicts two 2’s, one of these again predicts two 2’s. Hence, there are infinite branches containing only 2’s. Whether this is true for the 1’s is unknown. The colors used to shade the branches indicate the number of 1’s predicted by other 1’s.

Vedic Sphere is the sphericalization of Vedic Cube, a three-dimensional model of digital-root patterns in a 9 × 9 × 9 multiplication table. A digital root is found by repeatedly adding digits until one digit remains, revealing modulo 9 patterns. The two diagonal symmetries of the Vedic square extend in the XY, YZ, and XZ coordinate planes to form six symmetry planes: X=Y, Y=Z, X=Z, X+Y=9, Y+Z=9, and X+Z=9. On the sphere, these planes become intersecting circles and divide the surface into 24 spherical triangles. Built from acrylic mirrors, the model creates an optical illusion of a complete spherical structure through repeated reflection.

If we place identical coins as close together as possible on a table so that they touch but do not overlap, each coin is surrounded by six others. This is a simple example of a circle packing. But what happens if the circles have different sizes? The spiral shown here, by Peter Doyle, is a remarkable circle packing in which each circle is again surrounded by six neighbors, while the circles grow progressively larger toward the outside.

In 1904, the Dutch company Droste created a package for its cocoa featuring an image of a woman carrying a package of cocoa with an image of a woman carrying a package of cocoa with an image of a woman carrying a package of cocoa…
The Droste effect refers to an image that appears within itself. The next smaller version contains an even smaller version, and so on. Theoretically, this could continue indefinitely. This process is called recursion.

The Barth sextic, discovered by Wolf Barth in 1996, has 65 singularities (sharp points) in complex projective three-space; the real surface shown here has 50. It was later proved that no surface of degree six (where the sum of the exponents in any term is at most 6) can have more, making it an unbeatable world record. Whether today’s records for surfaces of higher degree can be surpassed remains an open question.
The image was created using the Surfer software.

This is a color symmetric hyperbolic fish tiling conformally transformed to a square. The production process is quite intricate. Marta generated an Euclidean version of the Fish Pattern in the iOrnament App, and carved soft carving material stamps from that. Form there a handcrafted three color real life print was created photographed and reimported into iOrnament as a symmetric image. Using a copy trace tool it was fed to the symmetry engine again and deformed into a hyperbolic tiling which then was once more conformally brought to a square frame.

This heart is more than a symbol; it is an algebraic surface defined by a polynomial equation (shown on the left).

Any point in space can be described by three coordinates, x, y, and z. The surface consists of all points whose coordinates satisfy the equation. The origin (0,0,0) does not belong to the surface; for example, the point (0,0,1) does.

Its pointed tip is a singularity, where the surface is not smooth.

The image was created using the Surfer software.

The pseudosphere is a surface of revolution with constant negative Gaussian curvature, generated by rotating a tractrix about its asymptote. The Dini surface (named after the Italian mathematician Ulisse Dini) is obtained by continuously twisting the pseudosphere. Although their shapes differ, both belong to the same family of surfaces and share the same constant negative curvature.

The quasiperiodic Girih Kolam is a mathematical knot. However, its black area can also serve as an overlapping unit cell, similar to a Gummelt decagon. The cell structure is made of 50 modified Penrose rhombi (HB tiles), arranged as a Cw3 cartwheel. Seven of the ten HB tiles in the center form a Cw1 cell, and 15 Cw1 cells cover the Cw3 cell. In any HB tiling, half of the elements must be mirror images. In a braided pattern, this half must be flipped to the back. When flipped, the red dots turn green and the green dots turn red. The four white-bordered dots mark a Penrose rhombus that is in an equivalence relation with the black Cw3 cell.

Unstable Structures explores how small changes can transform an ordered system. Inspired by mathematical concepts such as grid deformation, continuous transformations, and dynamic systems, the work shows how local perturbations can propagate through a regular pattern and generate new visual structures. The image was created as a digital artwork using vector graphics and geometric repetition, then progressively deformed to reveal the fragile balance between order, instability, and emergence.

