Dynamic Symmetry with Two PHI DS 4 to 5
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- Dynamic Symmetry with Two PHI DS 4 to 5
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4 to 5 rectangle (AR 1.25) with two vertical PHI rectangles (AR .618). 1.25 ~ .618+.618 = 1.236
Dynamic Symmetry with Two PHI DS 4 to 5
Basic Dynamic Symmetry Armature
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- Basic Dynamic Symmetry Armature
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Reciprocals: (Turquoise ) lines from corners to the top or bottom edge, perpendicular to the main diagonals (at right angles or 90 degrees)
Verticals and Horizontals: (Gray) through the intersections of the main diagonals with their reciprocals
Intersections of the diagonals with their reciprocals: potential focal points (sometimes called 'eyes') since the right angle ensures maximum thrust or movement
This is the construction for a generic rectangle; each aspect ratio gives rise to different points of intersection on the top edge.
See Full Armature Page:
Expanded Dynamic Symmetry Armature
Basic Dynamic Symmetry Armature
Dynamic Symmetry Armature 5:4 (1.25)
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- Dynamic Symmetry Armature 5:4 (1.25)
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- Aspect Ratio 5:4; eg, 8x10, 16x20 popular panel sizes
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The Basic Dynamic Symmetry Armature for panels with sizes, 8x10, 16x20 and any others with the aspect ration 5:4. The aspect ratio of 5:4 is so close to that of 14:11 that I might use this armature in the field for an 11x14 panel.
Dynamic Symmetry Armature 5:4 (1.25) Aspect Ratio 5:4; eg, 8x10, 16x20 popular panel sizes
Dynamic Symmetry Armature Root PHI (1.272..)
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- Dynamic Symmetry Armature Root PHI (1.272..)
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The aspect ratio of 11x14 is 14/11 = 1.2727..., close to the AR of Root PHI. Note how close this aspect ratio is to that of our common x10 or 16x20 at 1.25.
Dynamic Symmetry Armature Root PHI (1.272..)
Basic Dynamic Symmetry Armature Construction Step 1
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- Basic Dynamic Symmetry Armature Construction Step 1
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Step 1. Construct main diagonals from corner to corner. The diagonal leading from the top left corner down to the lower right corner is called the "Sinister" diagonal. The diagonal leading from the lower left corner up to the upper right corner is titled, the "Baroque" diagonal. (We also include one of the reciprocals here: the line perpendicular to a main diagonal from the opposite corner).
Basic Dynamic Symmetry Armature Construction Step 1
Dynamic Symmetry Armature 4:3 (1.333..)
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- Dynamic Symmetry Armature 4:3 (1.333..)
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- Aspect Ratio 4:3; eg, 6x8, 9x12, 12x16, 18x24
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The Basic Dynamic Symmetry Armature for panels with sizes, 9x12, 12x16, 18x24 and any others with the aspect ratio 4:3.
Dynamic Symmetry Armature 4:3 (1.333..) Aspect Ratio 4:3; eg, 6x8, 9x12, 12x16, 18x24
Dynamic Symmetry Armature Root 2 (1.414..)
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- Dynamic Symmetry Armature Root 2 (1.414..)
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- Root 2 with Aspect ratio 1.414...
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The vertical line dropped from the point where a reciprocal meets the top edge, marks off a vertical rectangle with the same aspect ratio as the whole rectangle (proved elsewhere by similar triangles on my blog for the general case). Just consider the right hand vertical rectangle for this discussion. Assume the large rectangle is 1xRoot 2 with the top edge of width Root 2 and the right edge of height 1 (with aspect ratio of small to large of 1 to Root 2). Let x be the distance between the upper right corner and the point on the top edge where the reciprocal meets it. Then this smaller vertical rectangle on the right has aspect ratio x:1 (smaller to larger). Since their aspect ratios are equal, we have x/1 = 1/Root 2 or x = Root 2/Root 2*Root2 = Root 2/ 2. So x = half of Root 2 or half the width of the rectangle. Root 2 is the Aspect Ratio for European A5 sizes, due to this characteristic of dividing the original rectangle in half and having the same proportions. We can fold papers in half again and again and continue to get sizes that are still in the A5 proportion. The aspect ratio of 11x14 is 14/11 = 1.2727..., close to the AR of Root PHI. Note how close this aspect ratio is to that of our common x10 or 16x20 at 1.25.
Dynamic Symmetry Armature Root 2 (1.414..) Root 2 with Aspect ratio 1.414...
Basic Dynamic Symmetry Armature Construction Step 2: Reciprocals
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- Basic Dynamic Symmetry Armature Construction Step 2: Reciprocals
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Step 2. Construct lines from each corner perpendicular (right angles aka 90 degree angles) through the main diagonals. These are called reciprocals. The intersections may be seen as good candidates for focal points since they have lines emanating from them at right angles. to each other on the diagonal. They are more dynamic than a checkerboard grid static in appearance.
