Paper Sphericon

I volunteered at an event for SUMM the other week where students were putting together different three-dimensional shapes (polyhedra, cylinders, cones, etc) from paper nets. Seeing kids building cones reminded me of sphericons. I hadn't designed a sphericon net before and thought it might make for a fun, slightly-more-advanced build.

Setup

For this activity, you'll want a sphericon net. For students with more experience in geometry, part of the fun can be designing their own nets. Otherwise, here is a printable net you can use to assemble the shape. I printed mine on cardstock to give it a little more structural rigidity.

If designing nets, students will want paper and pencils for calculation, a calculator (for computing a trig value), a ruler, and a protractor. Another option is to create the net in a tool like Desmos, which is what I did. (I color-coded the curves in the linked version to make it easier to interpret.) I exported from Desmos as vector file image (*.eps) so I had something that scaled nicely for making my pdf.

In any case, students will want some tape and/or glue to assemble the net.

Challenges

• If designing their own net, I would give students a model of the object to study. The only information they should need to get going is that one way to think about building a sphericon is:

  1. Build two identical cones. (What dimensions make sense?)

  

  2. Attach the two cones at their bases to make a bicone.

  

  3. Cut the bicone in half, from vertex to vertex.

  

  4. Rotate one half \(90^\circ\) relative to the other and reattach.

  (Since the cross-sections must line up before and after the rotation, this should be enough of a clue about what the height of the cone needs to be relative to its base diameter.)

• Assembling my net looks like this:



You can build it tabs-out (easier) or tabs-in (as pictured above). I haven't tried it, but it occurs to me that it might be nice to build a tabs-out version and then build a tabs-in version on top of it, which could make a lot of the taping and gluing easier than simply building a tabs-in model.

You don't technically need tabs, but the fact that the curvature at the cusps is positive while the curvature of tape is \(0\) means that taping across the cusp will result in bunching of the tape unless you manage to stretch or distort the tape just right. The triangulation of the tabs along the cusps is meant to make this problem less pronounced. (This was inspired by existing sphericon nets like these.)

Notes

When designing the net, the initial observation is that in order for the cross-sections to align before and after rotation in the steps 3-4 above, the cross-section must be a square. This indicates that the height of a cone must equal the radius of its base:

\[ h = r \]

If we take the cone and cut in a straight line down from its vertex to its base, we get a Pac-Man-shaped sector like the one pictured below.

The arc length \(l\) of circular arc must be the circumference of the base of the cone before cutting, so

\[ l = 2\pi r \]

The radius of this Pac-Man sector is the distance from the vertex of the cone down to its base, which is \(r \sqrt{2}\), since the altitude and radius of the cone form a \(45^\circ-45^\circ-90^\circ\) right triangle with legs of length \(r\). Thus, if the Pac-Man sector were a full disk, its circumference would be

\[ c = 2 \pi (r \sqrt{2}) \] The angle inside the Pac-Man sector in radians is thus \[ \frac{2 \pi l}{c} = \pi \sqrt{2} \] In order to make a sphericon net, we want to string four half-Pac-Men (half-Pac-Mans?) together as shown below.

The angle of each of the four sectors is \[ \theta = \frac{\pi \sqrt{2}}{2} \approx 127.28^\circ \]

Extra Challenges

The sphericon is a fun shape because it follows a wobbly path while it rolls. (We can actually see what its path looks like by studying its net!) There are other shapes we can create by starting with a simple object based on cones and then twisting one half relative to the other. MoMath has an exhibit based on this called Twist and Roll. Students might like to construct some of these other objects. Unfortunately, you likely won't be able to get away with a single net, but working out the design is a fun puzzle. Our sphericon was produced based on a square cross-section. Here is a shape made from a stack of two cones and a cylinder with a regular hexagonal cross-section:

Here it is after we rotate one half \(60^\circ\) relative to the other half:

You can see the design is based on two cones and a cylinder. Similar shapes can be produced for every regular polygon with an even number of sides using combinations of cones, chunks of cones (frusta), and cylinders. Once you've abandoned the pretense of using a single net, you can form the object by making half-cones, half-frusta, and half-cylinders of the right dimensions and patching them together.

This post was sponsored by the Julia Robinson Mathematics Festival