The best triply periodic minimal surface ever!

I put a bigger one here.

A pendant for metal printing.  I have trouble believing that these are photos and not renders...something about this object feels very unlikely.

A Klein bottle.
Warning: this Klein bottle does not open beers, it just looks cute. 
The Klein Bottle Opener is here on Shapeways or here on Bathsheba.com.
A bigger bottle is here.

Another projection of the 4-dimensional hypercube, this one close to vertex-centered. I love the shape of its hull: almost a rhombic dodecahedron, but skewed just enough to keep the central vertices from meeting.
The more usual projection is here, other polytopes are here.

There are six regular convex polytopes in 4D, which are analogous to the five Platonic solids in 3D. This is the fifth, the hyperdodecahedron, a remarkably beautiful object brought to my attention by George Hart.

Here it's shown in a Schlegel diagram so you can see all 120 dodecahedral cells, though most are transformed by perspective: in this projection, the only regular dodecahedra are the biggest one on the outside and the tiniest one at the center.

A bigger model is here.

A sculpture. Contrary to popular belief, it is not an algorithmic object.

There are six regular convex polytopes in 4D, which are analogous to the five Platonic solids in 3D. This is the fifth, the hyperdodecahedron, a remarkably beautiful object brought to my attention by George Hart.

Here it's shown in a Schlegel diagram so you can see all 120 dodecahedral cells, though most are transformed by perspective: in this projection, the only regular dodecahedra are the biggest one on the outside and the tiniest one at the center.

A smaller model is here.

A Voronoi network wrapped onto a Möbius toroid.

A delightful surface: the gyroid put through a simple inverse transformation. It's ellipsoidal on the outside, and there is a sphere taken out of the center, which is difficult to see; for otherwise it would be infinitely tiny inside, and therefore unprintable.

Found on an alien beach. Left by what creature?

A bigger model is here.

This Voronoi network has the symmetry of a snub dodecahedron...almost. It is my homage to the wonderful diatoms of the world.

A four-dimensional cube at pendant size.  Is it adorable?  Why yes, it is. Some feel it is a little large to wear as jewelry, so I've uploaded a smaller one here.

More polytopes are here.

The same as this model on bathsheba.com: here it is in some other materials.

Bigger version here.

A classic fractal.

This model does not print perfectly. Shapeways will do their best, which is darn good, but the smallest holes must be cleared by hand, and it takes time.

Enjoy, but please read the comments below before ordering.

The BBC feature The Code, which aired in the summer and fall of 2011, culminated in a treasure hunt. This is the treasure. The original was made in bronze-finish steel and silver; this recoloured model shows the structure. It is a rendering of the five Platonic solids, nested.
An uncoloured version is here.

There are six regular convex polytopes in 4D, which are analogous to the five Platonic solids in 3D. This one is the odd polytope out, the one without a 3D counterpart.

It has 24 octahedral cells, all shown in this Schlegel diagram. Like the pentachoron it's self-dual -- the only self-dual solid in any dimension > 2 that is not a simplex. And if that wasn't enough, it's also the only regular convex polytope in any dimension > 2 that tiles its space and is not a hypercube.

This was one of my first designs, from before I went to art school.
It's good to see it online.

A ribbon in space.
If you were wondering, it has two sides and therefore is not a Mobius strip.

Since this design looks best standing up, it has a mounting post that is 5/32" diameter. I would recommend a block of wood or blob of polymer clay as a base.

For the first time since before technology was invented, this sculpture is available. I've always liked it, so I'm happy to see it again!

I've seen enough to know where this is going...oh, just kidding.  It's a rhombic dodecahedron.  With an exotic symmetry group.  That's all.  Really.

A Merkaba, 9cm from tip to tip. (That's about 3 1/2", for us groundlings.)

There are six regular convex polytopes in 4D, which are analogous to the five Platonic solids in 3D. This is the sixth, the hypericosahedron, with 600 tetrahedral cells.

This was the hardest of this group to make a printable model of. For a Schlegel diagram one would need quite a large size to allow the amount of interior complexity required, and it gets difficult to build as well as expensive, so I used this face-first projection suggested by Henry Cohn. Some of the tetrahedral are collapsed and become planar, but on the plus side the complexity is on the outside where you can see it!

Found on an alien beach. Left by what creature?

A smaller model is here.

The design I'm trying to make is exactly twice as complex as this one. This stage of it seemed like a good place to pause for a breather.

It's a zero, and it's a one! This steel pendant IS geek chic.

There are six regular convex polytopes in 4D, which are analogous to the five Platonic solids in 3D. This is the first, the pentachoron or hyperpyramid, in a vertex-first projection. It has 5 tetrahedral cells, and like the tetrahedron is its own dual.

In every dimension there's one polytope like this: all triangles, self-dual, analogous to the tetrahedron. As a group they're called simplexes, so this is the 4-simplex.

A bigger gyroid.