Carlo H. Séquin
To be published in Educator's TECH Exchange, January 1996
Powerful computers together with sophisticated programs for geometric modeling and graphical rendering allow to create interesting and artistic displays in novel ways. Computers can play a role in the definition of new shapes, e.g., through the use of abstract geometrical generator functions, as well as in the realization of these shapes, e.g., by drawing exact blueprints or by milling parts. On the other hand, "sculptures" described in the computer need not be physically realized but can be enjoyed in their virtual form on a computer display screen or in an immersive virtual world. Emerging technologies will make it possible to touch these sculptures and to interact with them in cyberspace.
For thousands of years people have enhanced their environments with artistic artifacts, decorating objects for everyday use with patterns or drawings, or creating figurines or paintings to serve ritualistic purposes. The inspiration for pleasing shapes that give the beholder visual and/or tactile pleasure may often be drawn from one's environment. Shapes found in nature are stylized, or enhanced, and are placed in interesting constellations. The abstraction of natural shapes can be pushed to a degree that results in purely geometric patterns which may carry some aesthetic appeal without any direct reference to observable shapes in nature -- as in the line and dot patterns on Indian vases, or in the geometrical patterns in Moorish window grills. These patterns derive their visual appeal from their underlying symmetries and from an interplay of repetitiveness and variation; they often imply several interesting mathematical laws. The growth of our mathematical knowledge during the last few centuries permits us to create geometrical patterns of greater sophistication and in more than just two dimensions. The introduction of computers allows us to make these patterns more exact or far more rich than would be practical with traditional means. And finally, in the last few years, the computer graphics capabilities have gained enough power, so that high-quality renderings of such artistic artifacts can be produced in a fraction of a second. This makes it possible to generate dynamic displays that can be explored in interactive ways, thus creating projections of sculptures in virtual worlds. Unceasingly evolving technology will offer haptic feedback devices that will add a sense of touch to those environments. With that addition, virtual sculptures may then appear as real as the stone sculptures carved several thousand years ago.
The process of creating a piece of art has in principle two phases: an inspirational or design phase in which the shape of a sculpture is conceived, and an implementation phase in which that shape is realized in a particular material. Traditional sculpting often seamlessly blends these two phases; artists draw inspiration from the initial shape of a bone, root, or stone, and then try to bring out the form that they perceive contained within. In the processes of forming, sculpting, or polishing, the artist feels the material and gets stimulation for the further progress of the shaping process. As of today, when computers are involved, the two phases are much more clearly separated. A shape has to be defined and somehow represented in the computer, and then a method has to be found to realize this shape and to make it accessible to a potential audience.
One common way to represent the shape of a sculpture in computer-based form is to cover its surface with hundreds or thousands of points or "vertices" which in turn are interconnected by edges, thereby forming -- for instance -- a fine mesh of triangles. Each vertex has to be represented in the computer with three numbers, i.e., the x-, y-, and z- coordinates that define its position in space, and with an identifier that makes it possible to refer to that particular point. Each edge is then defined by reference to the two vertices that it connects; and a triangular facet can be described by reference to three vertices.
One way to make the described shape visible is to send the above data to a "rendering" program. Such a program assumes a certain camera position in space, and from there projects all visible triangular facets onto the computer display screen, properly taking into account the local intensity variations of the illumination, and possibly calculating shadows and/or reflections on shiny surfaces. The resulting visual effect can be quite realistic if enough computational effort is expended. Alternatively, the computer model can be used to produce exact blue prints or to guide numerically controlled machine tools to craft a physical artifact that realizes the designed model. The computer can play a role in both phases: in defining a shape as well as in realizing a conceived shape for enjoyment by others.
Typing thousands of vertex coordinates into a computer is a very tedious and error-prone process. It is more desirable to start with a suitable initial shape and make incremental changes with a graphical editing program. The starting shapes could be simple geometrical elements such as spheres, cubes, or cylinders, or they could be drawn from a library of previously modeled parts as are now being sold on CD ROMs by various companies.
Interesting shapes can also be captured from existing physical models with machinery costing several ten thousands of dollars. Typically a moving swath of laser light illuminates the model, while one or more light detectors catch the reflected light and determine the 3D positions of the targeted points in space. Sophisticated software then interconnects these data points into a mesh and simplifies the shape description into a geometrical model of acceptable complexity; this can then serve as a starting point for further virtual sculpting.
