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.