Here is a new method for folding an uncut circle into a 'sphere'.
1. Fold concentric circles
2. Divide disk into equal-sized pie-slices (any even number will do)
3. Draw zigzags along the radii
4. Reverse all folds in alternating pie-slices
(or in a less elegant version, just crimp the radii)
5. Fold up
6. Secure with pins
In this technique the sphere is not hollow. The main difference from the more usual way of making spheroid shapes in origami is that instead of folding up 'walls', one folds polar cross-sections of the sphere. In fact this sphere has the odd mathematical property, that its center is the center of the circle, its "surface" the edge of the circle, and all points in between on the sphere are in the same pair-wise order (in distance from the radius) as in the circle.
Greater accuracy and surface coverage can be obtained with some effort by using many 'wedges' (=pie slices in the original circle; looked at side-on they are 'arms' or 'meridians'). But since the height of the sphere shrinks in direct proportional to the number of wedge-pairs, you'll get a perfectly delineated ball only at infinity, and then it will also be solid--and the size of a point.
Added June 2010
Here are a few further points of interest about this form.
- This Sphere is related to the Concentric Winder. Imagine each of the layers of the Winder 'peeled': reverse-folded-along the same diameter line, at various angles, so that the Winder is 'rotated in space'. You will have something like this Sphere. (Some such idea was part of the conceptualization of this model.)
- Like the Winder, the Sphere also offers a negative illustration of the Albers Effect: Because of all the movement back and forth, the surface of each wedge can remain flat-on-average, without being forced to twist from its plane as it does with Albers disks or Yenn annuli.
- Less obviously, this Sphere is even more directly related to the folded Albers disk: for the latter too has circle-lines marking off approximate outlines of spheres. (Because of all the Albers twisting this is harder to see.)
- Perhaps the most surprising aspect, besides the directness with which the wedges rotate and condense into meridians, is that the initial paper circle has one curvature, the smaller sphere another, and yet the circle just magically adjusts its curve to oblige the needs of the sphere. Moreover it does so no matter how many wedges are put in, that is, regardless of the relative size of the sphere! Somehow everything always works out.
- I put my name on this Sphere, for its simplicity; that doesn't mean there are not more complicated things to be made by following its principles. This is new ground, there is considerable scope for variation--so go discover something for yourself!
Finally I have to admit---I got lucky here (though it wasn't just luck, as I was actively looking for a Sphere in the month before I found it). --If anyone should have discovered this principle before, it should have been Josef Albers 80 years ago. He after all was asking the right kind of questions, viz., what can you do to a sheet or a pattern that will cause the surface to change its expected behavior. This led to his discoveries of both the Concentric Circles fold and of the Hypar. In retrospect I see now that this Sphere is a cross of those two Albers shapes, though that's not at all how I came to it. But Josef Albers did not take this step: his loss, and my gain.