As shown on the Distance Calculation page, the distance of the Earth from the Sun, where Mercury is 1, is computed as follows:
((½ ( √3 + 1 )) ^ (½ ( √4 + 1 ))) * (½ ( √5 + 1 ))
Note: √x indicates the square root of x
There is something quite incredible about this derivation for Earth.
Note the repeating term ½ ( √x + 1 ) with 3, 4, and 5
First, it is quite amazing that the same form of:
½ ( √x + 1 )
would appear in three successive terms using the integers 3, 4, and 5 for x.
3, 4 and 5 are, of course, the key integers used in the most elementary example of the Pythagorean theory for right triangles, where
3² + 4² = 5².
The third term, ½ ( √5 + 1 ), is Phi!
More incredible yet is that the third term, ½ ( √5 + 1 ), is none other than the ubiquitous number 1.6180339…, better known as phi, or Φ. Phi has mathematical properties unlike any other number. The reciprocal of Φ is Φ-1, or 0.6180339… . When a line is divided at 1/Φ, the ratio of the small section to the large section will be identical to the ratio of the large section to the entire line.
This results in the incredible relationship of 1 and Φ between Venus (to the ¾ power) and Earth:
Phi appears in the design of many life forms!
Phi is the number behind the Golden Section, or Divine Proportion, and the Fibonacci series. It appears throughout in living organisms in everything from every key dimension of the human body, to the spirals of sea shells, to the arrangements of stems on a plant.
See the Links page for more on this.
Similarities in the derivation of Phi and the Earth’s distance.
It is also interesting that the derivation of Phi is so similar to the derivation of the distance of the Earth from the Sun. Consider the similarity in the formulaic construction, where the first two terms use an exponent and then a multiplier.
Phi is derived from:
Φ = 5 ^ .5 * .5 + .5 = 1.6180339…
While the distance of the Earth from the Sun, where Mercury = 1 is:
½ ( √3 + 1 ) ^ ½ ( √4 + 1 ) * ½ ( √5 + 1 ) = 2.583306…
The Earth embodies the two treasures of geometry
It’s as though Earth embodies the Pythagorean theory and the Golden Section in one beautiful construction.
Perhaps Johannes Kepler (Mathematician and Astronomer, 1571-1630) had greater insight into the universe than we may have suspected when he said:
Geometry has two great treasures: one the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.”
The ordinances of the LORD are sure and altogether righteous. They are more precious than gold. Psalms 19:9-10