You can find find more information on the sphericon shape from Paul J. Roberts http://www.pjroberts.com/sphericon/
This instructable is designed for folks with some handbuilding experience in ceramics. If you have some pottery skills but are new to handbuilding, try to practice some of the basic skills of slab work before attempting to build your own sphericon. There is a listmania of recommended books for handbuilders on Amazon.com here
Some new photos added, including a view of the bisqued piece made in the course of this instructable. Hey, and now we have a picture of the final item after firing to cone 10 in gas reduction! I wiped most of the glaze off before it went in the kiln.
Step 1: Roll out a slab of clay
Ideal thickness is influenced by the clay you're using and the size of the sphericon you are building. A larger shape requires a thicker clay slab. You can go much thinner with porcelain than with stoneware.
I'm using the house clay for this model, which is a high fire (cone 10) grogged stoneware.
Step 2: Mark out two circles of the same size
KEY: make sure you have the center point of each circle marked.
You only need about 3/4 of each circle, so it's OK if you include some thin spots or other defects. Just avoid these parts in the next step.
Step 3: Cut out 4 identical wedges (127 degrees each)
Simply match the point of the paper wedge with the center of your circle, and extend the sides as long as you need them to so it will fit.
Why 127 degrees? Well, it's 90 degrees * the square root of two. Geometry.