What is needed is Non-Euclidean geometry

Suppose for a moment that regular dodecahedra did fit together to 'tessellate' 3-D space. Each vertex would be the common property of 4 dodecahedra, and four edges would converge at each vertex. Their configuration would be just like what is called a tetrahedral bond in chemistry - think for example of the carbon atom surrounded by 4 hydrogens in methane. Looking down one of the bond directions, the other three are equally spaced with apparent angles of 120 deg between them. If the carbon is at the origin, we can take the hydrogens to be at (1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1). The cosine of the angle between any two of the bonds will be -1/3, giving us the angle as 109.471° = α. Here we have another measure of the failure, in actual fact, of dodecahedra to fit snuggly together. For the angle at the vertex of a regular pentagon is only 108°. Another case of 'angle deficit'.

But hold on - so far our unspoken assumption has been that we are working in Euclidean geometry. In non-Euclidean geometry, the angles of a triangle add up to less than 180° (in hyperbolic, negative-curvature geometry), or more than 180° (elliptic, or positive-curvature geometry) - and the more so, in both cases, the larger the area of the triangle. The same will go for for the internal angles of a pentagon - so what we need to get a pentagon with angles of 109.471° is elliptic geometry. Altogether, the angles will exceed 540° by 7.355° or 0.128 of a radian, implying that the area of the pentagon must be 0.128 times the square of the radius of curvature - or just 0.128 if we take the radius of curvature as one unit.

We can obtain the length of the side of the pentagon, by doing a bit of spherical trigonometry (which applies in elliptic geometry). The second fundamental formula of spherical trigonometry goes as follows:

cos(A) = - cos(B) cos(C) + sin(B) sin(C) cos(a).

Here A, B, C are the three angles of a triangle, and a is the length of the side opposite A (again we assume the radius of curvature is one unit). The formula is usually thought of as determining the angle A when a, B and C are known. However, it can also be used to determine a when A, B and C are known. In the present case, A is 72°, B and C are each α/2. We obtain

cos(a) = ( cos(72) + cos²(α/2))/sin²(α/2) = 0.9635

hence a = 0.2709.

In flat space, the volume of a dodecahedron whose side is a, is (2 + 3.5 φ) times a³ (here φ stands for the golden number), giving us in the present case 0.1524. This is of course only an approximation (an underestimate, in fact), for the dodecahedron about which we are now talking exists in positively curved space.

It is interesting to compare this with the volume comprised in a 3-sphere of unit radius, which is just 2π²: enough to fit in approximately 2π²/0.1524 ≈ 129 of our dodecahedra.