Last update: July 18 2013

Phi and the Metallic Mean Ratios


Anyone who's interested enough in math or geometry will eventually come across phi (aka the golden ratio). Like many who came before me, I was captivated by this particular ratio's natural beauty. As an artist I would often use phi and powers of phi in my artwork. As a result I occasionally discover something new about it.


 

 

Phi as it relates to the root grid.

 

Note how the intersecting circles which describe phi pass through square root of 2 and the square root of 1/2.

 

Also, it's hard to notice by looking at it, but the diameter of the yellow circles is the square root of 5. I mention this because a common way of deriving phi is the square root of 5 plus 1, divided by 2. This equation can be interpreted as a shortened version of the metallic mean equation (below).



silver mean ratios

Phi is part of a sequence called the metallic means (wiki link). Where n = any number, use n plus the square root of n + 4 divided by 2 to obtain the ratio. If n = 1, the result is phi.

Here's a slightly more detailed approach to the equation including details in how it relates to the geometry of the root grid:

 

  • First choose a value for cosine. Anything between 0 and 1 will do.
  • Then we get the tangent from cosine using the equation: the square root of 1 divided by cosine to the power of 2 minus 1
  • The diameter of the circles are obtained using the equation: the square root of tangent to the power of 4 plus 4
  • The y-offset (circle centers) are obtained using the equation: tangent to the power of 2 divided by 2
  • The metallic mean value is the farthest point of the circle. Use the equation: tangent to the power of 2, plus the square root of tangent to the power of 4 plus 4, divided by 2

 

Below you can see how the circles which describe the metallic means intersect with the square roots of the natural numbers and their reciprocals. All measurements are from 0.

 

square roots and silver mean ratios