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)
Step 4: Bevel the Curved Sides to 45 Degrees
I'm using a homemade tool -- maybe I'll get around to building an instructable on that someday. There are a couple of close up photos below of my homemade tool. Briefly, it's constructed of two pieces of aluminum holding a wire at the desired angle. The flat bottom of the tool rides on the work surface. The flat side rides along the precut slab of clay. The wire is extended out to slice into the clay edge at the proper angle.
Step 5: Measure the Straight Side of Your Piece
Step 6: Draw a Square With Each Side = Straight Side Length
Use scratch paper -- this is an excellent way to recycle unneeded phone books. A permanent marker won't leave unsightly ink on your wet clay. Any running ink will burn off, but I like things neat.
TIP: it is worthwhile to use a proper right angle to make your corners here -- it will improve the ease of matching up the pieces in the final steps. (Hint: you can use the corner of your scratch paper for this too.)
REPEAT to make a second square!
Step 7: Bend Each Piece to Form a 90 Degree Angle at the Point
Step 8: Score Curved Edge of One Piece
Step 9: Score and Apply Slip to Curved Edge of Second Piece
Step 10: Carefully Match Curved Edges, Aligning Straight Sides With the Square Template You Drew Previously
If you used plenty of slip, some should squeeze out. This is good!
When building a small sphericon, say 3" in diameter or less, you can get away with matching up the pieces in hand, rather than using the paper square for alignment.
Step 11: Repeat With Remaining Two Pieces
You may try gently paddling just the curved seam to strengthen it.
Step 12: Prepare the Squares Formed by the Straight Sides for Joining
Do you want your sphericon to rattle? Wrap a small ball of clay in some scrap paper before you mate the halves together.
Step 13: Mate the Two Squares, Offsetting the Corners Where the Curves Meet
Here's one way to think about aligning the pieces properly. Hold the scored & slipped half in one hand. Hold the scored half in your other hand. Line up the corners with the curved seams without touching the two parts together. Then twist one piece by one quarter turn, so that the curves no longer align.
There's a nice 3-d image here}
Step 14: Finish Your Piece
If you don't make the one air hole, your piece will begin to bulge out as the clay dries and shrinks, and may eventually crack.
Let it dry carefully, and glaze/fire according to your taste and the type of clay you used. If you do choose to glaze your piece, be aware that it could roll in the kiln. It is a good idea to use some little shims of soft firebrick to stabilize it. Or, woodfire and let the ash take over -- the wadding will hold it safely.
Why make this? It is not functional, just fun. Although, you could use it for a math lesson perhaps.
I hope you enjoy it. Please share your results! Post pictures in the comments and send a link to me on flickr if you're a member there too.