john sims “ mathart brain ”

John Sims “MathArt Brain”

Arts & Communications

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.

COMPOSITIONAL MODELS

THE GOLDEN MEAN

Art

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

Seurat“The Circus Sideshow”Golden Mean

Da Vinci“Vitruvian 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.

Kerry Mitchell“Mandel Lisa”

Salvador DaliFlamboyant and controversial Spanish surrealist painter who employed mathematics in some of his work.

Zarko D. Mijajlovich“Mathematical Landscapes”

Zarko D. Mijajlovich“Mathematical Landscapes”

Bathsheba Grossman

I'm an artist exploring how math, science and sculpture meet..

Bathsheba Grossman

Bathsheba Grossman

Robert Fathauer“Tree of Knowledge”

Michael Field“Armies of the Night”

George Hart“Aardvards”

Eric Landreneau“Icosahedral Extrusion”

Irene Rousseau“Hyperbolic Diminution-Blue”

Carlo Sequin“Hilbert Cube”

Carlo Sequin“Minimal Trefoil”

Carlo Sequin“Galapagos”

Carlo Sequin“Volution”

Doug Dunham“Five Equidistant Fish Patterns”

Anne Burns“Iterated Steiner Cells” Art+Math=X "Patterns in Nature” Conference

Doug CraftElements Square-Root of 5 2004-002 Water

“My 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 Tsai“Baire's Theorem”

Ann Burns“Fractal Scene”

Brent Collins“Music 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 Vasarely“Tridem K”

Victor Vasarely“Alome”

Victor Vasarely“Cheyt 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