Basic Dynamic Symmetry Armature Construction Step 2: Reciprocals
PHI Dynamic Symmetry Armature with a Golden Spiral Approximation (1.618..)
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- PHI Dynamic Symmetry Armature with a Golden Spiral Approximation (1.618..)
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- PHI Aspect Ratio
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Expanded+ Armature for our "Golden" paintings (or near-golden), superimposing the spiral onto the Expanded Dynamic Symmetry Armature. The golden spiral is approximated by the Fibonacci or Whirling Squares' Spiral. The whirling squares are constructed within rectangles as they rotate around a point which is sometimes denoted the eye of the spiral (in this case at the upper right). Note in passing that the squares formed here have side lengths, successive Fibonacci terms: 0,1,1,2,3,5,8,13 (in reverse), etc, where each term is the sum of the previous two terms.
Nice Diagram Here.
The ratios of successive Fibonacci terms gets closer and closer to the Golden Ratio (1.168...): 2/1 = 2, 3/2 = 1.5, 5/3 - 1.666.., 8/5 = 1.6, 13/8=1.625, 21/13=1.615..., 34/21=1.619..., 55/34=1.6176..,
Read More
For this expanded+ armature, the intersection of the main diagonal with its reciprocal coincides with the 'eye' or polar point of the spiral. We used this armature for the near-golden rectangular paintings in this exhibition.
The Golden Ratio has been hailed since antiquity by artists such as Leonardo Da Vinci as the most beautiful, perhaps because it can be seen throughout nature (galaxy and fern spirals, Nautilus Shell, the human body, etc.).
PHI Dynamic Symmetry Armature with a Golden Spiral Approximation (1.618..) PHI Aspect Ratio
Dynamic Symmetry Armature Golden Mean PHI (1.618...)
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- Dynamic Symmetry Armature Golden Mean PHI (1.618...)
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Basic Dynamic Symmetry Armature for panels with aspect ratio close to the Golden Mean of 1.618.. such as 10x16. The Golden Mean is defined by the proportion (a+b)/a = a/b and can be shown to be the number 1.618... where ... indicates continuing forever. It is denoted PHI. So the Golden Rectangle is also known as the PHI rectangle.
Dynamic Symmetry Armature Golden Mean PHI (1.618...)
Basic Dynamic Symmetry Armature Final Step
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- Basic Dynamic Symmetry Armature Final Step
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Basic Dynamic Symmetry Armature Final Step. horizontals and verticals through the intersections of the baroque and sinister diagonals with their reciprocals. The reciprocals to a diagonal are the lines from the corners at right angles with the diagonal. Baroque Diagonal: from lower left up to the upper right corner; Sinister Diagonal from the upper left to the lower right corner).
Basic Dynamic Symmetry Armature Final Step
Dynamic Symmetry Armature Root 3 (1.732..)
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- Dynamic Symmetry Armature Root 3 (1.732..)
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- Root 3 Aspect Ratio 1.732..
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Basic Dynamic Symmetry Armature for the Root 3 rectangles. Compound rectangles with enclosed Root 3 rectangles were popular with master painters.
Dynamic Symmetry Armature Root 3 (1.732..) Root 3 Aspect Ratio 1.732..
Dynamic Symmetry Armature Root 4 (2.0)
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- Dynamic Symmetry Armature Root 4 (2.0)
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- 1x2 Aspect Ration Root 2
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Basic Dynamic Symmetry Armature for the Root 4 rectangles. A popular 2 to 1 panel size; eg, 8x16, 12x24, 10x20.
Dynamic Symmetry Armature Root 4 (2.0) 1x2 Aspect Ration Root 2
Dynamic Symmetry Armature Root 5 (2.236..)
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- Dynamic Symmetry Armature Root 5 (2.236..)
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- Root 5 Aspect Ratio 2.236...eg; 10x22
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Basic Dynamic Symmetry Armature for the Root 5 rectangles. (Note: Root 5 = 2.236...). 10x22 or 5x11 would approximate this aspect ratio with 2.20 while the Root 5 panel has aspect ratio 2.236. The importance of the Root 5 proportion lies in its connections with the formula, PHI = (1+ Root 5)/2., so that PHI is the arithmetic mean of 1 and Root 5.
Dynamic Symmetry Armature Root 5 (2.236..) Root 5 Aspect Ratio 2.236...eg; 10x22
Dynamic Symmetry Armature with Similar Rectangles
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- Dynamic Symmetry Armature with Similar Rectangles
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Similar Rectangles (babies) in turquoise:Turquoise verticals are dropped from the edges as indicated. The reciprocals are now dashed, representing diagonals of the constructed vertical similar rectangles, baby rectangles inheriting the same proportion as the main parent rectangle,..
