Geometric Forms for the Collage of Backgrounds 
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While my earlier collages were figurative with a clear visual distinction between foreground objects and background, my work since 1997 has been exploring a new concept that I call The Collage of Backgrounds. This technique basically arranges natural abstract images  what would be considered backgrounds in my figurative collages  according to simple geometric rules based on Sacred Geometry and the Golden Ratio. Knowledge of the Golden Section, ratio, or proportion has been known for a very long time. The Egyptians knew about it and the Greeks learned about it from them. It is called phi, Φ, in honor of the sculptor Phidias, one of the architects of the Parthenon, and is approximated by the irrational fraction 0.6180339... Φ shows up throughout nature and represents one of the most fundamental numerical constants of the universe. For example, the finger bones are in approximate Φ ratio to each other, and the position of features on the human body is related to Φ.
The square is the fundamental geometric form used in my work, and squares combined with GR's in Golden Ratio proportion can create a wide range of possible rectangles and image aspect ratios. The following is an explanation and visual guide to the Golden Ratioderived shapes and geometric forms that I use in my collages, montages, and artwork in other media. You may aslo download this information as an Acrobat PDF file.
The Golden Rectangle and Proportional Squares 

The Golden rectangle (GR) and squares in Golden Ratio to the GR are the basic forms that I use for all of my artwork. The GR is a rectangle with an aspect ratio of 1 to 1.618, or 1 to (1+Φ).  
Coiled Golden Rectangles 

Below we see an interesting property of GR's that inspired one of my basic Collage of Backgrounds forms  the Coiled Golden Rectangle. When you cut out a square starting from one of the short sides of a GR (the gray square in GR #2 below), the square divides the long side at the Golden Ratio, and you will be left with another GR at a 90° counterclockwise angle (the smaller vertical white GR in #2 on the right). You can continue to cut out short side squares for each successively smaller GR (as in GR #3) and another smaller white GR will remain  once again rotated 90° counterclockwise. This process may continue leaving a smaller proportional GR as seen in GR #4. 1 2 3 4 

Below is a GR subdivided by Goldenproportional squares that converges on a point called the Eye of God  and hence the notion of a Coil. A superimposed logarithmic spiral also converges on this point, as does the intersection of diagonals (red lines, called radicals) from the primary GR, and the 1st subordinate GR. 
Below is a collage from the Elements in Golden Ratio series. This series of 84 collages explores the classical elements of earth, air, fire, water, and life (the 5th element that combines all 4 basic elements). The GR's coils from this series are also used as collage elements in the square root of five and mandala forms. 
Square Forms: The Offset Proportional Square and Mandala 

Below is a form diagram showing a square with three proportionally smaller squares I call an Offset Proportional Square. Each smaller square is in Golden Ratio to the next larger square. 
Below  AIR: Elements Offset Proportional Square 2004003, a collage of backgrounds combining images of earth, air, fire, and water. I used this particular idea as an organizing form and as a collaged element for other work in the Elements in Golden Ratio series. © 2008 Doug Craft 
The mandala is another of my square forms that uses a central square surrounded by proportional GR's. Mandalas are round or square designs that focus the viewer's attention on the center. As such they are often used as visual meditation objects. Below is the mandala form diagram showing how each of the peripheral GR's shares the same central square, forming four larger overlapping Golden Rectangles. The double square, a form thought sacred by the Egyptians, may also be found in the mandala form diagram. 

One Plus 3(Φ) Forms 
Left  These forms are based on a horizontal Golden Rectangle with two proportionally smaller vertical GR's on the ends, or a square with 3 vertical GR's joined together. Proportional diamonds (orange squares rotated 45°) are centered and overlaid on the background rectangle. The aspect ratio of rectangle is 1 to (1 + 3Φ), or 1 to 2.854. The overlaid diamonds create a bold angular break with the rectangular background. I have called these forms Seals (as in emblems or logos) and created an exploratory series of collages called Seals of the Great Kings using this format. Below is an example collage, WATER: Elements 1 + 3Φ 2004001, from the Elements in Golden Ratio Series. 

Tandem Golden Rectangles 

A Golden Ratio form used  unknowingly  in my earliest collages was the joining of two vertical GR's in tandem. My collages actually used the 8" x 10" format popular in photographic printing. Only later did I realize this size contained the Golden Ratio by combining two 5" x 8" Fibonacci number GR's. 
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