Exhibit of Mathematical Sculpture
Valencia, Spain, Nov. 2006 through Jan. 2007

Organized by the Universitat Politecnica de Valencia

Organizers: Javier Barrallo and Ricardo Zalaya

Participants: Helaman Ferguson, Bathsheba Grossman, George Hart, Rinus Roelofs, and Carlo Séquin

Sculptures by Carlo H. Séquin

"Hilbert Cube 512" (July 2005)
- stainless steel and bronze, 5" cube.
The motivation behind "Hilbert Cube 512" and similar works lies in the drive to find procedural formulations that extract the inherent symmetries and constructive elegance that lie beneath the best sculptures by highly skilled artists, and which also can be found in many natural artifacts and even in the physical laws of our universe.

"Hilbert Cube 512" in particular emerged from the challenge of taking the famous 2-dimensional Hilbert curve and exploring what can be done with this pattern in 3 dimensions. It has been generated by a recursive procedure that repeatedly places self-similar copies at the eight corners of a cube. There were many challenges in realizing the initial vague concept. Many combinations of splitting, twisting, and assembly of the individual recursive modules had to be tried out to meet all mathematical and aesthetic requirements. This would not have been possible without the help of computer-aided tools. The result resembles a cubist rendering of a brain, split into two distinct lobes that are only loosely connected to one another.

"Hilbert Cube 512" has been realized as a small metal sculpture with a novel rapid prototyping process available from ProMetal, a division of 'The Ex One Company' headquartered in Irwin, PA. In this process a 'green' part is first formed, composed of stainless steel powder and a selectively applied binder. This green part is then sintered, and the binder is drained out and replaced by liquid bronze. This makes it possible to fabricate very complex parts under direct computer control with no need for molds or for machining.

"Cohesion" (Fall 2002)
- bronze, 12" tall
In 1994 I started a collaboration with sculptor Brent Collins, stimulated by his "Hyperbolic Hexagon" that he had recently sculpted in wood. On the phone we discussed the relation of this sculpture to "Scherk's Second Minimal Surface", which is well know in mathematical circles. We contemplated several ideas how the paradigm underlying this sculpture could be generalized and extended. To try out and evaluated the many intriguing possibilities that we came up with, my students developed the computer program "Sculpture Generator I". This is a narrow special-purpose program, optimized to make smooth chains of holes and saddles. The user can manipulate a dozen sliders to specify the topology and geometry of this object, i.e., the order of the saddles used, their number in the chain, the amount of twist and total bending applied, the width and thickness of the surface itself, as well as the detailed shape of the edges formed. With this tool, new virtual shapes are formed in real time as the user moves any of the sliders, and it is thus possible to explore dozens of new ideas in just a few minutes.

"Cohesion" is one special instance emerging from this generator. Mathematically it is simply composed of two 3rd-order "monkey" saddles connected into a circle by their three arms with a 180 degrees of twist between them. All the other parameters were fine-tuned based on aesthetic considerations. The prototype was realized in ABS plastic on a rapid prototyping machine. Steve Reinmuth had found that these ABS maquettes could be used directly as the disposable originals in an investment casting process. A plaster shell is formed around them by repeatedly dipping them into colloidal silica slurry and fused silica stucco. This shell is then heated with great care to about 1600°F. The ABS plastic liquefies and is drained from the plaster shell, whereupon the hollow is refilled with liquid bronze. Reinmuth has cast several of my models in bronze and supplied them with precious patinas.

"Totem 3" (September 2004)
- bronze, 13" tall
Over the years, I gradually expanded some capabilities of "Sculpture Generator I". I added capabilities to wrap the hole-saddle chain around the toroidal loop more than once and to scale and stretch the sculptural forms in various ways. "Totem 3" is basically just a Scherk-Collins toroid with four 3rd-order "monkey" saddles and a total twist of 120 degrees. It makes use of the affine scaling capabilities in the program. By comparing "Totem 3" to "Cohesion" one can see how much the overall look of a sculpture can change as a result of varying just a few of the parameters in the generator. The prototype for the investment casting process was again made on the Fused Deposition Modeling machine. Steve Reinmuth did the bronze casting and applied the patina.

Because of the rapid feedback that a visualization program such as "Sculpture Generator I" provides, the designer can explore a much larger realm of geometrical possibilities. By examining hundreds of different parameter combinations, it becomes evident where the most constraining limitations are in the current program, and where it might be most promising to change the program and extend the range of sculptural shapes that can be generated. These new shapes then give new insights and generate new ideas. Used in this mode, the computer becomes an amplifier of an artists creativity. The virtual design space, unencumbered by physical limitations such as gravity, allows the artist to become a composer in the realm of pure geometry.

"Volution's Evolution" (August 2004)
- three bronzes, 5" cubes
"Volution" refers to shell-like modular sculptural elements. Each piece in this series is a constrained minimal surface embedded in a cube. The three bronze casts all have similar edge-patterns on the faces of a unit cube, consisting of two quarter-circles around opposite corners, with radii equal to half the edge length of the cube. On the inside of that bounding cube, the surfaces exhibit an increasing number of saddles and tunnels, thus evolving the genus of this surface. Each sculptural element on its own displays a remarkable variety of silhouettes, as it is laid down on different edges or stood on three of its protruding tips. The three elements together form a cohesive hyper-sculpture that gains an additional dynamic element from the increasing number of saddles and tunnels in this evolutionary sequence.