Dynamic Symmetry Armature with Similar Rectangles
Dynamic Symmetry Armature with Rebated Squares
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- Dynamic Symmetry Armature with Rebated Squares
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Rebated Squares: The pink sold lines are the sides of squares formed from the main rectangle plus a pink vertical. The dashed pink lines are the diagonals of the squares. Note that the turquoise reciprocals remain in this digram. They are used for the Similar Rectangle diagram, however.
Dynamic Symmetry Armature with Rebated Squares
Expanded Dynamic Symmetry Armature (with right Rebated Square)
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- Expanded Dynamic Symmetry Armature (with right Rebated Square)
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The Expanded Dynamic Symmetry Armature begins with the Basic Dynamic Symmetry Armature. Rebated squares are formed as follows to provide more structure for designing, when additional information is desired. .
The right rebated square in dashed red is formed by dropping a vertical from the top so that its top edge is the same length as the main rectangle's right edge.. Then the square's diagonals are drawn. The same process occurs on the left.
Sometimes the shorter term, Rebated Squares, is used, but another term is "Rabatment of the rectangle". The rebated square technique has a long history as a design tool. Read More.
Reciprocals: (Turquoise ) lines from corners through main diagonals, perpendicular to the diagonals (at right angles or 90 degrees)
Verticals and Horizontals: (Gray) through the intersections of the main diagonals with their reciprocals
Intersections of the diagonals with their reciprocals: potential focal points (sometimes called 'eyes') since the right angle ensures maximum thrust or movement
Basic Dynamic Symmetry Armature
Expanded Dynamic Symmetry Armature (with right Rebated Square)
Expanded Dynamic Symmetry Armature
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- Expanded Dynamic Symmetry Armature
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Combined Rebated Squares and Similar (baby) rectangles.
Expanded Dynamic Symmetry Armature
Expsanded Dynamic Symmetry Armature BW
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- Expsanded Dynamic Symmetry Armature BW
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Expanded Dynamic Symmetry Armature without the colors and dashed lines.
Expsanded Dynamic Symmetry Armature BW
Whirling Squares Upper Right
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- Whirling Squares Upper Right
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- PHI Aspect Ratio
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The Golden Rectangle consists of a square and a smaller (vertical) golden rectangle. One can divide up each subsequent golden rectangle into a square plus a smaller golden rectangle. And we can see the squares 'whirling' around a point that we might call the "eye". A spiral may be constructed by quarter circles within each square. Note how the eye is at the intersection of the main diagonal and its reciprocal.
Whirling Squares Upper Right PHI Aspect Ratio
GIF. Dynamic Symmetry Armature Construction
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- GIF. Dynamic Symmetry Armature Construction
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The animation of this GIF file only functions if looking at the High Resolution version, which only is available on a desktop. . Exhibits use hi res images, otherwise, CLICK ON THE HIGH RESOLUTION BUTTON BELOW TO SEE ALL CONSTRUCTION STEPS as an animated gif (5 seconds apart). You may also download the gif to your computer. Then if you open up the GIF on your computer you may select each image to hold and study - or download separately. OR take screenshots of each step in the 5 seconds for each image - for further practice at your desk. .
Step 1. Baroque Diagonal –Lower Left Corner to Upper Right corner (blue).
Orange Reciprocal Line from lower right corner (perpendicular to the diagonal at right angles- 90 degrees).
Step 2. 2nd orange reciprocal - from upper left corner through the Baroque Diagaonal at 90 degrees.
Steps 3 and 4. . Sinister Diagonal – Upper left Corner to lower right corner (blue).Two Reciprocal Lines (orange) perpendicular to the main diagonals at right angles from the other corners.
Image 4. All together. 2 main blue diagooals with 4 orange reciprocal (crossing) lines through them..
Image 5. BW. 3 by 4 dynamic symmetry armature.
GIF. Dynamic Symmetry Armature Construction
Fibonacci Whirling Squares
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- Fibonacci Whirling Squares
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- 21x34 Aspect Ratio 1.619...
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This rectangle measures 21x34, where 34 is the next term in the sequence (since 34 - 21+13).
Note the use of the lower right polar point ("eye") of the implied spiral here.
Fibonacci Whirling Squares 21x34 Aspect Ratio 1.619...
Basic 14-line Harmonic Armature
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- Basic 14-line Harmonic Armature
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the 14-line Harmonic Armature consists of main diagonal lines, lines (diagonal) from corners to midpoints on edges opposite to the corners and lines (diagonal) from edge midpoints to the midpoints of the adjacent edges.
Basic 14-line Harmonic Armature
Dynamic Symmetry Armature Root 5 with Overlapping PHI Rectangles (2.236...)
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- Dynamic Symmetry Armature Root 5 with Overlapping PHI Rectangles (2.236...)
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- Root 5 Aspect Ratio (2.236..)
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Note the inscribed PHI rectangles. This might indicate the connection between PHI and Root 5 geometrically. We also see at the ends, vertical PHI rectangles . And in the middle, a square.
