Phi and the Solar System
The dimensions of the Earth and Moon are in Phi relationship, forming a Golden Triangle
The illustration above shows the relative sizes of the Earth and the Moon to scale.
 Draw a radius of the Earth (1)
 Draw a line from the center point of the Earth to the center point of the Moon (square root of Phi)
 Draw a line to connect the two lines to form a Golden Triangle (Phi).
Using dimensions from Wikipedia and geometry's classic Pythagorean Theorem, this is expressed mathematically as follows:

Dimension (km) 
Proportion (Earth=1) 
Mathematical Expression 
Radius of the Earth 
6,378.10 
1.000 
A 
Radius of the Moon 
1,735.97 
0.272 

Earth Radius + Moon Radius 
8,114.07 
1.272 
B 
Hypotenuse 
10,320.77 
1.618 (Φ) 
C 
Hypotenuse / (Earth Radius + Moon Radius) 
1.618 (Φ) 

A²+B²=C² 
This geometric construction is the same as that which appears to have been used in the construction of the Great Pyramid of Egypt.
Source: Hidden Nature by Alick Bartholomew. Thanks to Sathimantha Malalasekera for bringing this to my attention.
Certain solar system orbital periods are related to phi
Certain planets of our solar system seem to exhibit a relationship to phi, as shown by the following table of the time it takes to orbit around the Sun:



Mercury 
Venus 
Earth 
Jupiter 
Saturn 
Power of Phi 
3 
1 
0 
5 
7 
Decimal Result 
0.24 
0.62 
1.0 
11.1 
29.0 
Actual Period 
0.24 
0.62 
1.0 
11.9 
29.5 

Saturn reveals a phi relationship in several of its dimensions

The diameter of Saturn is in a phi relationship with the diameter of its rings, as illustrated by the green lines.
The inner ring division is in a phi relationship with the diameter of the rings outside the sphere of the planet, as illustrated by the blue lines.
The Cassini division in the rings of Saturn falls at the Golden Section of the width of the lighter outside section of the rings.
Note: Phi grid showing Golden Ratio lines provided by PhiMatrix software.

A closer look at Saturn's rings reveals a darker inner ring which exhibits the same golden section proportion as the brighter outer ring.

Venus and Earth reveal a phi relationship
Venus and the Earth are linked in an unusual relationship involving phi. Start by letting Mercury represent the basic unit of orbital distance and period in the solar system:
Planet 
Distance from the sun in km (000) 
Distance where Mercury equals 1 
Period where Mercury equals 1 
Mercury 
57,910 
1.0000 
1.0000 
Venus 
108,200 
1.8684 
2.5490 
Earth 
149,600 
2.5833 
4.1521 
Curiously enough we find:
Ö Period of Venus * Phi = Distance of the Earth
Ö 2.5490 * 1.6180339 = 1.5966 * 1.6180339 = 2.5833
In addition, Venus orbits the Sun in 224.695 days while Earth orbits the Sun in 365.242 days, creating a ratio of 8/13 (both Fibonacci numbers) or 0.615 (roughly phi.) Thus 5 conjunctions of Earth and Venus occur every 8 orbits of the Earth around the Sun and every 13 orbits of Venus.
Mercury, on the other hand, orbits the Sun in 87.968 Earth days, creating a conjunction with the Earth every 115.88 days. Thus there are 365.24/115.88 conjunctions in a year, or 22 conjunctions in 7 years, which is very close to Pi!
See more relationships at the Solar Geometry site.
Relative planetary distances average to Phi
The average of the mean orbital distances of each successive planet in relation to the one before it approximates phi:
Planet

Mean distance in million kilometers per NASA

Relative mean distance where Mercury=1

Mercury

57.91

1.00000

Venus

108.21

1.86859

Earth

149.60

1.38250

Mars

227.92

1.52353

Ceres

413.79

1.81552

Jupiter

778.57

1.88154

Saturn

1,433.53

1.84123

Uranus

2,872.46

2.00377

Neptune

4,495.06

1.56488

Pluto

5,869.66

1.30580

Total


16.18736

Average


1.61874

Phi


1.61803

Degree of variance

(0.00043)

Note: We sometimes forget about the asteroids when thinking of the planets in our solar system. Ceres, the largest asteroid, is nearly spherical, comprises over onethird the total mass of all the asteroids and is thus the best of these minor planets to represent the asteroid belt. (Insight on mean orbital distances contributed by Robert Bartlett.)
2005 unveiled the discovery of a 10th planet called 2003UB313. It was found at a distance of 97 times that of the Earth from the Sun. Its ratio to Pluto would thus be 2.47224, much higher than any previous planet to planet orbital distance ratio. Could it be that this is actually the 11th planet and the 10th planet will be found at an orbit whose ratio is 1.52793 times that of Pluto, preserving the phi average? Time will only tell, but if it happens remember that you heard it here first.
The shape of the Universe itself is a dodecahedron based on Phi
New findings in 2003 based on the study of data from NASA's Wilkinson Microwave Anisotropy Probe (WMAP) on cosmic background radiation reveal that the universe is finite and shaped like a dodecahedron, a geometric shape based on pentagons, which are based on phi. The the Universe page for more.