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 and .

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 . This equation can be interpreted as a shortened version of the metallic mean equation (below).

Phi is part of a sequence called the metallic means (wiki link). Where n = any number, use 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 diameter of the circles are obtained using the equation:
- The y-offset (circle centers) are obtained using the equation:
- The metallic mean value is the farthest point of the circle. Use the equation:

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.