Contemporary art is profoundly opposed to the search for beauty; it creates its own languages and finds its own incarnations. But we do search for beauty, and we need beautiful objects and literature to lead happy lives.

Artistic pursuits are not alone in leading to the discovery of beautiful objects. Astronomy yields startling discoveries. The Hubble Space Telescope sent back images from nebulae and galaxies that enriched the iconography of the last century. Medical and biomolecular images yield surprising perspectives on organisms and their components. And there are fields of mathematics whose formalisms lead to astonishingly beautiful representations. The craze for fractal art in the early 1990s and the sudden popularity of algorithmic art in the mid-90s were directly related to the appearance of powerful hardware able to compute images of great complexity and depth.

It is fairly easy to generate superficial generalizations from these trends. But if we dare throw light only on the major streams of artistic consciousness, we must live with the consequences of adopting shallow heuristics and be prepared to see them as provisional: given that we are talking about fields of artistic endeavour whose very existence depends on the state of the art in several mathematical fields, artistic trends deriving from visualized equations can only be experimental at best; their strength is often tied to the abilities of artists to make those images part of an overarching visual discourse whose origins rarely lie in mathematical abstractions. The most famous example of such an artist is probably M C Escher. Escher was not a mathematician, even though he appreciated the interest his drawings generated in mathematical circles.

Jovian Skies
There are good examples for this thesis, like the images of the bombardment of Jupiter by the Shoemaker-Levy comet in 1995, which became part of the Netscape browser logo dramatizing the impact the web and its technologies had on the lumbering giants of telecommunications and information technology. Explosions brightening the Jovian skies might not be considered art or even good advertisement, but for a time, not many IT-literate workers escaped the fairly heavy-handed symbolism bombarding Netscape users after the latest Netscape browser installation.

Art and Algorithms
Mathematical and algorithmic art share common sources: a search for and comfort with intricate visual patterns, aesthetic pleasure woven into an appreciation of mathematical structures. It has been said that mathematics does not have an ontology, that it lacks an about-ness. Mathematical and algorithmic art turn mathematical operations on minimal geometric elements into objects of aesthetic appreciation, regardless of size, shape or dimensionality of the element. But like all computational arts and sciences, the operations are not bound to any particular substrate, be it organic or synthetic, biological or silicon-bound. The computational arts are not limited to mathematics as method or subject either - any of the hard sciences and engineering will do just fine. Protein shapes and the DNA helix will do just as well as exploding galaxies or complex knots and multi-dimensional shapes. Again, they become part of aesthetic experience.

Aesthetics and Dodecahedra
We have to separate very clearly aesthetic experience from artistic impulse: we can find a mountain valley beautiful; yet we are able to view the design of a Victorian bottle manufacturing plant as artistically valuable. Both, however, can be represented in an image using a single algorithm, even though the time to compute the image might vary widely. Like the proverbial tail wagging the dog, there are cases where the mathematical object becomes the substrate of experience and the computer-aided design (CAD) tool is just that, namely a tool: imagine a complex shape like this:

It is called a Schwarz D' surface modelled as a rhombic dodecahedron. The precise mathematical model is not really important here, although for the mathematically inclined among us I will say that this is a triply periodic minimal surface modelled using the public domain Evolver software. Now imagine that the dodecahedron above is not the end result of the artistic process, but someone wishes to create a hard real-life object by hand using some pliable metal alloy. It is possible to use 3D printing to achieve a rather impressive-looking result, although the costs of the 3D printer might make this enterprise prohibitive for an amateur.

Mathematics is Beautiful
Digital sculptors who model complex, abstract 3D shapes by hand can achieve very similar effects. They don't necessarily use either computers or equations to produce the first draft of the object, but they are likely to try and attempt to use mathematical formulas and 3D design tools like the GPL'd Blender or the public domain Evolver to refine and extend the model in question. The field between art and mathematics has of course been tilled before: the work of the second century architect Vitruvius, and Leonardo da Vinci's models of flying machines, (as well as his "Vitruvian" man), are outstanding pieces of engineering and anatomy, as well as prime examples of Renaissance art. Pythagorean ideas about the numerical proportionality of man and nature lent themselves to artistic exploration, although today most artists would look to sources very different from 6th century BC Southern Italy where Pythagoras spent his earthly days. But the idea that mathematical relationships underpin the cosmos naturally lent itself to artistic experiment, an idea that did not go unnoticed in the early days of computer art.

