  # Twelve and the Golden Ratio While writing my first book Interference, I discovered that the cardioid geometries found in humans and many animals can be described in terms of harmonic resonance. There is even a simple formula to describe this shape that balances a standing wave (π-resonance) with a golden spiral (Φ-damping).

We can describe this balance between π and Φ as:

f(x) = Wave / Spiral

f(x) = Res(x) / Dmp(x)

f(x) = x^2 / (Φ^x / √5), where Φ = 1.618033

This equation applied to the set of natural numbers produces a wave known as a first-derivative Gaussian distribution. It represents the change in velocity of a Gaussian "normal" distribution, which is found throughout nature and is the founding idea behind modern statistics. For instance, It is popularly known as the "Bell curve," which describes the distribution of intelligence in a population.

Anyway, when we plug the number x = 12 into the equation, this is what we find:

f(12) = 12^2 / (Φ^12 / √5)

f(12) = 144 / (321.996894 / √5)

f(12) = 144 / 144.001388 ≈ 1

Other than 1, the number 12 is the only number that balances between the harmonic series {1,2,3,...} and Fibonacci series {1, 1, 2, 3, 5, ...}, as approximated in the equation by the 12th power of the golden ratio divided by the square-root of five.

We can see this unique alignment at 12-squared or 144 in the two sequences:

Harmonic Series squared (a standing wave):
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196,...

Fibonacci Series (a spiral):
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...

The only two identical numbers in these two sequences is 144 or 12-squared. Thus, the square of 12 is the only point of intersection between a harmonically resonating standing wave and the base-Φ logarithmic spiral known as a golden spiral.

This is the same relationship represented in a dodecahedron which has twelve pentagonal sides. More importantly, it is the relationship found in the geometric form of a human embryo. Life is a balance between resonance and damping just like twelve.

As for today's date of 12/12/12, we can divide these as 12/12/12 = 0.083333333 which is cognate with 5/6 = 0.833333333 and 6/5 = 1.2. Now if that isn't a brain twister, nothing is.