All you need is a glue gun and 140 5 oz. Dixie Cups to achieve this rigid erection! Make several - then show off your amazing balls to friends and family! Then show them the thing you made out of 140 Dixie Cups! These Dixiespheres exhibit icosahedral symmetry, but other kinds are possible as well - I've included pictures of other Dixie Cup spherical constructions. I'd love to see any new ones you can come up with (especially if they exhibit some sort of symmetry), and I'd love to hear about how you use your Dixiespheres. I am a 30 year old man who makes things out of Dixie Cups!!! I'm starting to get a gut!!! I haven't taken a shower yet today...

Step 1: Constructing the Basic 7-cup Modular Unit

The Dixiesphere is composed of twenty spherical-hexagonal 7-cup modular units, which you will construct first. Each hexagonal modular unit consists of seven cups hot-glued together such that one central cup is surrounded tangentially by six more cups. (The spaces between cups will look like equilateral triangles with concave edges, and the circular bases and rims of the cups will look like hexagonal close-packings of circles, but they will posess a slight overall curvature, as the circles define the surfaces of spheres.) It is important that the central cup contacts each of its surrounding six cups at their bases. It's a tight fit getting six cups around a center cup. Work quickly while the glue is hot to ensure that all upper rims and lower bases of the cups are flush, and that there are no superfluous gaps between the bases of the cups. Make all twenty units before assembling the Dixiesphere. Do not micturate in the cups!

Step 2: Assmebling the Modular Units Into a Sphere

Glue two spherical hexagonal modular units together such that the gap created by the four cups resembles a square with concave edges. Here is a picture of where two hexagonal units meet. Two cups from one unit are glued at their bases to two cups of another unit. See the square? Glue three more modular units to the first two to form a spherical pentagon with a hole in its center. Repeat this process to form another spherical pentagon. Glue the two pentagons together. From this point on, it should be obvious where to add the remaining modular units - just continue to make more spherical pentagons and the structure will close upon itself to form a Dixiesphere.

Step 3: Some Thoughts and a Different Type of Dixiesphere

The finished spherical dodecahedron is impressively strong, but IT'S NOT A GEODESIC SPHERE! Quite the contrary, it is comprised of a lot of minor circles on a sphere, not great circles (aka geodesics). The cups are truncated cones (aka frustrums). I'm not really sure how to understand the structural dynamics or statics engineering of this type of construction. If the dixie cups were were made of thin-walled aluminum or sum shitz, and their bases were welded plates, would the hemispheres make good strong dome structures? I'm not equipped to consider this problem effectively. Any thoughts?
Now, here's a different variety of Dixiesphere at various stages in its construction.

Step 4: Dixiesphere2

Use a 6-cup spherical triangular modular unit for Dixiesphere2.

Step 5: DS2 Cont'd

DS2 consists of three intersecting great circles (geodesics). The somewhat "vaginal" openings do not occupy identical locations on the surface of the sphere; rather, they alternate higher and lower in their placement with respect to one another. You'll discover just what I'm talking about when you make DS2. Then you will be a rockstar, like me.

Step 6: Beyond Dixie Cups

Triangular prism carcboard mailing boxes! Any size will do, so long as they are equilateral triangular in cross-section. Ask and you shall recieve instructions - otherwise, I'm too lazy to type them.

Step 7: Dixie Cup Say What? ..Foo!

Use a very high-tack glue. This thing has a tendency to fall apart into a pile until the glue has dried. It's nice to be able to make adjustments as you complete the assembly.
The little hole you see between the three "turbining" triangular prisms is a flaw. The pieces should all be perfectly snug. Each prism is parallel to a face plane of a shape called the rhombic dodecahedron. A company called ULINE is one supplier of these mailing boxes. You could make this shape out of cheap acrylic optical prisms. Use capillary-action acrylic cement. This shape could serve as the base for a plastic-top coffee or end table. (You might decide to add interior reinforcement to the cardboard prisms.)