1. The answer is found in a synthesis of Russian mysticism, numerology, astrology, music, and Eastern metaphysics.

Notice that the HZ numbers, numerologically, all add up to 9. 864 HZ is 8+6+4 numerologically, which = 18. 1+8 = 9.

The following was not written by me.

Posted - 03 Jun 2008 :

Most things like this (such as the Enneagram itself) are not the relevant geometry but are distortions of the relevant geometry meant to highlight some properties of the actual math. The 72 is the number of vertices in the E6 polytope, if you add the 6 axes to the 72 vertices you get the 78 tarot cards. If you add the 4 axes of F4 to the 48 vertices of F4 you get the 52 playing cards; if you add the 4 axes of D4 to the 24 vertices of D4 you get the 28 Chinese Astrology points. Chinese Astrology also has the 12 "internal" points which are the D3 vertices; the 12 are also the Enneagram plus a second law of 3 like some researchers have, the 12 D3 vertices being a Law of 7 hexagon with a law of 3 triangle above the hexagon and another law of 3 triangle below the hexagon. Chinese astrology divides its 28 points into four directions worth of an octave like 7.

Theosophical addition is kind of taking multiple digit numbers and turning them back into one digit numbers like df did with the HZ numbers, like 10=1+0=1, 11=1+1=2. Here this lets you count to 72 by repeating the numbers 1 through 9. 10=1, 11=2, 12=3, 13=4, 14=5, 15=6, 16=7, 17=8, 18=9, 19=10=1, 20=2, 21=3, etc. There's a nice one for powers of 2 using the law of 7 rearranged in the Celtic Enneagram order of 124875 since after 1,2,4,8, you have 16=7, 32=5, 64=10=1, 128=11=2, 256=13=4, 512=8, 1024=7, 2048=14=5, etc.

Music is more complicated as explained here:

"the 7 and 5 structures in music

correspond to different projections of the Witting polytope

from 8 real dimensions into lower dimensions.

In some sense, the 7-structures are related to the 24-cell

and therefore to the ordinary 4-dim subspace

while

the 5-structures are related to the 600-cell

and therefore to the Golden sqrt(5) 4-dim subspace

of the 8-dim Witting polytope.

From the 4(complex)-dim point of view,

the 7-structures are REAL

and

the 5-structures are IMAGINARY."

That Witting polytope is E8 which has 240 vertices and 8 axes and contains all that E6,F4,D4 and D3 stuff I mentioned before.

There's a pattern with the Fibonacci sequence on the outside which can give you the Enneagram's law of 7 and law of 3 also:

It has the Fibonacci sequence in the upper right quarter and the rest comes from opposite numbers adding to 9 and opposite parallel lines of numbers adding to 9. There are 24 outer points, the 24 D4 vertices.

The colors just seem to show besides the 12 pattern, the law of 3 and the law of 7 with the law of 7 broken in half (beginning with the 2 point as in 285-714).