Sacred Number

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 Koran, Earnest 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.

McClain Yantra for 432,000. Only this yantra can yield a hexagonal region where tones and their reciprocals co-incide. The hexagon has 37 tones because 37 is a hexagonal number, viz chapter 10 and the cube of Metatron.
[from p 90 of  Meditations on the Koran by Earnest McClain, Nicolas-Hays, Ney York, 1981]

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, all 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, subversive to the traditional ideas behind 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 [1728] 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.

The square root of two has a massive significance in geometry, musical harmony and ancient symbolism. I have a little piece of geometry about this that happened the other day:

A unit radius has been drawn so that, using a rational Pi of 22/7, one eighth of the perimeter is 11/14 in length. We know that a very good approximation to the square root of 2 is 140/99 and that the vertical shown must be the reciprocal. On can notice therefore that to transform 11/14 into 99/140 we have multiplied by 9/10, the minor wholetone ratio (lost in our equal temperament).

This is an interesting relationship because it employs the key ancient approximations of Pi and root two with a linkage straight out of ancient tuning theory - a tradition so well developed that it was arguably the coded source of most surviving mythic material such as the Old and New Testaments, Gilgamesh, etc. [Much of what remains is then absorbed in the massive ancient speculations about precession of the equinoxes].

The Biblical source of 99/140 concerns Abraham and circumcision. As Ernest McClain shows in Meditations on the Quran,  Abraham's decision to circumcise himself at the age of 100, and similarly Ishmael at age 13, uses circumcision as a code for removing a little bit from a known rational ratio, 10/14 by removing a hundredth part to obtain 99/140. The 13th fifth in music causes the Pythagorean comma, a musical sin that must be corrected, symbolised again by removing something small "like" a foreskin. More of Ernest at www.ernestmcclain.net.

In the diagram, a geometric arrangement has achieved the same as this rule of thumb but also shown a relationship between a circle and an inscribed square. Multiplying by 14/11, the sector becomes a unit length and the vertical just 9/10, the radius 14/11. It is ideal (as shown by Neal in the Scottish Brochs) that the radius have 7 in its length. Here there is 14 and we also see the 11 as "pre-removed" to obtain the unitary circumference (of 8 units long).

It is very unlikely that the diagram above was not known by  the specialists of the Megalithic and Middle Eastern cultures as it reveals rational order between two irrationals, linked to a harmonic ratio.