The third, shorted, period Na to Ar + K

Nasingle cloud in face of [Ne] tetrahedron
Mgtwo single clouds at 180° touch [Ne] tetrahedron along C2 axis (figure)
Al three clouds in D3h touch tbp deformed [Ne] tetrahedron (figure)
Si  Td
P  Td single, double occupation same radius, should be -> C3v
S  Td single, double occupation same radius, should be -> C2v
Cl  Td single, double occupation same radius, should be -> C3v
Ar  Td
K  single cloud in face of [Ar] tetrahedron (figure)
These computations make it clear, e.g. with Mg, Al, and K, that the "construction" of an atom with more than two shells of nonoverlapping spheres needs symmetry adjustments. These are far from artificial, since e.g. the Al atom really forms a D3h AlH3 and AlF3 molecule, which have a trigonally deformed Ne core. The plot for K marks the end of a painstaking stacking of tetrahedra ("qualitative" Kimball fans have claimed at several places that Kimballs model cannot be used beyond K (or even Ar) for this reason). A continuation by tetrahedra brings us at variance with the Periodic Table, since we have to accommodate the 10 3d electrons before we can complete the 4s24p6 shell at Kr. That can be done by abandoning tetrahedral for a trigonal-bipyramidal arrangement up to Ni or Zn. We will do so in the spherical atoms but first fuse the inner tetrahedra to a common, energy equivalent outer sphere, each, centered at the nucleus.

Except for Al all the above atoms have a tetrahedral [Ne] core. Al needs a trigonal deformation for the D3h arrangement of three half filled valence clouds. We have used a deformed Ne core with trigonal bipyramidal form. Let's compare the two Ne cores to give us an idea of how much we may manipulate the 2s2p6 density packed into Kimball spheres:

Ne coreEtotR1 R2R3 axialVneVeeVirial
Td-128.9080.145796 0.624762 -309.58851.77252.00000
tbp-128.7750.145963 0.6297910.604871-309.51251.96132.00000

The difference here is for all numbers negligible in the face of what the Kimball model is expected to achieve. However, that is a highly significant result. It shows that the next shell below the valence electrons may easily be polarized in the external ligandfield. The idea, that an inner tetrahedral shape is maintained, as often depicted in qualitative "Kimball-models" is certainly inadequate and not supported by higher level QC computations.
Here is the tbp arrangement as projection, left xy, right xz plane:

How this looks like with a reasonably complete HF computation is shown below: This is a partially transparent surface plot of the topmost totally symmetric A' orbital of AlH3 with Gaussian09 optimization and model chemistry HF/6-311+G(2d,p) (Etot = -243.6403074 Eh; the HOMO above is of course E'). Under the valence electron surface a deformed Ne-core is clearly visible. It exhibits an elongation in the z-direction, as expected.
Na to Si again from Kimball-Model in chemistry didactics at ETH, Zürich
This picture violates simple criteria of symmetry for Mg and Al, and is unacceptable for an educational purpose. It makes believe, that electronic shells are not polarizable, almost like hard solids. The contrary is true: They must deform in the external field. Compare with the figures in Mg and Al at the top. However, it is not necessary, to take the artists pictures so seriously. They just want to convey the impression, that below the valence electron clouds there is a more or less untouched Ne core. Whether this is a tetrahedron of Kimball spheres or with a symmetry-adapted shape does not make any significant difference in energy, in fact it amounts to about 1/1000!