Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1571369.

This model is featured in figure 3.25 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

Dual polychora

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1608466.

These models are featured in figure 4.4 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

Shaded curvature surfaces

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1622906.

This model is featured in figure 5.6 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Note that I digitally modified the photograph of the model shown here to colour the knot.

Trefoil torus knot

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1668608.

These models are featured in figure 6.14 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Klein quartic pairs of pants and hexagons

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:2463217.

Stages of the development of the dragon curve fractal, extruded through space.

Developing dragon curve

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1668620.

This model is featured in figure 6.18 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Figure 8 knot complement triangulation

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:3908970.

These gripping gears mesh with and rotate around each other, but with no axles and no frame. 

This version has "pass-through" holes, that allow a solid object (e.g. a screwdriver) to pass between the gears.

YouTube video: https://youtu.be/RBZG8M8_a8Y

To make these, you will need to print four copies of each of the files. Two of each file are bolted together using standard M3 nuts and bolts to make each gear.

This is joint work with Sabetta Matsumoto and Will Segerman. Thanks to ICERM (https://icerm.brown.edu) for their hospitality during some of my development of this design.

Gripping gears

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1668585.

These models are featured in figures 6.4 to 6.7 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Mystery surface

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1551848.

This model is featured in figure 1.24 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). See also [Solid comma symmetry spheres](http://www.thingiverse.com/thing:184692).

Comma symmetry sphere *44

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:394634.

This model is featured in figure 3.31 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

This sculpture was inspired by a question of [Vi Hart](http://www.youtube.com/user/Vihart). As far as we know, this is the first sculpture (in fact, physical object) with the quaternion group as its symmetry group. The quaternion group {1,i,j,k,-1,-i,-j,-k} is not a subgroup of the symmetries of 3D space, but it is very naturally a subgroup of the symmetries of 4D space. The monkey was designed in a 3D cube, viewed as one of the eight cells of a hypercube. The quaternion group moves the monkey to the other seven cells. Radial projection moves the monkeys onto the 3-sphere, the unit sphere in 4D space, then stereographic projection moves the monkeys to 3D space. The distortion in the sizes of the monkeys comes only from this last step - otherwise all eight monkeys are identical.  For more details, see Vi's and my [paper](http://archive.bridgesmathart.org/2014/bridges2014-143.html), or Evelyn Lamb's [blog post](http://blogs.scientificamerican.com/roots-of-unity/2014/05/19/a-hypercube-of-monkeys-quaternion-group/) at Scientific American.  
This is joint work with [Will Segerman](http://www.willsegerman.com). Also available from [Shapeways](http://shpws.me/t2sb).

Also check out the interactive animated online version at [monkeys.hypernom.com](http://monkeys.hypernom.com). (Use the WASD keys, arrow keys, numbers 1-6.)

More fun than a hypercube of monkeys

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1570138.

This model is featured in figure 3.17 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

5-cell

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1551805.

This model is featured in figure 1.15 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Cube with two kite panels

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1570123.

This model is featured in figure 3.14 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

Tesseract

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1551849.

This model is featured in figure 1.25 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). See also [Solid comma symmetry spheres](http://www.thingiverse.com/thing:184692).

Comma symmetry sphere 8*

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1696383.

This model is featured in figure 7.4 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Developing Hilbert curve

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1571375.

This model is featured in figure 3.33 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

More fun than a 24-cell of monkeys

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:78218.

[Video](http://www.youtube.com/watch?v=QhXjevOY_uk).  

This is intended to work with [Triple gear](http://www.thingiverse.com/thing:66708). The axle threads through the central hole in the gears. Rotating the axle makes the gears turn. The gears sit on the baseplate, which prevents them from rotating around the axle itself. The axle has a 5mm diameter cylindrical hole in the bottom, for fitting onto a motor.  

This is also available at [Shapeways](http://shpws.me/nIjE).  

If you watch the video, you'll see that the motion isn't perfectly smooth. Possibly some improvements could be made to the shape of the baseplate and/or axle and/or triple gear to make it smoother.

Base plate and axle for Triple gear

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1571379.

This model is featured in figure 3.34 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

More fun than a 120-cell of monkeys

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1668598.

This model is featured in figure 6.12 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Hinged flat torus

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1739545.

Four identical parts go together to make a symmetric sliding mechanism. It is a difficult puzzle to put them together, and even more difficult to have them slide together so that they all have the same offset relative to each other.

This worked well printed vertically with no support.

This is joint work with Saul Schleimer.

Available from Shapeways at http://shpws.me/MCng

Tetrahedral racks

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1145123.

This model is featured in figure 3.22 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

This is joint work with Saul Schleimer.

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.

Video: https://www.youtube.com/watch?v=MyUfAs30yZk
Shapeways: http://shpws.me/DqGB

Half of a 120-cell

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1608483.

This model is featured in figure 4.8 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). This is joint work with Geoffrey Irving.

This was printed on an SLS machine. I have no idea if it will work on other kinds of 3D printer. Let me know if you can get it to work!

Hinged geodesic saddle

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:202774.

This model is featured in figure 3.12 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

[Video](http://www.youtube.com/watch?v=VX-0Laeczgk).  
Stereographic projection is a map from the sphere to the plane: Imagine straight lines going from the north pole of the sphere down onto the plane. Each line hits the sphere in one point and the plane in one point. Stereographic projection maps that point on the sphere to the point on the plane.  
If you put a light source at the north pole of this sculpture, the rays of light effectively do stereographic projection! The curves on the sphere cast shadows, mapping them to a straight line grid on the plane.  
Stereographic projection is also available at [Shapeways](http://shpws.me/q41V).  