The simplest shape, Volution_0, is topologically equivalent to a disk. The twelve quarter-circles on the surface of the cube form a continuous, closed edge that defines the rim of this highly warped disk. Fitting the disk to this contorted edge loop results in a dramatic saddle surface with twisted canyons on either side. The bronze cast uses two subtly different patinas to make the two-sided nature of this object more apparent.
In the next evolutionary step, represented by Volution_1, two central tunnels have been added, lying side by side, and forming a short-cut connection between pairs of ear-shaped flanges with the same surface color. In adding those tunnels, care has been taken to maintain the strict D2 symmetry that is inherent to all three sculptures. Objects belonging to this symmetry group have three mutually perpendicular axes of two-fold rotational symmetry.
Finally, in Volution_5, four more tunnels have been added to the second shape, enhancing the genus of this surface to a value of 5. This means, that if the rim of this surface would be extended and closed into a big spherical dome, the resulting surface would be topologically equivalent to a donut with five holes (or equivalently, a sphere with five handles stuck on). Again, D2 symmetry was maintained while these tunnels were added.

CAD technology was used to define and optimize the shapes of these sculptures. The geometrically significant fundamental domain of each of these symmetrical objects was first described as a simple polyhedral object that implicitly defines the intended surface connectivity and topology. These objects were then subjected to a few subdivision steps and the resulting triangle meshes were smoothed out in Brakke's Surface Evolver into close approximations of minimal surfaces. The optimized meshes were then thickened to a few millimeters by creating offset surfaces on both sides of the original mathematical manifold. These solid shapes were saved as .STL-files and sent to a Stratasys Fused Deposition Modeling machine. The three master patterns, each 5" on a side, were made from ABS plastic with this layered manufacturing technique. These plastic originals were then used in an investment casting process, where they were burned out from a plaster-of-paris shell and replaced with molten bronze. Steve Reinmuth was the artist who provided these bronze casts with their wide variety of beautiful patinas.

"Arabic Icosahedron" (April 2005)
- 3D-print, 5" diameter
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 a sphere - or, more precisely, an icosahedron. A trefoil knot can be constructed so that it is relatively flat and of roughly triangular shape. Thus we can start with a polyhedron made from regular triangular faces and replace its faces with trefoil knots that interlock along the edges shared by two adjacent triangles. In particular, four trefoil knots can be joined in a tetrahedral formation. Eight knots can form an octahedral shape, and twenty knots can make the "Arabic Icosahedron", which I first depicted in virtual form in 1983.

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 can link with two lobes each resulting in a much tighter meshing. The amount of warping associated with each of these linkages, and the nature of the weave - whether it forms a strict "over-under-over-under" pattern or a "2-over - 2-under" weave as in the depicted sculpture, are also an important design variables.

The design was generated with the student-built SLIDE CAD system at U.C. Berkeley. The resulting geometry was captured in a finely triangulated STL-file and sent to a 3D printer made by Zcorporation. In this layered manufacturing process, subsequent layers of plaster powder, each less than 1/100 of an inch thick are selectively infiltrated with an adhesive. In the end, the loose powder particles are brushed away, while the glued-together particles form the desired sculpture.


Carlo H. Séquin is a professor of Computer Science at the University of California, Berkeley. He received his Ph.D. degree in experimental physics from the University of Basel, Switzerland in 1969. From 1970 till 1976 he worked at Bell Telephone Laboratories, Murray Hill, N.J., on the design and investigation of Charge-Coupled Devices for imaging and signal processing applications. At Bell Labs he also got introduced to the world of artistic computer graphics in classes given by Ken Knowlton and Lillian Schwartz.

In 1977 he joined the faculty in the EECS Department at Berkeley. He started out by teaching courses on the design of very large-scale integrated circuits (VLSI), thereby trying to build a bridge between the Computer Science Division  and the Electrical Engineering faculty. In the early 1980's he collaborated with Professor Dave Patterson, to introduce the `RISC' concept (Reduced Instruction Set Computers) to the world of microcomputers,  demonstrating that computers don't have to be complicated to be powerful. During this work on the RISC microprocessor chips, he realized that what can be built depends strongly on the available tools; consequently he became a tool-builder and started to work on computer-aided layout tools for integrated circuits.

In the mid 1980's Séquin started to transfer some of the insights gained while building CAD (computer-aided design) tools for designing IC (integrated circuits) to the areas of mechanical engineering and to architecture. During the construction of the new building for Computer Science at Berkeley,  Séquin  and his students developed an interactive “WalkThrough” program that allowed them to visualize in virtual form the planned building.

Séquin's work in computer graphics and in geometric design have provided a bridge to the world of art. In 1994 he started a collaboration with Brent Collins, a wood sculptor living in Gower, MO, who has been creating abstract geometrical art since the early 1980s. Their teamwork resulted in a program called “Sculpture Generator 1” which allowed them to explore much more complex ideas and to design and execute them with much higher precision. In this collaboration Séquin has found yet another domain where the use of computer-aided tools can be explored and where new frontiers can be opened through the use of such tools.

Since 1997, Séquin has been a regular participant at many annual Art-&-Mathematics conferences, where he explores ever new aspects of 'Artistic Geometry' -- either by distilling out the artistic potential of a mathematical construct, or by discovering the inherent mathematical regularities in pre-existing pieces of abstract art.

In this work I see myself as a composer in the realm of pure geometry. The driving motivation behind much of my work lies in finding a procedural formulation that can reflect the inherent symmetries and constructive elegance that seems to lie beneath many natural artifacts and that seems to be inherent to the physical laws of our universe.”

Page Editor: Carlo H. Séquin