Students and faculty take advantage of Union’s 3D printing facilities to create physical models that assist with visualizing mathematical concepts in three dimensions. Here is a gallery of several examples, though more are being created all the time!
The knot shown is one of the many examples encountered in MTH325: Knot Theory. It is called a torus knot because it lives on the surface of a torus (doughnut shape), at least before it is thickened. Torus knots are classified by the number of times they wrap around the torus in the “wide” way and the “narrow” way.

The monkey saddle is a classic example of a surface with a degenerate critical point. Students in MTH115: Calculus III learn how to compute critical points and classify them as local maxima, local minima, and saddle points. (If you’re not sure why it’s called a monkey saddle, think about the tail!)

The pseudosphere is in a sense the exact opposite of the ordinary sphere. While a sphere has constant positive curvature, the pseudosphere has constant negative curvature. This is studied in MTH448: Differential Geometry.

Students learn how to compute the volume of 3D solids in MTH117, like the Steinmetz solid shown above. It is obtained by taking the intersection of two perpendicular cylinders. It’s easiest to compute the volume of 1/8 of the object, as shown.
