The usual version of a Möbius strip has as its single boundary curve an unknotted loop. An unknotted loop can be deformed into a circle, with the strip deformed along with it.

In this version, the boundary of the strip is the circle in the middle, and the surface "goes through infinity", meaning that the grid pattern should extend outwards all the way. To save on costs, I've removed the grid lines that would require an infinite amount of plastic to print.

Note: this is an updated version of the model from the one shown in the YouTube video.

This was designed with Saul Schleimer.

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.

Pipes of constant radius around the curves.

Self referential cube.

The 120-cell is one of the six regular 4-dimensional polytopes. This is the result of taking the edges of the polytope, radially projecting them onto the unit 3-sphere in 4-dimensional space, then stereographically projecting the result into our 3-dimensional space. We remove the half of the design in the hemi-3-sphere nearest to the projection point, which makes printing the image in 3-dimensional space feasible.

This design is based on an idea originally due to Geoffrey Irving (http://naml.us/~irving/).

This shows (a somewhat idealised version of) the path of a juggling club as it is thrown from one hand to the other. See http://shpws.me/a4k3 and http://shpws.me/a4kw for larger, hollow versions, the first suitable for printing in the "Strong and Flexible" materials, and the second for stainless steel.

This is made by gluing two copies of the Round Möbius Strip along their boundaries. A Klein bottle in 3-dimensional space has to intersect itself, and in this case it intersects itself along a straight line.

Note: this is an updated version of the model from the one shown in the YouTube video.

This was designed with Saul Schleimer.

This steampunk style knotted cog was procedurally generated using 3-dimensional spherical geometry, then stereographically projected into our (mostly) Euclidean universe.

Other sizes:
www.shapeways.com/model/231026/knotted_cog__large_.html
www.shapeways.com/model/231045/knotted_cog.html
www.shapeways.com/model/277265/knotted_cog__smaller_.html
www.shapeways.com/model/232385/knotted_cog__small_.html

This is a baseplate and axle for the Triple gear (full size version). The axle has a 5mm diameter hole in the bottom, for fitting onto a motor.

A version of http://www.segerman.org/autologlyphs/dodecahedron_photo.jpg
projected onto a sphere and embossed.

This is made by gluing two copies of the Round Möbius Strip along their boundaries. A Klein bottle in 3-dimensional space has to intersect itself, and in this case it intersects itself along a straight line. This was designed with the assistance of Saul Schleimer.

This model is both the Half of a 120-cell and Half of a 600-cell together in the same space, showing how they are dual to each other.

This is a collection of common example surfaces from classes in multivariable calculus. Much larger versions of these surfaces are available here.

Contour lines, together with 8 radial curves make up the surfaces. All surfaces are plotted in such a way to show values of z in [-2,2]. The hyperbolic paraboloid is further cut along a cylinder of radius sqrt(2). The equations of the surfaces are:

• Elliptical cone: z = +- sqrt(2x^2 + y^2)
• Hyperboloid of one sheet: z = +- sqrt(x^2 + y^2 - 1)
• Hyperboloid of two sheets: z = +- sqrt(x^2 + y^2 + 1)
• Circular paraboloid: z = x^2 + y^2
• Elliptical paraboloid: z = 2x^2 + y^2
• Hyperbolic paraboloid: z = x^2 - y^2
• Sphere: z = +- sqrt(1 - x^2 - y^2)
• Ellipsoid: z = +- sqrt(1 - (x^2)/2 - y^2)

This steampunk style knotted cog was procedurally generated using 3-dimensional spherical geometry, then stereographically projected into our (mostly) Euclidean universe.

Other sizes:
www.shapeways.com/model/231026/knotted_cog__large_.html
www.shapeways.com/model/231045/knotted_cog.html
www.shapeways.com/model/277265/knotted_cog__smaller_.html
www.shapeways.com/model/232385/knotted_cog__small_.html

This was generated by Mathematica using the following code:

f[u_, v_] := {u, v, u^2 - v^2}; scale = 40; radius = 0.75; numPoints = 24; gridSteps = 10; curvesU = Table[scale*f[u, i], {i, -1, 1, 2/gridSteps}]; curvesV = Table[scale*f[j, v], {j, -1, 1, 2/gridSteps}]; tubesU = ParametricPlot3D[curvesU, {u, -1, 1}, PlotStyle -> Tube[radius, PlotPoints -> numPoints], PlotRange -> All]; tubesV = ParametricPlot3D[curvesV, {v, -1, 1}, PlotStyle -> Tube[radius, PlotPoints -> numPoints], PlotRange -> All]; corners = Graphics3D[Table[Sphere[scale f[i, j], radius], {i, -1, 1, 2}, {j, -1, 1, 2}], PlotPoints -> numPoints]; output = Show[tubesU, tubesV, corners] Export["MathematicaParametricSurface.stl", output]

This is the trefoil knot, printed out in a shape that will smoothly roll across the table!