With math-based art, creativity is still an important part of the equation. Of course no one can become a Bathsheba Grossman or Paul Nylander overnight, but if you're interested in learning some of the techniques that make math-based art possible (and don't have $1000 to spend on a copy of Mathematica) then Shapeways has the tool for you.
Voxels are exciting building blocks for 3D printing since they are more flexible than commonly used polygons or NURBs curves. In polygon modeling holes are difficult to generate with triangular meshes, but since voxels have depth, holes and other complex surfaces are less of a problem – making voxels perfect for complex surfaces.
In mathematical art, voxels are used quite frequently since intricate, intertwining surfaces are fairly common. Voxels are also great at generating very fine details and ornate patterns, but unless you have a background in advanced mathematics, knowing what formulas to use can be a daunting task.
ShapeJS solves the problem by providing a number of preset algorithms which can be modified on the fly. Examples include gyroid shapes, objects with reflective symmetry, and boolean derived surfaces. Clicking on an example file's thumbnail will open up the web-based interface which consists of lines of code on the left and the derived voxel shape on the right. Users can tinker with the code by modifying the variables in the program and then hit "run script" to process the geometry. The resulting model can then be rotated In 3D space for approval. If the results are satisfactory, the final product can be uploaded to Shapeways and 3D printed in a variety of materials.