Math

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John Sims MathArt Brain
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PHI is the Divine Ratio and the Golden Mean
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Luca Pacioli

• "Without mathematics there is no art."

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THE GOLDEN MEANNature

• The Golden Mean, 1.61803398874989, represented
  by the Greek letter phi, is a naturally occurring
  number, like pi, that repeatedly occurs in
  various relationships. Like pi, it is an
  irrational number. Unlike pi, it clearly and
  regularly appears in the growth patterns of many
  living things, like the spiral formed by a
  seashell or the curve of a fern.

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COMPOSITIONAL MODELS
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THE GOLDEN MEANArt

• The Greeks discovered they could create a feeling
  of natural order, as well as structural
  integrity, in their works. Artists since have
  used it for the same reason, to create a feeling
  of natural order in their works. It is thought by
  many people to describe the most aesthetically
  pleasing rectangle.

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Golden Rectangle Modern artists use it, and
even the ancient Greeks used it to develop the
facade of the Parthenon.
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Golden Mean

• The Fibonacci Series and the Golden Mean are
  intimately connected. The Fibonacci Series
  numbers increase at a rate equal to (actually,
  oscillating round) the Golden mean.

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THE FIBONACCI SEQUENCE FOR VISUAL LAYOUT
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A rectangle whose sides are related by phi (such
as 13 x 8) is said to be a Golden Rectangle. It
has the interesting property that, if you create
a new rectangle by swinging the long side around
one of its ends outward from the rectangle, to
create a new long side, (in combination with the
short side), then that new rectangle is also a
golden rectangle. In the case of our 13 x 8
rectangle, the new rectangle will be 21 x 13. We
see that this is the same thing that's going on
in the Fibonacci Series.
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The Golden Rectangle
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The Golden Rectangle
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The Golden Rectangle
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The Golden Rectangle
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The Golden Rectangle
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The Golden Rectangle
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The Golden Rectangle
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The Golden Rectangle
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The Golden Rectangle
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The Fibonacci Sequence If you dissect a work
like Perugino's Madonna Enthroned with Child and
the Saints John the Baptist and Sebastian you
will notice that the saints are set into
rectangles which reflect a .618034 ratio of the
total width of the work, measuring from each side
inward.
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The Golden Rectangle in Nature
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SeuratThe Circus SideshowGolden Mean
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Da VinciVitruvian Man Golden Ratio
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Da Vinci St. Jerome Golden Mean
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At left, Edward Burne Jones, who created "The
Golden Stairs" at left, also meticulously planned
the smallest of details using the golden section.
Golden sections appear in the stairs and the ring
of the trumpet carried by the fourth woman from
the top. Can you find more examples?
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• This self-portrait by Rembrandt (1606-1669)... is
  an example of triangular composition. A
  perpendicular line from the apex of the triangle
  to the base cut the base in golden section.

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PENTAGON AND THE GOLDEN RATIOMichelangelo Holy
Family
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Leonardo DaVinci used phi when examining artwork
for the human body. The famous painting the "Mona
Lisa" shows phi, as does a wide variety of
artwork throughout time.
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Da VinciDrawing studies of the human face is an
expression of the Golden Ratio of the Golden Mean
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Human beauty is based on the Divine
ProportionThe blue line defines a perfect square
of the pupils and outside corners of the mouth.
The golden section of these four blue lines
defines the nose, the tip of the nose, the inside
of the nostrils, the two rises of the upper lip
and the inner points of the ear. The blue line
also defines the distance from the upper lip to
the bottom of the chin.The yellow line, a golden
section of the blue line, defines the width of
the nose, the distance between the eyes and eye
brows and the distance from the pupils to the tip
of the nose.The green line, a golden section of
the yellow line defines the width of the eye, the
distance at the pupil from the eye lash to the
eye brow and the distance between the
nostrils.The magenta line, a golden section of
the green line, defines the distance from the
upper lip to the bottom of the nose and several
dimensions of the eye.
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Kerry MitchellMandel Lisa
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Salvador DaliFlamboyant and controversial
Spanish surrealist painter who employed
mathematics in some of his work.
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Zarko D. MijajlovichMathematical Landscapes
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Zarko D. MijajlovichMathematical
Landscapes
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Bathsheba GrossmanI'm an artist exploring how
math, science and sculpture meet..
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Bathsheba Grossman
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Bathsheba Grossman
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Robert FathauerTree of Knowledge
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Michael FieldArmies of the Night
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George HartAardvards
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Eric LandreneauIcosahedral Extrusion
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Irene RousseauHyperbolic Diminution-Blue
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Carlo SequinHilbert Cube
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Carlo SequinMinimal Trefoil
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Carlo SequinGalapagos
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Carlo SequinVolution
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Doug DunhamFive Equidistant Fish Patterns
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Anne BurnsIterated Steiner Cells ArtMathX
"Patterns in Nature Conference
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Doug CraftElements Square-Root of 5 2004-002
WaterMy collage, photography, and painting
explores sacred geometry with forms based on the
Golden Ratio.
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Brian Dance of the Sugarplum Fairy, variations 5
(from Tchaikovsky's "Nutcracker")This still
image is a visualization of sounds and short
pieces of music numeric models of sound and
melody, mapped into color.
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LunYi TsaiBaire's Theorem
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Ann BurnsFractal Scene
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Brent CollinsMusic of the Spheres
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Piet Mondrian "The proportions and rhythm of
planes and lines in architecture will mean more
to the artist than the capriciousness of nature.
In the metropolis, beauty expresses itself more
mathematically
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R. Buckminster Fuller (1895-1983) R. Buckminster
Fuller was an architect, engineer, and more who
had a keen interest in design and technology. He
is best known for his geodesic domes.
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Johannes Kepler Well known for his work in
astronomy, Kepler also had a keen interest in
geometric tesselations and polyhedra.
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M.C. Escher was not
very good at mathematics in school, and was a
graphic artist by training and profession. Early
in his career, he spent much of his time in
Italy, where he made a number of more-or-less
traditional woodcuts. After a trip to the
Alhambra, Spain, Escher became fascinated with
tessellations. It was at this time, in the
1930's, that his work began to turn away from
traditional subjects to mathematical and fanciful
ones
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Escher
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Max Bill Moebius "I am convinced it is
possible to evolve a new form of art in which the
artist's work could be founded to quite a
substantial degree on a mathematical line of
approach to its content."
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Victor Vasarely Op ArtHe uses the coloring
of simple geometric shapes, often in arrays, to
suggest motion and concave/convex effects on a
flat canvas.
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Victor VasarelyTridem K
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Victor VasarelyAlome
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Victor VasarelyCheyt M
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Benoit MandelbrotMathematician who was
largely responsible for formalizing and
popularizing the concept of fractals. He
discovered the Mandelbrot set, the best-known of
fractal objects. He also coined the term
"fractal", derived from the Latin word "fractus",
meaning fragmented or broken.
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Doug Harrington Fractals
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Doug HarringtonFractals
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Doug HarringtonFractals
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Doug HarringtonFractals
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Doug HarringtonFractals
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Segmented Wood Turning based on Math
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Richard Pagano
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Kevin Neelley
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Kevin Neelley
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Kevin Neelley