Hydrocarbon based bipeds - Solve this!

In a hasty attempt to remove comments about me from the top of the page, I pose this puzzle(?)based on something I did about 10 years ago:

A lot of us may know/remember that the bond angle of methane - CH_4 is approximately 109 degrees 28 minutes. This is created by 4 H atoms in 3d space repelling each other to produce a symmetric arrangement about the central C atom. Now tell me what is the bond angle of a hypothetical CH_(n +1)molecule in n dimensional space?
(Note for those who don't like chemistry: The chemical notation is used only to simplify description of what I would have to otherwise state as the angle subtended by the edge of a regular n dimensional hypertetrahedron at its center).

~ grenade
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In general, a hyper- prefix stands for a something with a generalised number of dimensions. ie a 2 dimensional (hyper)cube would be a square. A 3 dimensional (hyper)cube is a cube. The hyper prefix, however begins to become non-trivial when go into a larger number of dimensions. Just as a 3D cube is made by using 2 parallel 2D cubes that are offset in the 3rd D, and then by joining pairs of corresponding vertices by edges, the 4D hypercube is made by 2 parallel 3D ubes that are offset along an imaginary 4th axis (that's perpendicular to x, y and z) and then joining the corresponding vertices of the cubes.

A regular tetrahedron is a little more simple. In n dimensions, the regular n-hypertetrahedron consists of n+1 points that are equidistant from each other. So a 3D regular tetrahedron is a normal regular tetrahedron. A 2D regular tetrahedron is an equilateral triangle. A 1D regular tetrahedrom is a line segment (which is also a 1D cube). A 0D tetrahedron is a point (which is pretty much a 0D anything - well, anything that's not nothing, anyway).

I am afraid I am unable to give a better explanation within the confines of a blog editor, but I would recommend reading Martin Gardner's Mathematical circus, one of whose chapters has an introduction to hyperspaces.
~grenade
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From the lack of responses, I assume that the jump into hyperspace was made too far. So I will pose a simpler(?) question from the aforementioned Martin Gardner book:
Imagine an infinite checkerboard. Circumscribe the black squares with circles. These circles will intersect the white squares, and it is a matter of simple arithmetic to check what fraction of the white squares' areas are left unoccupied by the circles (or discs, if you would prefer).

Now extend it to 3D. We have an infinite "3d chessboard" where black and white cubes repeat alternatingly in all 3 directions. The black cubes are circumscribed by spheres whcih overlap the white cubes. It is a pretty hard exercise to calculate what volumes of the white hypercubes are left unoverlapped by the spheres, because the spheres intersect each other, and the math becomes messy.

My question is this: In 4D, what fraction of the hypervolume of the white hypercubes is left unoverlapped by the hyperspheres circumscribing the black hypercubes in an infinite 4d chessboard (alternating black and white 4d hypercubes along 4 mutually orthogonal directions)?

How about n-dimensions? n>4?
~grenade