Update: added a version of the file with a wider base for easier printing.

Stereographic projection

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:184692.

These show a selection of the spherical symmetry groups - the ways in which an object can be symmetrical by rotating about and reflecting through a central point. See the [wikipedia page](http://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups) for more details, or better, the book [The Symmetries of Things](http://books.google.com/books/about/The_Symmetries_of_Things.html?id=EtQCk0TNafsC), by John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss.   

The labelling of the different symmetry types on the rendered image is in [orbifold notation](http://en.wikipedia.org/wiki/Orbifold_notation). All of the polyhedral types are shown (3∗2, 332, ∗332, 432, ∗432, 532, ∗532), as well as representatives of the infinite families of cyclic and dihedral types (88, ∗44, 8x,  8∗, 222, 223, 228, 2∗4, ∗224).

Solid comma symmetry spheres

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1570254.

These models are featured in figure 3.24 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

Dual polyhedra

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1570237.

This model is featured in figure 3.23 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

Half of a 600-cell

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1668611.

This model is featured in figure 6.16 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Figure 8 knot complement

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1622876.

This model is featured in figure 5.4 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

This is joint work with Keenan Crane.

Topology joke

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1551860.

This model is featured in figure 1.28 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). See also [Solid comma symmetry spheres](http://www.thingiverse.com/thing:184692).

Comma symmetry spheres (7 infinite families)

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1551864.

This model is featured in figure 1.29 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). See also [Solid comma symmetry spheres](http://www.thingiverse.com/thing:184692).

Comma symmetry spheres (7 oddities)

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1608596.

This model is featured in figure 4.21 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

This is joint work with Roice Nelson.

{5,3,4} hyperbolic honeycomb

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1622871.

This model is featured in figures 5.1-5.3, 5.10, and 5.12 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Ball joint chain

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1608454.

This model is featured in figure 4.3 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

Curvature surfaces

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1668575.

These models are featured in figure 6.1 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Twisted strip surfaces

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1622912.

This model is featured in figure 5.11 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Symmetric figure 8 knot

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1608450.

This model is featured in figure 4.2 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). This is joint work with Geoffrey Irving.

This was printed on an SLS machine. I have no idea if it will work on other kinds of 3D printer. Let me know if you can get it to work!

{3,7} tiling

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1696417.

This model is featured in figure 7.7 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). It is based on the Stanford Bunny, source: [Stanford University Computer Graphics Laboratory](http://graphics.stanford.edu/data/3Dscanrep/).

"Bunny" Bunny

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1622908.

This model is featured in figure 5.7 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Three torus knots and a torus link

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1668603.

This model is featured in figure 6.13 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Klein quartic

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1622884.

This model is featured in figures 5.5 and 5.6 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Hinged flat square surface

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:66708.

[Video](http://www.youtube.com/watch?v=I9IBQVHFeQs).  

In this unusual mechanism three gears mesh together in pairs, and yet they can turn!  

If you take three ordinary gears and put them together so that each gear meshes with the other two, then none of the gears can turn because neighbouring gears must turn in opposite directions. Triple gear avoids this problem by having the three "gears" arranged like linked rings - the gears then rotate along skew axes, and the opposite direction rule no longer applies (although see also Oskar van Deventer's [Magic Gears](http://shpws.me/lrqz) for another possible solution).  

This is joint work with Saul Schleimer. We were inspired by another of Oskar's designs, his [Knotted Gear](http://shpws.me/lmqP), which consists of two linked rings that gear with each other, and of course we wondered if it would be possible to do three linked rings!  

[Paper](http://arxiv.org/abs/1304.6859) describing the mathematics behind the Triple gear, and how we designed it.  

[Triple gear (solid)](http://www.thingiverse.com/thing:79310) is a non-hollow version, which should make it easier to produce on a home printer.  

[Baseplate and axle for Triple gear](http://www.thingiverse.com/thing:78218) is designed to work with this, and gives a way to drive the gears using a motor.  For axles without the baseplate or motor fitting, see [15cm axle for Triple gear](http://www.thingiverse.com/thing:79768) and [30cm axle for Triple gear](http://www.thingiverse.com/thing:79781).  

Triple gear is also available at [Shapeways](http://shpws.me/mQ6F).  

Things that could be improved upon:   

* Make the motion smoother  
* Less wiggle room for the gears - at the moment they can move quite far out of the 3-fold symmetric position  
* Can the gears be made to rotate at different speeds? 

Triple gear

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1608610.

This model is featured in figure 4.23 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

{3,3,6} hyperbolic honeycomb

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1551869.

This model is featured in figure 1.37 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

"Sphere" Sphere

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1551792.

This model is featured in figure 1.11 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Cube with kite panel

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1608613.

This model is featured in figure 4.23 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

This is joint work with Roice Nelson.

{6,3,3} hyperbolic honeycomb

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1569896.

This model is featured in figure 2.2 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

Dented icosahedron and icosahedron

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1668594.

This model is featured in figure 6.10 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

Seifert surface for the (3,3) torus link

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1608479.

This model is featured in figure 4.6 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). 

Geodesic domes

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:79781.

[Video](http://www.youtube.com/watch?v=QhXjevOY_uk).  

This is an axle for driving the [Triple gear](http://www.thingiverse.com/thing:66708).

30cm axle for Triple gear

Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:79310.

This is a version of [Triple gear](http://www.thingiverse.com/thing:66708) which is not hollow - each gear is solid. This should hopefully make it easier to print on a home printer.

Triple gear (solid)

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