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Software

Rhinoceros
3D CAD modeling software for Windows.  This is my main workplace: in addition to good NURBS modeling capability, it has strong scripting support which allows it to interface easily with other software.  It has a free demo and an invaluable support newsgroup.

Surface Evolver
Freeware for studying and creating minimal surfaces, relaxing knots, and many other things that are over my mathematical head.  Also very useful for conditioning meshes.

Qhull
Mathematical freeware for computing convex hulls and Voronoi networks.  Again, much of what this does goes way over my head; but I can report that the parts which I can understand work well.

Protein Explorer
Nice freeware for 3D viewing of proteins and other large molecules.

Kinemage
Another free protein viewer, which writes files in a useful well-defined format.

Mathematica
Needs no introduction, but I want to mention that it has the ability to write meshes compatible with 3D printing.  Much could be done with this.

Chaoscope
Ineffably gorgeous strange attractors.

Rendering

Maxwell
A renderer that's intended to work by simulating the physical properties of light.  It has settings like a physical camera, and specifies materials by their optical properties, which makes it a more real-world-related experience than Renderman-based tools.  I find it easier to use.  It's also very slow and development proceeds unevenly, so use caution.

Rhinoman
A free Renderman plug-in for Rhinoceros.  Three cheers for Brian!

AIR
A very powerful renderer at a bargain price.  AIR is Renderman-based and has a steep learning curve – too steep for me.

3Delight
A freeware Renderman renderer.  Put it together with Rhinoman, and you're looking at a lot of renderosity for no money.  Again not simple to use.

3D Printing

Prometal, a unit of Ex One
Direct metal 3D printing.  I do most of my sculpture with this process, and the accuracy, freedom and strength of these parts is excellent.  This is mainly an industrial/wholesale company, so not the perfect place to get started with just a few models.

Scicon Technologies
This service bureau also has metal 3D printing, by the 3D Systems SLS process.  Part quality is very good, and this company may be more friendly to short runs and one-offs.

Ionic Models
A good deal for plaster (ZCorp) models, and a great place to get started with RP.

3D Art to Part
A plaster (ZCorp) service bureau.  Generic service, great prices.

3DFactoryUSA (Cad Blu)
A jewelry-oriented service bureau: small high-resolution models on the Viper and Invision HR machines, with the option of precious-metal casting for Invision models.  I've seen variable quality; I think they may use several foundries which are not all equally good.

American Precision Prototyping
Beautiful stereolithography parts, and the pricing is reasonable.

Rapid Prototyping Resources

The Worldwide Rapid Prototyping Directory
The indispensable guide to the industry.  The directory of service bureaus alone is worth the trip.

Computer Sculpture
A sculpture-oriented site with much information about milling and other CAD/CAM sculpture methods.  Also sells many types of equipment, and is a good discount source for Rhinoceros.

The Rapid Prototyping Mailing List
This industry mailing list used to be indispensable, but was killed by the owner's refusal to control spam on it; this was finally solved but too late for most members.  The list still exists but is now very low-traffic.  It does searchable archives.

Sculpture Sources

C.A. Brown
A foundry pouring bronze, aluminum, silver, gold, and doing moldmaking at jewelry to medium sizes.  Steve does great work at great prices, and is very experienced with burning out ZCorp patterns.  He's cast hundreds of pieces for me.

Treasures
Mike Barajas, a full-service jeweler of great resource and technical prowess in Scotts Valley, CA.

M5 Industries
Special effects for feature films, commercials, and anything else you need them for, both CG and real-world.  Industrial prototypes also a specialty.  The Mythbusters guys work here when they're not on TV.

Laser Glass

Laser Crystal Awards
High-end laser etching, featuring the most advanced laser machines in the US.  2D photos or 3D designs, quantities one to thousands, they can do it.

Art & Geometry

ISAMA
Nat Friedman's brainchild, the International Society for the Arts, Mathematics, and Architecture.  Runs an annual conference, and has lots of good math/art links.

Bridges Conference
Another art/math conference.

The Geometry Junkyard
Coming from the math side, more links than you can shake a tetrahedron at.

Mathematical Sciences Research Institute
This is a strangely organized site – find things in it using Google, not its own search engine – but there's an awful lot of math information hidden in it.

Mathpuzzle
If you like math puzzles, this is where they are.

Puzzle World
A deep site about mechanical puzzles: theory, practice and sources.

Pacific Puzzle Works
Puzzle World is no longer receiving updates, and its data will eventually be transferred here.  For now, go there to view Lee Krasnow's world-famous wooden box puzzles.

Artists

George Hart
Excellent mathematical sculptor – the reigning Prince of Polyhedra – and his site has many mansions.

Reinmuth Bronze Studio
An artist for the ages.  It seems petty to mention it, but he's also an extraordinary technician, doing all his own casting.

Brent Collins
Awe-inspiring mathematical woodcarver.

Bjarne Jespersen
His brain is the complement of my brain.

The Pain – When Will It End?
Buy Tim Kreider's books.  There's no better value in genius per dollar.

This is by no means an exhaustive list of mathematical artists, or even of my favorite artists. I hope you won't be offended if you are one and are not listed here.  There are so many noteworthy people in this field that I can't possibly take on the job of listing everyone; I've just put a few here whose sites are exceptionally useful or hard to find.  You'll find many more artist links in the Art & Geometry resources above.