To many artists, the act of creating a sculpture may be as important as the pleasure of regarding the final piece. There is joy in working with the medium: chiseling away on a marble block, bending a band of hot metal, sanding down a piece of wood, or grinding patterns into a metal surface.
Computers can give similar pleasures related to the power of interactive creation. While one cannot yet feel the emerging forms, one can nevertheless change their shapes interactively. With suitable interactive editing software, a surface can be modified locally by moving vertices with some pointing device, such as a mouse or a 3D wand. Simulated knives or drills can remove parts of the current shape; but "material" can just as easily be added with other virtual operations. We can glue on new features, or blend or fuse different shapes together. Alternatively, the whole embedding space of the sculpture as a whole can be stretched, bent, or twisted [{1} Sederberg, pp151-160]. Thus the computer can become a medium, akin to some kind of virtual clay of infinite flexibility, in which arbitrary shapes can be defined -- with no mess, and where mistakes can readily be undone ...
The use of such editing software is at least two-fold. First, the computer can be used as a virtual prototyping tool. I recently collaborated with Brent Collins, an artist who creates wood sculptures of smoothly blended assemblies of saddle surfaces [{2} Francis and Collins, pp 313-320]. Over the phone we discussed one of his recent creations, the Hexagonal Ring of Saddles (Figure 1), and from our discussion emerged a possibility of adding an extra saddle into the ring, thereby giving the latter an extra 90 degree twist and making the whole surface single-sided like a Moebius band. It took Brent several days to create a rough mock-up from sections of plastic piping from which he could judge whether the idea had artistic merits and was worth pursuing. In a suitable computer graphics environment, this construct and several other related possibilities could be explored in less than an hour.
FIGURE 1: Hexagonal Ring of Saddles, wood sculpture by Brent Collins.
Initial shape definition or subsequent shape modification can also be done in a more abstract way: by algorithmic computer programs. Here the connection with Mathematics becomes obvious. The "artist," rather than affecting the form directly, specifies procedures, rules, or constraints, based on which a shape -- or a whole family of shapes -- are then generated. This approach can be particularly powerful, since tens of thousands of points can easily be calculated in a very short time, and the generated shape need not correspond to any physically realizable object; disconnected clouds of shapes or self-intersecting surfaces, e.g., Klein bottles, can just as easily be generated (Figure 2).
FIGURE 2: Skeleton of Klein Bottle, computer generated virtual sculpture by C.H. Sequin.
The algorithmic generator approach has other interesting possibilities. The defined shape can be the result of an optimization process that could not be carried out without a computer, i.e., the final shape could be entirely the result of some mathematical concept (Figure 3). In principle such shapes are defined with infinite precision, and it is only a question of the computational effort one is willing to expend to view these shapes in any desired detail. Moreover, once a generic program is written, many different shapes of a similar nature can be generated by just re-running the program with different parameter settings. As an example in two dimensions, think of the infinite variety of colorful fractal patterns derived from the famous Mandelbrot set [{3} Peitgen and Richter, pp 80-86] by just choosing a different zoom-in window in the display plane.
FIGURE 3: Minimum Variation Surface of Genus 3, computer optimized shape by C.H. Sequin.
After a computer model of a sculpture has been defined, we still have the problem of getting the shape out of computer memory to make it somehow visible to possible observers. This can happen in many different ways. For constructivist sculptures, the computer could print out the blueprints for sheet metal parts, or the exact cutting angles and locations for tubular sections, from which the envisioned shape can then be assembled.
Other sculptors prefer to use subtractive processes. The actual sculpture itself, or a mold for a cast (Figure 4), could be cut from metal, wood, or plastic by numerically controlled milling machines.
FIGURE 4: Umbilic Torus NC by H. Ferguson, cast from a computer controlled, machined mold. [{4} Ferguson, p6]
However, this approach has its limitations; Helaman Ferguson likes to carve stone sculptures of substantial sizes which represent precise mathematical surfaces [{4} Ferguson, whole book]. He uses a drill suspended by six cables under tension, the length of which is monitored by computer and from which the coordinates of the tool tip position can be calculated [{5} Ferguson, pp109-116]. The computer also calculates the distance from the tool tip to the desired mathematical surface and displays this value on a readout on the drill (Figure 5); this makes it possible to start the sculpting process by carving several hundred reference holes to a precise depth.
