Golden Ratio Gallery
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The objects here are all golden ratio (GR) objects. The rods have GR major/minor radii elliptical cross-section. Similarly for the parabloids and hyperboloids. The multi-colored wall hanging is like the “Garden Sculpture” and is made of golden rectangles.
Start with 0 and 1, add them together to get 1, giving the sequence 0 1 1. Add the last two in the sequence together, giving 0 1 1 2. Again, add the last two, giving 0 1 1 2 3. This procedure give you the Fibonacci series: 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 ...
The last two numbers have the ratio 1.62 and if you carry it on to very large numbers, the ratio of the last two is (sqrt(5)+1)/2 or 1.618033989...(Golden Ratio)
One dimension (1-D): divide a line in two parts such that the ratio of lengths of the long segment to the short segment is equal to the length of the line divided by the length of the long segment. The ratios are the golden ratio (GR).
1/GR = GR - 1 = 0.61803...
GR * GR = 1 + GR = 2.61803...
GR * GR - GR - 1 = 0 (and, technically, (1-sqrt(5))/2 = -0.61803 is a solution also).
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Rosie’s Angels
Don Lacy had this made from sassafras left over from building his house.
I call the genre “wall furniture”, a new art form.
The inside of the large golden rectangle and the outsides of the 2 small golden rectangles have a 2:1 ratio. Thus, the angle of the small tilted one can be shown to be 38.17 degrees with respect to the bottom.
The thicknesses of each piece is 1/11.090 times the largest side. 11.090 is GR^5 (shorthand for golden ratio to the fifth power).
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GBox1GoldSphereYellow
An 11x17 giclee print entered in BAA 2006 Member’s Show. (I didn’t make the GoggleWorks Show.)
The gold color and the golden ratio boxes are the important features. I used the same color method in “Reflections From a Gold Sphere”
There is a marble-textured version hanging on the wall in “AAA Studio” (studio.jpg).
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Lacy Red 1
Instructions from which Don Lacy made the wooden object (see “Rosie’s Angels”).
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Garden Sculpture
Each piece is a golden rectangle.
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Golden Spiral
2-D Golden Spiral: spiral increases in size by GR every 90 degrees. The outer spiral is 90 degrees ahead of the inner spiral.
This semi-abstract sold at the Reading Museum Show in 1998.
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Golden Squares
Golden rectangle sections are added in a complex sequence. 2-D sine wave patterns done by a program were added for decoration.
The main square was repeated in a reduced square spiral.
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Cow & Cat Jumped Over The Moon
Golden spirals, a large one for the cow and a small one for the cat, and whose sections are “reversed” left-to-right and top-to-bottom every 90 degrees. I call this a “golden flake”.
The main square is duplicated in a reduced square golden spiral.
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Rocket
Connected golden ratio rectangles: the model for the “Garden Sculpture”.
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Fiboshells
Fibonacci shells “welded” together (some fibonacci stacks use gravity).
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Christmas Cone
The cone has a GR height/base ratio. The colored spheres are on a GR spiral. The brown truncated cone base is similarly formed.
The star consists of 8 golden triangles.
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Mirror Test
Reduced cube golden spiral.
The smaller cubes are being viewed in one mirrored side of the large cube.
Start with a 2-D reduced square golden spiral and expand the squares into cubes above and below the plane (see “Reduced-Square Golden Ratio Spiral”).
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Fibo Block 1 Tan Red Yelow Map
Same as “Fibo Grav 1”, except the large glass block has been turned to the top to enhance refraction. The “hole” in the top is not physical but is refraction.
Photos of the photo are on the walls.
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Fibo Irregular Grav4
Silver, glass and checkered fibo blocks using gravity.
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Fibo Grav 1
Two silver & two glass fibo blocks using gravity.
A “fibo block” is made of sides in the ratios of 3 consecutive terms in the fibonacci series. I generally use 5 8 13, 8 13 21, 13 21 34. (See “AAA Studio” for the fibonacci series explanation.)