There are a number of popular methods to create digital sculptures: we would find that mathematically inspired sculptures are prototyped by hand, recreated in the computer with the help of datasheets and 3D modelling software and then molded or printed with the help of 3D printers. Then again, if scientific experiment or observation is our vantage point, we don't need to generate the datasets, but we can try and find datasets in the scientific world: there are large 3D datasets available from a number of scientific fields whose datapoints can be etched into crystals using lasers.

Mr. Universe Has a Point
Open datasets are a bit hard to come by, though, and indeed it is a major problem for digital sculptors to get new scientific datasets that describe something as complex as a chemical molecule or a galaxy. Astronomical data can be fairly easily procured, although they don't always make for particularly interesting objects: galaxies are fairly hard to distinguish for the untrained eye. But there is the rich and age-old field of astronomical simulation; from planetary systems to large-scale structures in the universe models are modelled with the help of fairly straightforward data. I am talking about static modelling, not videos, of course: the big advantage is that as a form of art, astronomical models can be generated using non-visual data. Speculations about the structure of the universe can only be sustained by fairly esoteric data and a lot of ingenious theoretical exploration. The results can be fascinating.

In any event, algorithmic art takes a very similar approach, although any attempt at similitude with cosmic processes tends to be purely accidental. They do exist, though; great attractors, for example, have been found to explain supra-galactic structures. Fractal art and any 2-D and 3-D graphs using complex analysis are seeded with few straightforward initial conditions the results of which are often aesthetically intriguing. One can create various natural-looking objects like planets and landscapes or even human constructs like houses and cities. The results are varied and make interesting objects of study in their own right. But we have to carefully distinguish equations used in such methods working within 3D space from the algorithms tracing the objects within their virtual electronic environments; their origins have little to do with artistic or mathematical concerns.

3D Space and the People Living in it
The 3D graphics algorithms are completely separate from the complex analysis or the topologies used in these environments; these artistic endeavours usually exhaust themselves in tuning the algorithms producing the picture. Device drivers and 3D packages whose quality is independent of the mathematics at the command of the artist render the actual image. Once several objects are created, for instance in Lindenmayr systems growing artificial flowers within a terrain-describing algorithm, the artist's role has relatively little to do with the algorithms themselves. She is painting an image using a computer. The computer is projecting the image to the screen, while keeping the algorithms away from the artist. In this sense, we are not talking about algorithmic art anymore, but about straightforward computer graphics.

But the digital sculptor working with 3D media does not produce images resident in computer memory alone. Apart from producing models that are being refined using geometry and 3D CAD software, she could add another step and make the end result a real, hard object. Computer graphics residing on the web or in videos and movies obey a slightly different set of rules: the end result has to be made as realistic as possible to suspend disbelief from the side of the audience. Fractal art, terrain generation and other artistic objects are centered around the act of viewing at a cinema, TV or computer monitor. When the result ends up in an art collection or a private home, in other words, if the idea evolving on the computer is embodied in wood, steel, glass, or crystal, we are back to the realm of fine art. It stands to reason that the end result is of course something to be sold - open source art would not make much sense. The datasets and the modelling software are a different matter.

Laser Etching, 3D Printing and Metal Sculpture
Artistic metal objects modelled by CAD or mathematical software can then be cast directly via a 3D metal printing process. This might sound a little like gobbledegook squared with monkeytalk, but casting bronzes and other metal objects has been a stock in trade for many sculptors for centuries. The major difference to traditional techniques lies in a much-shortened pattern production process. Molds don't have to be produced by hand; the metal objects are layered using tiny metal granules in a process that takes days rather than weeks. The process resembles printing insofar as metal granules or rather metal powder is sprayed in 0.04-0.07 mm layers. After some heating and melting and adding of bronze to remove the spaces between the metal granules, the metal object is ready for the artist to produce the texture and add coloration. The metal sculpture can be as complex as a mathematician can imagine - a dodecahedron is not considered a complex in this context.

Laser etching is another fascinating process to produce 3D sculptures without involving a casting process or any kind of manual craftsmanship. Tiny point patterns are created by focused laser-beams. They heat and crystal by using pulsed lasers.

Laser etching itself was invented during the last years of perestroika in the former Soviet Union. Clear crystals whose surfaces are flat, and preferably not curved or spherical, are perfect for this kind of work. The work for these designs is different since the laser-etching process should be done entirely by creating each point according to the image in the artist's mind or the data model used. Starscapes lend themselves quite naturally to this, but geometrical shapes can be turned into rather impressionistic 3D shapes when laser etching is used. The artist has to be at pains to avoid creating too many points in close proximity of each other, since the crystal would be damaged by the heat fractures that make up the etched 3D sculpture.

