I'm 7 years too late, but to whom may find this useful:

What that Wikipedia image is depicting is the optimal color solid (or Rösch-MacAdam color solid), which is the theoretical gamut of surfaces, not the visible gamut.

The visible gamut is bounded by the spectral cone and the inverse spectral cone. In a linear color space, such as CIE 1931 XYZ or LMS, the spectral cone is the surface formed by the set of rays that start at black ((0, 0, 0)) and pass through one spectral color (the XYZ coordinates of the spectral colors can be found on the CIE website). The inverse spectral cone is the symmetric of the spectral cone with respect to central grey ((0.5, 0.5, 0.5)). The volume that these two cones enclose is the visible gamut. The locus where they intersect is the set of the most chromatic colors that we can see.

The optimal color solid is tangent to the visible gamut's boundary at the blackpoint and the whitepoint, but it is pretty far from it in the highly chromatic colors, especially in the reds and cyans. This is because surfaces cannot reflect a single wavelength of light and be bright at the same time. But light sources can.