FIGURE 5: Ferguson's setup for computer assisted sculpting.
Finally new "solid free-form fabrication" technologies are emerging, such as stereo-lithography or selective laser sintering, in which arbitrary 3-dimensional shapes are built-up layer by layer. In each thin layer, a computer-controlled laser beam polymerizes some liquid plastic resin or fuses some ceramic powder, thereby adding another contour to the emerging shape. Today these processes are still slow, expensive, and very limited in overall size.
Given all the difficulties with physical realization, are there other ways for us to enjoy our possibly quite elaborate computer-generated shapes ? We can start by rendering them on a display screen. Using suitable illumination, shading, and shadow models, we can produce displays that looks like photographs of a "real" sculpture. But our computer description is much more than just an "image" of a sculpture; it is a full 3-dimensional model. With enough computer power, we can manipulate this model interactively. We can turn it around faster than we could walk around a large 3D sculpture, and we can look at it from angles from which we might never see the real sculpture. We can also view the sculpture under different lighting conditions, we might change the material from which the sculpture is "constructed," or we might see dynamic reflections of passing clouds in its surface.
Moreover, these virtual sculptures can be dynamic; they can vary as a function of time or in response to user actions such as touching hot buttons or pushing on parts of the sculpture. Figure 6 shows a mechanical assembly that can unfold in one smooth motion from a closed octahedral shape to an open cuboctahedron (truncated cube). For this piece, all angles and dimensions have been worked out properly, so that such a mechanism could actually be constructed. But in our virtual machine we did not have to worry about the technical challenges of meeting any strength criteria and of providing a dynamic drive for the whole contraption.
FIGURE 6: Octabug, a realizable mechanical design that changes shape between a closed octahedron and an open cuboctahedron.
Moreover, virtual sculptures in the computer need not be bound by our normal physical laws. Parts can float in space, they can change their shapes and even their volumes, and individual pieces could readily penetrate one another. Figure 7 shows a "magnetically levitated" virtual sculpture in which the eight triangular plates pass through each other and through the central point periodically, so that the whole sculpture turns inside-out.
FIGURE 7: A non-physical geometric constellation of an octahedral jitterbug.
The above dynamic sculptures could simply move in a periodic manner as a function of time. Alternatively, the user could control the shape transformation with an interactive slider. Similarly, the observer could control the morphing operation that changes one object into another one or one face into another as seen in the form of eye-catching graphics in some recent TV ads. More sliders could control more complicated articulated assemblies such as a whole dancing robot with many different joints.
The technology for immersive virtual reality environments is under rapid development. Better and less expensive head-mounted displays offering stereo images and "quadraphonic," position-dependent sound, as well as improved input devices and position sensors are emerging every year. Once the problems with resolution, speed, and tracking lag have been overcome, the observers will truly have a sense of being immersed in a synthetically created space.
Sensations will not just be limited to video and audio channels; gloves with pressure transducers will make you feel objects as you "touch" them. Excursions to a virtual museum can then become a deceptively real experience. The visitors could touch the virtual 3D sculptures and slide their hands along their surfaces; they might even be able to produce a realistic "ringing" when they strike a metal sculpture.
The most fascinating aspect is that such a virtual environment is not restricted to simulating reality. Invoked reactions could be exaggerated -- a small push could result in an unexpectedly large effect, or the reaction could be totally unreal. A virtual sculpture could interact with the observer; it could respond to touch or to the human voice, responding with a change in shape or color, or by producing interesting sounds, music, or voice output. Sculptures could display "feelings;" they could be given individual "personalities," thus reacting in varied ways to approaching observers. The possibilities are boundless and mind-boggling.
The combination of art, math, computers, and of other emerging electronic media will offer new possibilities for providing intriguing shapes, sequences of motion, or interactive behavior. These geometrically conceived artifacts may or may not satisfy some yearning for beauty, but, at the very least, they will stimulate our curiosity and cater to our general appetite for novelty.
Technology still has a few years to go before we will see flawless and convincing virtual environments at affordable prices. The editing and sculpting tools outlined earlier do not yet exist in the public domain, and significant development is required to develop easy-to-use and intuitive interfaces that will make it as natural to play with these virtual shapes as it is to knead a lump of clay. The problem of creating sculptures in such virtual environments and the potential for new art forms is still poorly understood. We have just started to scratch the surface of a whole new artistic universe.
Page Editor: Carlo H. Séquin