## Beyond PHI?!

The golden mean (phi, Φ, 1.618) is a fundamental harmonic. In the randomness (chaos) that is found in nature and the cosmos phi is found to closely parallel much of All That Is, which no other number can claim.  The √5 component of phi (with √2 and √3) remind us of the harmonics based on 3, 4, and 5.

The spirals, five-fold nature, and proportion of phi each accord with our Human model, other life forms, and layout of our solar system. There is magic here; the Template of Spirit, and mathematics; the Mind.

I’ve found out a whole lot of neat math things about this number, which just might be something no one has looked at before.  For example:

 Radical value Equivalent Radical value Equivalent 1 + 1√5 2 Φ –(1 – 1√5) 2 1 Φ 3 + 1√5 2 Φ2 3 – 1√5 2 1 Φ2 4 + 2√5 2 Φ3 –(4 – 2√5) 2 1 Φ3 7 + 3√5 2 Φ4 7 – 3√5 2 1 Φ4 11 + 5√5 2 Φ5 –(11 – 5√5) 2 1 Φ5

The tables, much of it generated by tracing the 5-pointed star, show us “Mathemagic”.  We find the complex value of 1 + √5 initiating patterns. We find √5 ubiquitously in the star, in the mathematics of its trigonometry (eg. Cos 36 = (1 + √5)/4 ), and in the Icosahedron and Dodecahedron.

The beauty of these two bodies is enhanced by their stellating into each other, demonstrating their duality and harmony.

Just as easily we find √3 in the equilateral triangle and more elegantly in the 6-pointed star. √2 and √3 also “come alive” in the tetrahedron, cube, and octahedron. These three forms can be related to the other two Platonics, showing a harmoninc relation in 3D between phi and the other two proportions.

In higher geometries like 8- and 12-pointed stars can be found further proportion corresponding to √(2 + √2) and √(2 + √3), reminiscent of the half- angle trigonometric values for the angles contained therein: 22.5° and 15°.

Further, we can explore 7- and 9-fold geometries. Seven is the first number to not readily divide into a circle. It is also a very profound Mystery number. Significantly, in the Great pyramid, the slope angle (corresponding closely with Stonehenge regarding its latitude and alignment) is near to being an exact division of a circle by 7, while also fitting in well with both phi and pi proportions! Contemporary Stonehenge and Karnak (Egypt) concur with regard to latitude selection and solar alignment, and seven; showing perhaps a common science and tremendous understanding of geography and astronomy.

Construction of 5 and 7 pointed stars based on radii of 1, PHI, and 2.  (5 star 100% accurate, 7 star 99+)

DaVinci seems to have used 7-fold geometry in his Vitruvian man. So too the builders of Stonehence. And even 8- and 9-fold.

7, 8, and 9 fold division of Stonehenge and the location of the standing stones.

We can do certain mathemagic with 7, and find parallel with 5 and 9. And in 9-fold geometry we find something very interesting; a further numeric proportion which behaves something like phi. Its value is 2.8794. A radical/whole solution is elusive, as it relates to trisection, which no-one have effectively worked out yet. But this proportion (which I call psi, Ψ) suggests higher order. Suggests that there are many numbers beyond phi. As Gauss perhaps discovered in 17-fold geometry…

Spiral generated in 5 and 9 pointed stars, in proportions of phi and psi.

Just some of the proportions of psi found in the 9 pointed star.

 2 x Trig value Function of Ψ 2Cos 10° 1.9696 √(Ψ + 1) 2Cos 20° 1.8794 Ψ – 1 2Cos 40° 1.5321 √((2Ψ + 1)/Ψ) 2Cos 50° 1.2586 √((2Ψ – 1)/Ψ) 2Cos 70° 0.6840 √(Ψ + 1)/Ψ 2Cos 80° 0.3473 1/Ψ

Many numbers beyond phi?!

What a concept that is!  So much left to be discovered.  Dimensions of depth, and an understanding of life.

@paul stang 2009