Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tessellations, fractals, anamorphic art, and more
 

Mathematical Works :: Carlo Séquin

Since high school I have been fascinated by geometry. I enjoyed constructing the more complicated Platonic solids with ruler and compasses, as well as reading about the 4th dimension. While at Bell Labs in Murray Hill, I was introduced to the field of Computer Graphics, and later developed the Berkeley UniGrafix rendering system, so that I could depict objects more complex than I could build. Since then, the focus of my work has been on computer-aided design (CAD) tools – for engineers, architects, and artists. When creating abstract sculptures I see myself as a composer in the realm of pure geometry. The artistic achievement then lies in finding a procedural formulation that can reflect the inherent symmetries and constructive elegance that seems to lie beneath many sculptural master pieces as well as at the foundations of the physical laws of our universe.
Carlo Séquin

 
Recursive Figure-8 Knot

In some depictions of a mathematical knot, some of the meshes formed between the criss-crossing strands resemble the overall outline shape of the whole knot. It is then possible to fit a reduced copy of the knot into every one of these meshes and reconnect the strands so as to obtain again a mathematical knot consisting of a single closed strand. Then this process can be continued recursively resulting in a self-similar pattern. This general process was applied to the 4-crossing Figure-8 knot. But rather than performing this process in a drawing plane as outlined above, subsequent generations of reduced knot instances were placed in planes that are roughly perpendicular to one another, resulting in a truly 3-dimensional sculpture.

Arabic Icosahedron

Moorish patterns found in the Alhambra often depict lattices of interlocking knots. Here such a pattern composed of interlocking trefoil knots has been wrapped around an icosahedron. Each of the 20 faces is replaced with a trefoil knot, which interlocks along the triangle edges with three adjacent trefoils. The exact nature of the linking between adjacent trefoils leaves some freedom to the designer: In the simplest case two adjacent trefoils interlock with just one lobe each. In the "Arabic Icosahedron" they are linked with two lobes each, resulting in a much tighter meshing.

125 tetrahedra in 25 projected 5-cells(I)

The 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine.

125 tetrahedra in 25 projected 5-cells (VIII)

The 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine.

K12 embedded in Genus 6

Plaster model, hand painted, 5" tall, 2004

The complete graph K12 connects 12 vertices with 66 lines. Of course, in the plane this graph cannot be drawn without many crossings. A surface needs to be at least of genus 6 to allow a crossing-free embedding. With this model I have attempted to find the embedding of highest possible symmetry. The model has the 12-fold symmetry of the oriented tetrahedron. The 44 countries bounded by the 66 lines, and colored differently, are all 3-sided. "My professional work in computer graphics and geometric design has also provided a bridge to the world of art. In 1994 I started to collaborate with Brent Collins, a wood sculptor, who has been creating abstract geometrical art since the early 1980s. Our teamwork has resulted in a program called ?Sculpture Generator 1? which allows me to explore many more complex ideas inspired by Collins? work, and to design and execute such geometries with higher precision. Since 1994, I have constructed several computer-aided tools that allow me to explore and expand upon many great inspirations that I have received from several other artists. It also has resulted in many beautiful mathematical models that I have built for my classes at UC Berkeley, often using the latest computer-driven, layered-manufacturing machines. My profession and my hobby interests merge seamlessly when I explore ever new realms of 'Artistic Geometry'."

Birds in the Sky

The surface of a sphere is divided into 24 identical regions with the same symmetries as an oriented octahedron. These tiles have bird-like shapes inspired by the work of M.C. Escher. Half the tiles are yellow and have a relief pattern that clearly identifies them as birds. The other 12 tiles are blue without a special relief pattern; they can thus be seen as either the shadows or profiles of birds, or alternatively as blue sky background.

Knot divided

Can a DIVIDED KNOT be NOT DIVIDED? When carving this sculpture out of a 10x10x12 foot block of hard compacted snow, we started with the simplest possible knot: the overhand knot, also known as the trefoil knot. We then split lengthwise the whole ribbon forming the three big loops. But there is a twist that may lead to surprises: The original knotted strand was actually a triply twisted Moebius band! Thus the question: Does our cut separate the structure into two pieces, or does it form a single, highly knotted twisted strand? Read more about this snow sculpture.

Poincare FishDish

A tiling with regular heptagons does not fit into the Euclidean plane, since 3 times the dihedral angle of the heptagon exceeds 360 degrees. But if we are willing to introduce a progressive scale factor, then the whole hyperbolic plane can be fit into the Poincaré disc. Here is a visualization of a {7,3} tessellation where 3 heptagons join at every vertex, using a tiling motif inspired by the famous Dutch artist M.C. Escher. Each heptagon is cut into 7 identical pizza slices with irregular boundaries in the shape of fish that properly interlock with one another. See more tiling patterns on the Poincare disc.

Lizard Tetrus

24 Lizard tiles, inspired by one of the many planar tilings by M.C. Escher, are mapped around a rounded tetrahedral frame of genus 3. This tiling is a contorted version of the pattern of 24 heptagons displayed on the surface of the marble sculpture "Eight-fold Way" by Helaman Ferguson. That sculpture celebrates Felix Klein's famous "Quartic Curve" which achieves the maximal symmetry of 168 automorphisms possible on a genus-3 surface. Read more about patterns on the Tetrus surface. Thanks to Pushkar Joshi and Allen Lee for their help with mapping Escher tiles onto the tetrus.

Hilbert Cube 512

"Hilbert Cube" is a space-filling recursive curve in 3 dimensions in analogy to the famous Hilbert curve in the plane. Special care has been taken never to place more than 3 coplanar line segments in sequence. At the largest recursion step the geometry has been slightly altered so as to obtain a closed loop. In the proper parallel projection one can see that the 2 halves of this sculpture are connected by only 2 tube segments. This piece of art gives me the association of an abstract, constructivist model of the human brain. See more views of the "Hilbert Cube 512".