There are three basic ways to make a sphericon model: folded paper, solid, or wire-frame.
To build my solid sphericons, I followed the usual method. This is to first make a couple of cones with 90-degree apex angles. Glue these together by their bases. When dry, cut the assembly in half again, this time along a plane through their apexes. Note that when you do this, you will get, as the cut surface, a perfect square.
After you again have two pieces, sand the newly-cut surfaces flat again. Turn one of the pieces 90 degrees from its original position, and glue it back onto its partner. You now have a solid sphericon.
As you may find out, working with these odd shapes presents certain logistical problems. The first one I encountered was that it was hard to hold on to the material to cut the cones. I eventually set up a faceplate in the lathe, and used screws from the backside of the faceplate to hold the material while it was turned. I used a metal lathe, which made it pretty easy to get the 90 degree apex angle by adjusting the compound to 45 degrees.
A more difficult problem was devising ways to clamp the pieces together without damaging them, as they have several "sharp" edges that you can't really put a clamp on directly. The first clamping, putting the two cones together, requires that you apply pressure to the point ends of the cones. You don't want to do this, because it will ruin the cone points. I had to make up two blocks with cone-shaped recesses in them. (I also drilled a hole right in the point of the recess so I was sure it wouldn't damage the point.) Sorry I don't have a photo of these blocks, but they are pretty easy to imagine. I do have a photo below of how they were 'repurposed' to good use. I could then put one cone point down in the bottom block, put the second cone point up on top of it, and the top block on top of the whole stack. This gave me flat surfaces to apply clamping pressure on.
An interesting side note at this point on clamping is that I have successfully used gravity as my clamping pressure for many larger projects that need lots of pressure, or pressure over a larger area (such as laminating). I do this by jacking up a vehicle, then placing the project under it, and lowering the vehicle. You get 1500 or so pounds of force doing the job (more if you have one of the behemoth SUVs now lumbering about the country).
After you get the cones put together, then cut apart, you have an even more awkward clamping job to do. You again have a situation where you don't want to apply clamp pressure to a sharp edge, this time the arcs. I knew that I needed the inverse shape of the two half-cones I was assembling. But how could I make this shape of a recess? After much thought and soul searching, I realized that I had all the materials I needed, they just needed to be 'reconfigured.'
Since I had no further use for the two clamping blocks I had used to join the cones, I glued them together face to face, so that I had a single hollow block. (This is how the two blocks from the first clamping operation were 'repurposed'.) In the center of this hollow block was a space the shape of two cones, base to base. I now cut this block apart again (using a chain saw, just for effect), through the apexes of the cones. I now had the two clamping blocks I needed (see photo at right)! I again parked my car on top of the assembly for a day or so, and did the final finish work.
When I started to make other variations of sphericons, I realized I needed a new way. After some more thought, I devised a mandrel for the lathe that would spin a half of a sphericon at a time. However, the flat side of the part was now 90 degrees rotated from my original faceplate. Doing it this way, I only had one glue joint instead of two, and I can make sphericons with potentially any number of sides on their cross-sections, not just squares. Look on the Series page to see some of the interesting shapes that result!