Creating the point clouds is rather sophisticated work; it needs several software tools to achieve the desired effects in the crystal. It starts off from a raw point cloud drawn by the artist using CAD software. The point cloud must then be conditioned to avoid stresses to the glass and to adjust it to the material that the laser etching process has to use.

Even though scientific images are fairly popular, one could quite easily use 3D layering to produce photographs or painting-like effects. One can only dream what the whirlpool galaxy would look like as a 3D sculpture.

The weird and wonderful world of mathematical art has found many new expressions with the appearance of new technologies and packages making 3D design accessible to artists. Biology, manufacturing and modern astronomy all combined in fascinating new world that seems far removed from the Vitruvian man the Renaissance. The very small and the very large become artistic subjects rather than objects of appreciation. We use digital sculptures to throw new light on natural law.

Surface evolver
Whenever geometry is mentioned in polite society, a certain quietly desperate tone tends to enter the conversation. It is not that people tend to explode in rage at the memory of dry mathematics teachers - it is just that anything forcing the imagination to conform to rules seems difficult to be fond of.

Mathematicians dealing with visual shapes are a fairly strange breed. Although their ancestry goes back to the 3rd century BC Greek mathematician Euclid, one of the greatest synthesizers of mathematics, it is not necessarily axioms and deduction that pre-occupy topologists and or experts in computational geometry. Rather uniquely among most scientists, their research almost entirely centers on their own creations. The bodies they imagine are built according to almost impossible constraints; those constraints are called "energies", although other constraints are possible. It often seems that that such objects cannot exist in nature, but only in people's (mathematical) imagination, but the simpler versions can exist in nature for considerable amounts of time.

The Evolver software models, for instance, surface tension energy of soap bubbles or interfaces between fluids of different surface tensions. Gravitational potential energy can be modeled using point constraints that minimize gravitational effects on particular vertices and edges of the object concerned. The user can also introduce global constraints like volume and pressure to build complex objects.

Why is Evolver important? It is one of very few packages that permit the design of extremely complex geometrical objects with hundreds of surfaces and edges that show realistic changes when a larger number of physical constraints are introduced. It does so using complex geometries and topologies that are not usually implemented in packages like Blender 3D or commercial software.

Evolver was written by the US mathematician Ken Brakke; he made it available as public domain software. There are parallel versions of it and Linux is of course supported. It has its own, very simple scripting language and is fairly addictive to the mathematically minded.

Blender 3D
3D design occurs in almost every walk of life. George Lucas would not be able to translate his visions of planet Tatooine on screen and neither would an architect be able to get work if she didn't do her design on Microstation or a similarly powerful application. 3D design is the natural habitat of game designers and animators.

Blender 3D is unusual among GPLed projects since it has an almost cult-like following; groups of developers and users meet regularly for conferences; the application has been ported to 9 operating systems and it should be one of the very few animation programs running equally happily on Linux, Mac OS X and Windows. Books have been written about it and tutorials are freely available on the web.

It is possible to do three-dimensional fractal generation and terrain modeling; humans and animals that the designer might put together using organic animation tools populate the resulting landscapes. If certain tools are missing or a tool chain needs to be scripted, the Blender APIs are available. Various texture production tools as well as ready-made textures are available with the tools.

Raytrace rendering is possible; a large number of tools like soft shadows and ambient occlusion maps have been made available with the new rendering engine. Even a game engine and a video sequencer are part of newer Blender releases.

The history of Blender is rather more interesting than that of most GPLed applications: it started out as a commercial application and for a while it looked as if it was going to be a major success. Coming out of the Dutch 3D animation company NeoGeo, Blender grew out of a total 3D toolset rewrite led by the software engineer Ton Roosendaal.

The company named NaN ("Not a Number") led Blender development from 1998. The toolset became free, although the services and products based on it were definitely commercial. The company attracted substantial venture capital investment and looked set in early 2000 to replicate the success of similar companies like Trolltech (of Qt toolkit fame) and other gaming toolkit producers. At one stage Blender 3D had more than 250,000 users, more than matching the original vision of bringing 3D modeling to the masses.

NaN, alas, folded in early 2002, but the so-called Blender foundation and some initial seed money made sure that the rights to the source code were secured and some of the original NaN employees continued to contribute. Blender has been going from strength to strength since.

Frank Pohlmann

References
Blender
www.blender.org
www.blender3d.com

The Surface Evolver
www.susqu.edu/facstaff/b/brakke/evolver/evolver.html

3D Geometry and Art
www.bathsheba.com

3D printing
www.dimensionprinting.com
www.3dsystems.com

3D datasets
www.3dcafe.com/asp/meshes.asp