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 Ratio-derived 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).

Benzoic Acid Crystal 2006-036, © 2008 Doug Craft

Mancos Shale, Utah, 2004-036, © 2008 Doug Craft
Image of Artwork by Doug Craft


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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° counter-clockwise 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° counter-clockwise. This process may continue leaving a smaller proportional GR as seen in GR #4.

1 Image of Artwork by Doug Craft 2Image of Artwork by Doug Craft 3Image of Artwork by Doug Craft  4Image of Artwork by Doug Craft

Below is a GR subdivided by Golden-proportional 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.

Image of Golden Rectangle -based form used by artist Doug Craft

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.

AIR: Golden Rectangle Coil 2004-004,© 2008 Doug Craft
Image of collage Golden Rectangle by artist Doug Craft.


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The Square Root of 5 Ratio and Rectangle

The figure below shows that every GR can coil in 4 different directions and has 4 Eyes of God. The 4 Eyes of God form the vertices of the tan GR in the center with sides in 1 to 2.236, or square root of five ratio to the larger GR. On either side of the center GR we see 2 vertical green rectangles - and these have an aspect ratio of 1: 2.236. I call these green rectangles Square Root of 5 Rectangles.

Below we see an interesting property of GR's when you out a square starting from one of the short sides of the GR (the gray square in the 2nd GR below), you will be left with another GR at a 90° angle (the vertical white GR to the right). You can continue to cut out short side squares for each successively smaller GR and another smaller GR will remain as seen in the 2 right-hand GRs..

Image showing the Square Root of 5 Rectangle by Doug Craft

The diagram above shows how the Square Root of 5 rectangle is formed by adding 2 GRs to the central square. You can also see that this rectangle is formed by 2 GRs that share a central square.

AIR - Square Root of 5 Collage 2004-003, © 2008 Doug Craft
Collage by artist Doug Craft

Image of panorama photo by artist Doug Craft


Double Arch, Arches National Park, Utah,

Square Root of 5 Panorama 2003-021,
© 2008 Doug Craft

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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 SquareEach smaller square is in Golden Ratio to the next larger square.

A square divide into Golden Ratio proportional squares by artist Doug Craft

Below - AIR: Elements Offset Proportional Square 2004-003, 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

Offser Proportional Square colles by artist 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.

A square divived into Golden Proportions and usd by Artist Doug Craft for Mandala collage forms


Below
is an example mandala collage, EARTH: Elements Mandala 2004-004.  Note the use of offset proportional squares in the corner positions. © 2008 Doug Craft

Mandala collage by artist Doug Craft based on the Golden Ratio

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One Plus 3(Φ) Forms

Image template for 1 + 2Phi collage formes by artist Doug CraftLeft - 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 2004-001, from the Elements in Golden Ratio Series.


One plus 2Phi collage by artist Doug Craft


EARTH - Elements One + 3(Phi) 2004-003,
© 2008 Doug Craft

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Double and Triple Golden Rectangles

These wide aspect ratio forms are primarily used when I create panorama images, either traditional landscape panoramas, or triple Golden Rectangle photomontages:

The double Golden Rectangle (below) has an aspect ratio of 1 to (2 + 2Φ), or 1 to 3.236.  Oddly, this aspect ratio can also be described as 1 to (1 + square root of 5)!
Images showing double Golden Rectangle template used by Artist Doug Craft

The triple Golden Rectangle form diagram with an aspect ratio of 1 to (3 + 3Φ), or 1 to 4.854:
Image of triple golden rectangle template used by artist Doug Craft

Lake Powell Inflow, Utah, Double Golden Rectangle Panorama 2003-013, © 2008 Doug Craft
Double Golden Rectangle panorama image by artists Doug Craft

Microcosmos Number 1, Triple Golden Rectangle Photomontage, 2002-001, © 2008 Doug Craft
Triple Golden Rectangle Photomontage by artist Doug Craft

Mount Evans, Colorado, Triple Golden Rectangle Panorama 2003-027, © 2008 Doug Craft
Triple Golden Rectangle panorama image by artist Doug Craft

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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.

Image of tandem Golden Rectangles used in collages by artist Doug Craft

Agricultural Suspension, collage, 1980, © 2008 Doug Craft
Image of tandem Golden Rectangle collage by artist Doug Craft

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