2: Cut a 1" length of 1" rod. Mark a vertical diameter on one end, and a horizontal diameter on the other. Grind off material such that the ground faces become half-ellipses, so that the curved “end” touches the tip of the horizontal diamter and the straight “end” is the vertical diameter.

(It might be easier and/or safer to do the grinding first, then the cutting. It is also possible to cut the diagonals rather than to grind them off.)

3: Extrude a 1" circle, a 1" square, and the 1"x1" triangle. Intersect them all at right angles (e.g. cylinder along the x axis, square prism along the y axis, and triangular prism along the z axis).

This is a very cool site that contains drawings a ton of polygons. It also has 3D color pictures that can be rotated of polyhedrons of varying complexity, I found one that looks like the Epcot center.
On to the goods

1.) Omni Cork Plug

2.) Put a screw in the bottom of a 1x1 cylinder and hold it verticly ^ (when its done) sorta like that drawn and mill out the triangle shape.

3.) Extrude a 1in. Dia. Cylinder, with a 60 deg. taper, 1in.

Looking at this site i described above i noticed that on the square face the piece removed by the triangle is a perfect parabola.

Epcot’s Spaceship Earth is the world’s largest complete geodesic sphere. There are many other geodesic domes scattered around the world and a sphere or two, as well.

If you were to extend the length on the main cylindrical feature the extremes would be parallel, but if you to expand the height and width so that the poece would always be a perfect square the sides of the parabola would just be intersecting with the corners of the square shape and would continue in a parabolic “line”

This one is the probably the best. Note that the mathworld link says there is an infinite family of shapes that satisfy the challenge. The one with minimum volume has triangular cross sections.

I think the mimimum volume cork plug can be (approximately) modelled in Inventor as a loft from a thin rectangle one inch long by 1/1000th inch wide to a one inch diameter circle. The rectangle is sketched on a plane one inch from the circle and the long centerline of the rectangle projects onto the circle’s diameter. I used a model made this way to generate an STP file from which we made a “minimum volume” cork plug on a Viper SLA machine at Emerson.

Of course, Alan’s procedure is more practical. Just round the elliptical edges a bit and you end up pretty close to the minimum volume.

GW is right. The curved edge is elliptical, not parabolic. Referring again to the the mathworld site, an ellipse is a section through a cylinder (a circle is the special case in which the section plane is perpendicular to the cylinder’s axis), while a parabola is a conic section (section through a cone).

Correct. Specifically, a parabola is the special case where the section plane is parallel to one “edge” of the cone, so that the curve never closes on itself: Conic Section -- from Wolfram MathWorld

If you make this like the way everyone else is wouldn’t there me more lines on the triangle? They wouldnt be hidden lines because they are outside of the projection of the triangle face.