The name of the Ka’aba means “cube” whilst the building itself is evidently not a cube (with identical side lengths). In Sacred Number and the Origins of Civilization much was made of the fact that the square root of three appears between opposite corners of a cube with unit length (the root of two arising between the opposite corners of its square sides).
Root three appears as a very important proportion in the Gothic style of building whilst many sacred cubes are noted in sacred building in general, the Temple of Solomon and Leto [Egypt], Hindu temples, the Delphic altar to Apollo, and the cube of the New Jerusalem.
In his third book, Meditations on the Quran, Eanest McClain suggests that the original prototype of the cube is the cube of 60, which was the base number of the Sumerian and then Babylonian civilisations. From this base, still in use today (within time and angular measure) McClain shows the ability of 603 to generate a tone map that defines a hexagonal array of musical notes, all within a single octave but held between 216,000 [=603] and 432,000, a very significant number in, for example, the Vedic tradition as being the length of the Kali Yuga or dark age.
We need to remember that a hexagon is a cube seen from one corner (and projected upon a flat surface) and that, in McClain’s world, many ancient texts were written with numerical tuning theory in mind as a key plot generator. This last point has not been acknowledged yet by the worlds of scholarship or religion because the implications are too profound, being subversive to traditional ideas about religious texts; as communications from God or gods (rather than intimations of the divine world).
Returning to the Ka’aba, McClain gives a splendid account of its possible origins, the background of Mohammed’s re-purposing of the shrine and the meaning of its non-cubic design.
Its width, length and height were probably intended to be 10 by 12 by 16 and McClain points to these proportions as being similar to the temple of Poseidon in Plato’s myth of Atlantis, namely “6:3 plethora (full)” meaning expressive of 6:5:4:3 in that 5:6 lies between width and length and 3:4 between length and height. These ratios are minor third and perfect fourth.
One should note that the proportion between the width and the height is 6/5 times 4/3 which equals 8/5 the synodic period of Venus [584 days] in terms of the “practical” year of 365 days. The variation of width, length and height can lead to a greater tonal achievement that may be significant.
Why then should the polar radius of the Earth be, according to the ancient model, 3456 royal miles (as I pointed out in Sacred Number)? 3456 is 12 cubed  times 2 and “the doubling of a [cubic] altar” was considered a challenge for the ancients, proved theoretically impossible in modern times. New Jerusalem was declared in Revelations to be a cube of 12,000 per side and as I found, one royal mile [8/7 miles] equals 6000 Greek feet of 176/175 English feet. This makes 1728 royal miles equal to the volume of a cube side length 2 royal miles of 12,000 such feet. [This I found in potentio within the layout of Washington DC].
The volume of a relatively small cube of 12 royal miles, projected to form a hexagon upon the surface of the Earth, generates an allusional volume one half of the polar radius. This relates to both tuning theory and the doubling of the altar. Such a cube has only powers of 3 and 5 plus 2 that figure within 10 = 2 times 5, and 12 = 4 times 3. However, 2 royal miles is 1 over 1728 of the polar radius, that is 1 over 123 of its length.
Taking up the “doubling of volume” for a 12 side cube, the result is a cube 12 times the cube root of two in side length. The latter is very nearly 1.26 [actually 1.25992105, one part in 16,000 different] and this raises some interesting “ancient metrological” considerations, for John Neal’s Standard Canonical transform is 126/125 or 1.008 of a foot. We can then see that 1.25 is 5/4 [the major third] and that this multiplied by 126/125 yields a highly accurate ability to create a cube that has doubled in volume. 5/4 times 126/125 is 1.26 which then times a side length of 12 gives 15.12 for the double volume cube’s sides.
It appears then that this metrological ratio happens to be a practical means to the “doubling of the cubic altar” problem.