Circle packing can be seen as the art of placing tangent circles on the plane, leaving as little unoccupied space as possible. The so called Apollonian Gasket recursively fills the space between tangent circles. But this method leaves large empty spaces inside the initial circles. The Steiner Chain method puts a chain of tangent circles between two circles. Here, I combined two methods and applies them repeatedly. Moreover, in order add more exploration possibilities, transforms that preserves circles and tangency are applied : circle inversions and mobius transforms.

Doyle Spirals are a particular case of circle packing: each circle is surrounded with 6 tangent circles. When the parameters are well chosen (and this is the hard stuff), this circle packing can tile the plane. Applying to this tiling geometric transformations preserving the tangency property, one can create elegant patterns. This work was initiated while looking at the beautiful and challenging images created by Jos Leys.

Slinky, the metal spring toy that walks down stairs has been my key inspiration. When squeezed it forms a cylinder with an ellips as cross section. The endings remain circular, parallel or anti-parallel. The parts are assembled to make flexible structures. They led to the discovery of several transforming polyhedra. 

A particular decoration of the Spectre monotile reveal intriguing hexagonal cluster patterns on its tilings, which are reminiscent of basalt column. It turns out the pattern naturally lifts to a pleated surface in 4D. I reprojected the surface to 3D with some custom code and then created this artistic render with Blender, the 3D computer graphics software, adding some texture to the surface so as to imitate rock and a character for immersion, both found on BlenderKit under a Royalty free licence. One of the original Spectre tiles has been drawn in white chalk, can you spot it? 

Lissajous figures, named after the 19th century French mathematician Jules Antoine Lissajous, are curves determined by the intersection of two perpendicular oscillating movements. The horizontal (x) and vertical (y) position of each point is calculated with a formula that compares the frequencies of two sine waves. Here the waves are modulated, in other words, the frequency of one sine wave controls the amplitude of the other. In this artistic representation, additional lines are drawn between pairs of vertices located close enough in the space, to form a web like representation of the surface.

This image shows the interior of Adhara, a mathematical kaleidoscope. The cube, one of the five Platonic solids, has 48 symmetries. Three mirrors arranged at precise angles generate these symmetries through repeated reflections, while a fourth mirror creates an infinite repeating pattern.
This image is part of the Visible Mathematics exhibition at the UNAM Institute of Mathematics in Mexico City.

A Möbius strip is the simplest non-orientable surface, with only one side and one boundary curve. The twisted wire shown here is precisely that boundary curve. When dipped into soap solution, the film naturally forms a minimal surface, minimizing its area while spanning the wire. This photograph beautifully illustrates the interplay between topology, geometry, and physics, resulting in a form that is as visually striking as it is mathematically elegant.

This image is based on the Ulam spiral, in which integers are arranged along a square spiral and prime numbers are highlighted. Although primes can seem irregular, they often align along diagonal paths, revealing unexpected patterns in their distribution. Darker regions indicate areas of greater local density, while nearby prime pairs suggest the persistent mysteries of number theory. Created digitally with generative code, the image transforms a mathematical structure into a visual field of pattern and rhythm.

This digital work is based on an iterative construction of a wild knot built from a chain of mirrored spheres. Each sphere reflects the others, producing smaller images of the whole arrangement inside itself. As this process repeats, the form becomes nested and self-similar, suggesting an object with endless detail. Mathematically, the limit is a circle embedded in space that remains infinitely knotted at every point. Created with custom generative code and rendered in POV-Ray to simulate reflections.

This digital image depicts the Dadras–Momeni attractor, a form produced by a three-dimensional system of ordinary differential equations. Starting from many different initial conditions, the simulated paths gradually converge toward the same intricate structure. The image shows how simple deterministic rules can create complex, unpredictable-looking behavior over time. Created with custom generative code, it translates mathematical dynamics into a visual record of motion, order, and instability.

The image shows six iterations of a variation of the Ammann-Beenker tiling substitution rule overlapped with transparency.
Ammann-Beenker tilings are the 4-fold (4D) analog of Penrose tilings. They were studied independently by Ammann, an amateur mathematician, and Beenker, De Bruijn’s student. They can be constructed via a substitution rule consisting in a subdivision of the tiles followed by an inflation, and the projection method by projecting a slice of a higher dimensional periodic lattice onto an irrational subspace (from 4D for the Ammann-Beenker tilings and 5D for Penrose’s)