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Mathematicians at Play: 3-D Printing Enters the 4th Dimension

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I was at a math conference last week, and one of the other attendees brought a puzzle.

I am a pretty slow puzzle-solver, so it will be a while before I figure out how to assemble those five pieces to get this.

Three views of the assembled puzzle.

Saul Schleimer, a mathematician at the University of Warwick who was at the conference with me, and Henry Segerman, a mathematician at the University of Melbourne, are the co-creators of the Thirty Cell puzzle. They are both theoretical math researchers who also enjoy using 3-D printing—a technique for manufacturing a three-dimensional object from a computer program—to create mathematical art and visualizations. (In August, Scientific American featured some of Segerman's sculptures in a slide show from the Bridges math-art conference.)

This puzzle is a projection of a four-dimensional shape into our three-dimensional world. To explain how the projection was created, Schleimer brings it down a dimension and starts with a three-dimensional cube. Imagine a cube sitting inside a sphere. Now put yourself at the middle, holding a flashlight. The light projects all the edges and vertices out to the surface of the sphere. "We replace the usual cube that we know and love with a roundy cube on the sphere," says Schleimer. This process is called radial projection.

Left: a cube inside a sphere. Right: radial projection from the center of the cube onto the surface of the sphere.

From there, a process called stereographic projection places the "roundy cube" onto a flat two-dimensional plane. To visualize this, imagine that the sphere has a plane running through its equator. A line connecting the north pole of the sphere to a point of the cube on the sphere's surface then intersects the equatorial plane at one point.

Left: Stereographic projection from the circle to the line. Right: Stereographic projection from the sphere to the plane.

The collection of all those intersection points is the stereographic projection of the cube. (In general, stereographic projection can be defined for many different planes through the sphere, but the equatorial plane is the one used in this case.)

Stereographic projection of the "roundy cube" to the two-dimensional plane.

For the puzzle Segerman and Schleimer created, the whole process goes up a dimension. The accompanying informational sheet describes it succinctly: "When assembled the 30-cell puzzle is a part of the stereographic projection of the radial projection of the 120-cell in four-space to the three-sphere to three-space."

The 120-cell is one of the six convex, regular polytopes in four dimensions. These are the four-dimensional equivalents of the regular polygons (such as the equilateral triangle and square) and Platonic solids, a class of three-dimensional figures. Schleimer says that dimension four is ideal for interesting regular polytopes because it has enough examples to be interesting, but not too many. In two dimensions, there are infinitely many regular polygons, and in five and higher dimensions, there are only three different kinds of regular polytopes. There are five three-dimensional Platonic solids, and dimension four has six regular polytopes. The 120-cell is one of those six. It is made of 120 dodecahedral cells, with four meeting at each vertex.

Schleimer says that he and Segerman started working on this model when they were studying a topological object called the Hopf fibration. The 120-cell arose naturally in their work, and they decided that they wanted to try to visualize it as an actual three-dimensional object, not just a computer representation or theoretical object in their minds. After creating the first printed model on a 3D printer, Schleimer and Segerman discovered new aspects of the shape. "In a way, producing the 3-D model helped us find some cool stuff that we didn't realize was there. We understood it somewhat, and we produced this awesome toy, and we've learned new math," says Schleimer.

Segerman's and Schleimer's first experiments with the 120-cell led to a few different 3-D models. Creating a projection of the whole 120-cell was prohibitively expensive, so they have been experimenting with different subsets of the object. "We're having a good time finding chunks to stereographically project," says Schleimer. One of the models was this set of three rings, which he calls a "fidget."

It's not a puzzle, but it's fun to play with, and there are several interesting configurations for it. Segerman demonstrates them and explains a little more in this video.

As Segerman says, the fidget led him and Schleimer to develop the puzzle version of the 120-cell, which I now get to play with.

3-D printing is not new, but it is getting more popular, and the number of media—metals, plastics, sugar, and so on—that can be used as "ink" is growing. It is being used for creating engineering prototypes, model skulls for paleontology research and even artificial blood vessels. Segerman wrote an article for the Mathematical Intelligencer about using 3-D printing in math (final version here, free preprint here).

Segerman and Schleimer use the company Shapeways to print their models. They use programs such as Python, Adobe Illustrator and Rhino to create files of an object that they send to Shapeways to translate into very precise 3-D models. Shapeways uses the computer files to program a laser to fuse powders into the shape of a 3-D object. It can even print objects with multiple interlinked components, such as the the fidget above. Another popular type of 3D printer, MakerBot, melts new layers of a material over previously deposited ones, so the models must be supported during the entire process. Shapeways doesn't have that constraint, but its printers are more expensive. The company lets people upload their models and then ships the printed material out to them, rather than having users own printers themselves.

The puzzle Schleimer sold me is made of tough but slightly flexible nylon. He and Segerman also printed a larger one in bronze for a friend, but unfortunately no one has been able to put it together yet.

"The math is fine, but the physics isn't fine," says Schleimer. The plastic models have an almost imperceptible amount of flex in them, but the bronze is too rigid. Segerman and Schleimer have made the puzzle available on the Shapeways website. (The website is experiencing Hurricane Sandy-related problems, so you'll have to bookmark the link for later. UPDATE Nov. 2: The website is up again.) There, you can order one if you'd like to enter an extra dimension yourself.

Images and video courtesy of Saul Schleimer and Henry Segerman.

The views expressed are those of the author and are not necessarily those of Scientific American.

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