Divine Proportion

When I first started sculpting marble, I noticed a phenomenon happening as I took repeated measurements.  I noticed patterns first appearing on the plaster then emerging in the marbles. They would appear as triangles first, then pentagrams emerged.  It was a fascinating lesson to be learned.

Notice with each the Lines of Coincidence, where things converge along the lines.  And at the ends of the pentagrams. Those are the Points of Coincidence.

The Pentagon and the Golden Triangle

Is the perfect ratio for producing logarithmic spirals. In a rectangle where the ratio of the larger side to the smaller one is the Divine Proportion.

Pentagon & the Golden Triangle

In constructions of tiled pentagons, every segment is smaller than its predecessor by a factor equal to the Divine Proportion.

(the ratio of "a" to "b" is the Divine Proportion, the ratio of "b" to "c" is the Divine Proportion, and so on.

The Golden Rectangle

In a rectangle where the ratio of the larger side to the smaller one is the Divine Proportion, the ratio of the sides of the "daughter rectangles" will still conform to the Divine Proportion when squares are cut from the original rectangle. Connecting the points where this series of "whirling squares" divides the sides will generate a logarithmic spiral that coils inward.