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John Sims MathArt Brain

PHI is the Divine Ratio and the Golden Mean

Luca Pacioli

- "Without mathematics there is no art."

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

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.

Golden Rectangle Modern artists use it, and

even the ancient Greeks used it to develop the

facade of the Parthenon.

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.

THE FIBONACCI SEQUENCE FOR VISUAL LAYOUT

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.

The Golden Rectangle

The Golden Rectangle

The Golden Rectangle

The Golden Rectangle

The Golden Rectangle

The Golden Rectangle

The Golden Rectangle

The Golden Rectangle

The Golden Rectangle

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.

The Golden Rectangle in Nature

SeuratThe Circus SideshowGolden Mean

Da VinciVitruvian Man Golden Ratio

Da Vinci St. Jerome Golden Mean

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?

- 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.

PENTAGON AND THE GOLDEN RATIOMichelangelo Holy

Family

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.

Da VinciDrawing studies of the human face is an

expression of the Golden Ratio of the Golden Mean

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

Salvador DaliFlamboyant and controversial

Spanish surrealist painter who employed

mathematics in some of his work.

Zarko D. MijajlovichMathematical Landscapes

Zarko D. MijajlovichMathematical

Landscapes

Bathsheba GrossmanI'm an artist exploring how

math, science and sculpture meet..

Bathsheba Grossman

Bathsheba Grossman

Robert FathauerTree of Knowledge

Michael FieldArmies of the Night

George HartAardvards

Eric LandreneauIcosahedral Extrusion

Irene RousseauHyperbolic Diminution-Blue

Carlo SequinHilbert Cube

Carlo SequinMinimal Trefoil

Carlo SequinGalapagos

Carlo SequinVolution

Doug DunhamFive Equidistant Fish Patterns

Anne BurnsIterated Steiner Cells ArtMathX

"Patterns in Nature Conference

Doug CraftElements Square-Root of 5 2004-002

WaterMy collage, photography, and painting

explores sacred geometry with forms based on the

Golden Ratio.

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.

LunYi TsaiBaire's Theorem

Ann BurnsFractal Scene

Brent CollinsMusic of the Spheres

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

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.

Johannes Kepler Well known for his work in

astronomy, Kepler also had a keen interest in

geometric tesselations and polyhedra.

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

Escher

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."

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.

Victor VasarelyTridem K

Victor VasarelyAlome

Victor VasarelyCheyt M

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.

Doug Harrington Fractals

Doug HarringtonFractals

Doug HarringtonFractals

Doug HarringtonFractals

Doug HarringtonFractals

Segmented Wood Turning based on Math

Richard Pagano

Kevin Neelley

Kevin Neelley

Kevin Neelley

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