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`phi = (1 + sqrt(5))/2`

1.618033988749895

The **Golden Ratio** can be defined explicitly by looking at two objects whose sizes conform to a unique ratio. Two quantities, A and B, are in golden ratio if the ratio of A/B (where A>B) is the same as the ratio of their sum, ( A+B), to the larger of the two, namely A.

The ratio is made explicit by:

`(A+B)/A` = `A/B`

Many examples in architecture have applied the Golden Ratio to achieve what many consider an aesthetic perfection. Most notably, numerous of the ancient Greek structures, such as the Parthenon, employed the Golden Ratio in their design. Interior designers toady may likewise use the Golden Ratio as a design element.

We think of it most commonly in defining ratios of rectangular length and width; however, the Golden ratio has been applied in many geometrical application to include regular pentagons, decagons, as well as the polyhedron, the dodecahedron.^{(1)}

^{(1) }a dodecahedron is a regular polyhedron with twelve faces, where all the faces are regular pentagons.

DERIVING the GOLDEN RATIO ==

Here is the expression that gives rise to the golden ratio:

`phi` = `(A+B) / A` = `A/B`

This gives us:

`(A+B) / A` = 1 + `B/A` = 1 + `1/phi`

So, 1 + `1/phi` = `phi`

Multiply both sides by `phi`:

`phi * (1 + 1/phi)` = `phi * phi`

`phi` + 1 = `phi^2`

rearrange that equation to give: `phi^2 - phi - 1 = 0`

Solve using the quadratic formula and you get:

`phi = (1 +- sqrt(5))/2`

Since the ratio is intended to be the ratio of positive things, like lengths, only the term that results in a positive value is relevant, so,

`phi` = `(1 + sqrt(5))/2`

The Golden Ratio can also be expressed as the infinite series:

`phi = 13/8 + sum_(n=0)^infty((-1)^(n+1)(2n+1)!) / ((n+2)!* n! * 4^(2n+3))`

The Golden Ratio is seen in nature in geometries that are considered aesthetically pleasing, like the shell of the Nautilus, in crystals (seen in atomic structures such as the magnetic resonance of spins in cobalt niobate crystals)

As you can see in the picture, applying the Golden Ratio to a shape that in this example looks like the shell of a Nautilus has very unique geometric relationships. Each of the squares are in the Golden Ratio with the next smaller square. So the ratio of the sides of the square (abch) to square (cdei) is equal to `phi`.

Looking at the ratios of lengths of the squares:

`phi` = `"ab"/"ic" = "ic" / "ef" = "ef" / "gh"` … and on and on for adjacent squares in the shape continuing to get smaller and smaller and continuing to infinitely smaller squares.

Another interesting attribute of this shape is that all of the rectangles in the geometric structure are themselves rectangles whose sides are in Golden Ratio:

`phi` = `"bd"/"ab" = "df" / "hf" = "ie"/"ih" = "ih"/"hg"`… and on and on for all successively smaller rectangles in the geometric structure, continuing on to infinitely small rectangles.

And finally it is interesting to note that even though the rectangles spiral inward, the two diagonal blue lines intersect the corners of every rectangle in the picture. That includes all the rectangles as they continue towards infinitely small rectangles.