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 borromeanring-minimal surface in honeycomb style. Now i have one in a half size too.

This is a graph embedded in 3-dimensional space as a subset of an "octahedral lattice", which is related to the tessellation of space using octahedra and tetrahedra. The graph has a fractal structure, formed by a process of repeated substitution. Each vertex at each step of the construction is degree 4, and is replaced at the next step by 6 vertices arranged in an octahedron, with certain choices of edges connecting them to each other. Each edge is replaced at the next step by 2 parallel edges. We begin the construction with the first step being the edges of an octahedron, and this is the result at the fourth step.

author George Hart

This is a graph embedded in 3-dimensional space as a subset of the cubic lattice. The graph has a fractal structure, analogous to the fractal structure of a step in the construction of a space filling curve, but with greater connectivity. This greater connectivity makes the physical sculpture considerably more robust than the analogous sculpture of a step in the construction of a space filling curve would be. Each vertex at each step of the construction is degree 3, and is replaced at the next step by 8 vertices arranged in a 2 x 2 x 2 cube, with certain choices of edges connecting them to each other. Each edge is replaced at the next step by 4 parallel edges. We begin the construction with the first step being the edges of a cube, and this is the result at the fourth step. The spacing between the vertices varies in order to highlight the fractal structure.

This is a plastic version of a popular stainless steel QR-Code tag that you can find here: http://www.shapeways.com/model/167409/f669f17fc5aefc1e18b8b8d91fbc03aa?li=

Can you imagine a modern person without an iphone, windows phone or a smartphone running on android?.. Many of these smartphones come with already installed decoding-apps, and when they don't one can always download one of hundreds of free apps, which means that literally every second person can decode a QR-tag.

QR-Code tagging is becoming a popular trend in media and everyday life. They are not just a new generation of bar codes. QR is more than that. QRs are nice and sexy, and most of all QR codes look so interesting that you want actually to DECODE them!

All this makes a personal QR-code tag an absolutely cool widget for you or a great gift for your friends. It doesn't just look good, it FUNCTIONS! Encode a message, a link to your website or online portfolio, your additional emergency info, or any any other text in this little tag, or use it as a one-of-a-kind business card!

Moreover: you decide what color you want it. But, please, note that the dark "dots" should be dark enough in comparison to the light-colored "surface".

Please note, that with extensive use the colors will fade out over time and scratches will appear.

Hypotrochoid gear system part 1 of 2. In order for you to combine two colors in the gear system, it is split into two products. You can find part 2 of 2 here:
http://www.shapeways.com/model/151829/hypotrochoid_3_to_5_part_2_of_2.html?gid=ug26333

This is a spherical function 'designed' using Functy (3D graph rendering) and intended to test out full colour sandstone. The object can be described entirely using the following functions.

Radius = (0.01*((1+cos(12*(a+(p*2)))*sin(p))+(1+cos(3*a+p))))

(R, G, B) = ((1+cos(p*2))/2, 0.3, cos(12*(a+(p*2)))>0.9)

The result is surprisingly effective, and although the blades get very thin at the edges, the printer has dealt with them wonderfully. It feels brittle, but not dangerously so. The colours are also much more vivid than I was expecting. Great!

Alas, it has no practical value that I can think of. Pushing mathematical functions into reality feels like an end in itself.

A classic 'chaos' attractor :) Eg. look at http://mathworld.wolfram.com/LorenzAttractor.html PS: 3/3/2013 - have updated the model to make it printable - sorry that the earlier version had problems.

A Konoid is an artistic representation of a mathematical object called Sphericon.

It’s hard to believe that being placed on a slanted flat surface it rolls freely in a straight line.
Well it wobbles a little, but it will not stop.

From Wolfram MathWorld:
A sphericon is the solid formed from a bicone with opening angle of 90 degrees (and therefore with a=r=h) obtained by slicing the solid with a plane containing the rotational axes resulting in a square cross section, then rotating the two pieces by 90 degrees and reconnecting them.

A sphericon has a single continuous face and rolls by wobbling along that face, resulting in straight-line motion. In addition, one sphericon can roll around another.

This implicit surface was generated and a shell was added to the isosurface in MeshLab.

Please check out all of the other implicit surfaces in my shop. implicit surfaces at vertigo polka

The hyperboloid x^2+y^2-z^2=1 as a doubly ruled surface.

A linking stars model based on a dodecahedral symmetry

This spherically twisted projection of a 5-cell pentachoron is ideal as an attractive pendant or desktop trinket. The gracefully curved contours of the tetrahedral cage create a whimsical, dream-like quality that is fascinating and beautiful. This stylish piece is available in gleaming Sterling silver as well as Stainless and plated steel.

This winged model was generated with a single mathematical expression using the implicit surfaces filter in MeshLab.

A nice simple torus with a basic minimal surface component.