Related Shapes

In the course of my goofing around with these sphericon things, I've run across some other interesting, related shapes.

The first one I found, called a femisphere, was the same basic shape, but had no straight lines on it. Instead of the straight lines that could be used to define the surfaces on a regular sphericon, this one used a circular arc, of arbitrary radius. While this doesn't change the basic shape, it drastically changes the overall appearance of the solid. You can see more about this attractive shape at

Another related shape that has really piqued my interest is called a hexasphericon. The information about this shape came from a rather involved (from my perspective) discussion on "The Differential Geometry of the Sphericon" that used to be posted at the American Mathematical Society's web site, but has now been removed. I believe that it qualifies as a member of the sphericon family, as it exhibits several of the same qualities:

It is different in several ways, including the fact that you can have right- and left-handed versions, as you'll see later.

The paper pattern for making a hexasphericon is given below:

You can also draw your own pattern any size you want. The width of the rectangle can be whatever you want. This becomes the radius for the two arcs. The length of the rectangle must be the same as the arc length. The angle of the arcs is pi times the square root of 3 divided by two radians, or about 155.88 degrees.

You may download an AutoCAD version of this pattern here. You'll need AutoCAD or something that can read its files to use this, but if you have that, you can resize to suit or interrogate the shape.

In order to make one of these using the paper pattern, you'll need two copies. Cut out carefully. Tape the two together. Surprisingly, it isn't obvious how to do the taping; take some time to figure it out before you start taping, or you'll end up cutting the tape and redoing it, like I did.

I had to actually make one of these with paper before I could even visualize what they looked like. Now I see that they can be made as follows:

  1. Take a cylinder, and "sharpen" a cone on each end, like you would sharpen a pencil, only with an apex angle of 120 degrees, as shown at right.
  2. The length of the unsharpened side of the cylinder should be the same as the length of the side of the cone on the end. So, after you have both ends sharpened, you should be able to look at the side of the cylinder and see a regular hexagon. (Note: this is similar to requiring that you be able to see a perfect square when looking at the pre-cut double-cone when you are making a standard sphericon.)
  3. Cut the cylinder down its axis. The face of the cut should be a perfect hexagon.
  4. Take one half of the cylinder, rotate it 60 degrees along its planar face, and reattach it to its mate. Here is where you can create either right- or left-handed versions, depending on which direction you rotate the half. You can also create the right- or left-handed versions with paper depending on which direction you curve the paper when you attach them together - either toward you or away from you.

Here is what I ended up with, made from some 2x4 scraps. As with a regular sphericon, it is hard to photograph well enough to readily describe its 3-dimensional shape. I made a few small errors in manufacturing this one, but the next one should go smoothly. An interesting object. Next I need to make a wire frame model of this thing and dip it in bubble solution . . .

Another question that now comes up is: If a regular sphericon has a regular square surface when cut in half, and a hexasphericon has a regular hexagonal surface when cut in half, is there a continuing series of shapes that have 8-, 10-, 12-, etc sided surfaces when cut in half that also exhibit sphericon-like properties? If so, what do they look like??? It seems like there must be a limit. What about a 200-sided surface? Might there be a series in here somewhere that would tell us?

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