The exact golden ratio could be used, in which case the ratios would be 1:1.61803:2.61803. Integers are nice to work with in drawings and “fibo block” has a nice ring to it.
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Dodecahedron
A dodecahedron has 12 pentagon sides. A pentagon is made of 3 golden triangles.
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Pyramid
A golden pyramid is a pentagon-based star with the tips folded until they touch.
One side is left off here to view a star in the center.
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3D Golden Spiral
Two pairs of 2-D golden spirals, in sixteen segments, are connected by golden rectangles based on the length of each segment.
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Fibo Flush Grav 1
Silver & glass fibo blocks using gravity.
The glass blocks are dark with subtle color changes due to internal reflections.
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Reduced-Square Golden Ratio Spiral
Start with a GR rectangle (large black). Cut off a square (black), leaving a (small) black GR rectangle. Draw crossing red diagonals on the large and small black GR rectangles. The crossing point defines the green rectangle which can be shown to be a GR rectangle. Continuing the process, we get the green square, blue square and orange square which form a reduced square spiral with the original black square. (A reduced cube golden spiral is made by expanding squares into cubes, half of each cube above and below the plane of the squares. See “Mirror Test”)
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18-54 POVRay Golden Prism
This is a POVRay example. The top (green) side is removed to see a reference centered red cube and its shadow. The colors are the same as the MPF example except half of each face is a goldenrod color. It uses data transformed from the Wings 3D model.
Front (Y-): Orange
Back (Y+): Yellow
Left (X-): Red
Right (X+): Blue
Bottom (Z-): Purple
Top (Z+): Green
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18-54 Golden Prism Layout
This layout was made using the 18 small and 54 large templates shown in “Golden Parallelograms” (18s_54s_18l_54l.png). Tracing this on to thin cardboard, you can make 4 sides of a 18_54 golden prism. (You must turn the templates over for the 3rd and 4th sides.) The remaining 2 sides are rectangles with one long side and one short side.
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Golden Parallelograms
These are the four golden parallelogram templates made of two golden triangles each. Each type, 18 or 54, is in the ratio 1.61803:1. These can be used to make physical models of the 4 types of golden prisms shown in “Golden Prism Models” (models.jpg). “18-54 Golden Prism Layout” (18_54_layout.png) is one example.
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Golden Prism Models
Shown is a photo of the four cardboard models of golden prisms.
Two of the three pairs of faces for each golden prism is made of pairs of golden triangles. The description number is the unique half-angle (in degrees) of the 2 golden triangles making up the parallelogram face, 18 for the “acute” golden triangle (with angles 36, 72 and 72 degrees) and 54 for the “obtuse” golden triangle (with angles 108, 36 and 36 degrees).
The first number describes the smaller face with sides GR and one. The second number describes the larger face with sides GR^2 and GR. (GR = 1.61803... and GR^2 = 2.61803...)
The third pair of faces were picked to be rectangles with sides having a GR^2 ratio for 18-18, 18-54 and 54-18 prisms and picked to be a 36-144-36-144 parallelogram for the 54-54 prism (with the sides also having a GR^2 ratio). A rectangle is not an option for the latter case.
See “18-54 Golden Prism Layout” (18_54_layout.png) and “Golden Parallelograms” (18s_54s_18l_54l.png) for examples of the sides with a GR ratio.)
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18-54 Golden Prism (WPF)
This is an example of a 18-54 Golden Prism model rendered using Microsoft .NET’s Windows Presentation Foundation (WPF). It uses data transformed from the Wings 3D model.
Front (Y-): Orange
Back (Y+): Yellow
Left (X-): Red
Right (X+): Blue
Bottom (Z-): Purple
Top (Z+): Green
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Gold Prisms (Wings 3D)
This is a composite of screen captures from 3D modeling program Wings 3D. Eight x,y,z points describe each prism. (Awk programs were written to convert these points into triangles or faces for Povray scripts and WPF XAML files.)
The composite views are similar to those in “Golden Prism Models” (models.jpg).
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