# A from Golden Ratio With B

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The Golden Ratio has been used for thousands of years in architecture, engineering and art to define an aesthetically pleasing ratio of sizes and other attributes.

This equation takes input length B, assumed to be the smaller of two quantities in the Golden Ratio, and reports what  the value A should be in that ratio.

Since phi = A/B, where A > B > 0,

phi is computed as phi = (1 + sqrt(5))/2

Then A = phi * B

# Notes

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

## INTERESTING ADDITIONAL CHARACTERISTICS OF A GOLDEN RATIO EXAMPLE

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.