I first happened upon the sphericon shape in an October 1999 Mathematical Recreations article by Ian Stewart in Scientific American magazine. The article demonstrated how to make them, as well as giving some of their interesting properties. I don't believe you can find an authentic web version of this article, so I have reproduced that article here in both HTML and Adobe Acrobat .PDF (105k) formats. You can also do a Google web search on the word "sphericon" and you'll find lots of interesting stuff. (Don't be fooled by the Macintosh computer game using this name.) Try the Puzzlecraft or Wolfram pages as good ones to help you visualize this oddity.
I thought it would be interesting to make one, so I fired up the old lathe, and did so.
The first one I made was out of some old wood I had lying around. It was only about the size of a tangerine, but was quite intriguing. Its shape was odd and hard to comprehend, and it rolled very wobbily across a flat surface, or down an incline. I next got hold of some clear cast acrylic plastic, and made one out of that. While the joints didn't turn out very well (lots of air bubbles around the edges), it showed me that a see-through one would be interesting.
About that time, I was involved in cutting down a fairly good sized almond tree. From the trunk, I got a couple of pieces of good-sized solid wood, and thought I could make a nice, big sphericon out of them. So, I put the chunks in a paper bag in the garage for a few years to let them dry out slowly. I eventually made a very impressive sphericon with a 6" square middle section (see photo at left). Most people who see it like to hold it, turn it over and over, and stroke its single, sinuous surface, trying to figure out what the shape really is. The local woodturning club was very intrigued.
I began to wonder if I could make a wire frame version of a sphericon (only the edges would be solid, all else would be open). If I could indeed do this, and I dipped this thing in bubble solution, what shape would the bubbles be? As kids, we have all seen strange curved bubbles. We even had a cube that we dipped, and got a very unexpected set of bubble surfaces!
I just can't quit. I'll continue dreaming up variations on the theme. I am working on connecting with a glass artist, who might be willing to make some out of glass, either solid or hollow. They should be lovely.
I also happened up on a sphericon look-alike, made as art. It was made by local artist Roger Berry, and is called Eclipse. It is a fairly large (about 10 feet tall) spherical structure in front of the Crocker Art Museum in Sacramento, CA. Someone actually told me about this, and wanted me to go have a look at it. Unfortunately, it is not a sphericon. It is merely a set of connected bands that were wrapped around a nonexistent sphere.
All of these variations got me thinking - What is a sphericon? How similar to the solid does something have to be to still be considered a sphericon? Does it have to roll like one, or have the surfaces that a solid one does? Does it only require two curved edges at 90 degrees? I'll have to ponder that one for a while.
In the meantime, what about other, related shapes?