Divine Proportion

2012-03-24T13:29:00.000-07:00

Visual Thinking

2012-03-17T14:31:00.001-07:00

Visual Thinking is a work from which many others emerge. Kress and Van Leeuwen, for example, cite this work as crucial to all of their theories of Visual Design. Kostelnick and Roberts devote much to Gestalt psychology in their writing about visual language. Schriver concentrates on the visual design of text and typography largely for print, and she uses gestalt principles extensively in a section called Seeing the Text. Mark Johnson positions his theories under Arnheim’s umbrella of thought.

Arnheim begins this work by positioning his thought within the structure of Western philosophy, that is, the dichotomy between rationalism (Plato) and empiricism (Aristotle). He comes down strongly on the side of the empiricist and writes:

Aristotle asserts that an object is real to us through its true and lasting nature, not through its changeable properties. Its universal character is directly perceived in it as its essence rather than indirectly collected through the search of common elements .

Arnheim establishes the notion that the visually thinking mind is not simply mechanically recording images and regurgitating them repetitively. He insists that perception is intelligent.

Roy R. Behrens comments on this work:

“In that book, Arnheim intended to narrow the gap between scientific and artistic knowledge, to use scientific findings to better understand the arts while preserving the equally pivotal role of subjectivity, intuition, and self-expression. In a subsequent book, titled Visual Thinking, published in 1969, he challenged the age-old distinctions between thinking and perceiving, and between intellect and intuition.

Contending that "all perceiving is also thinking, all reasoning is also intuition, all observation is also invention" (1974), he attacked the established assumptions that words, not images, are the primary ingredients of thinking, and that language precedes perception. Rather, Arnheim argued, "the remarkable mechanisms by which the senses understand the environment are all but identical with the operations described by the psychology of thinking" (1969). Like scientific discovery, he wrote, artistic expression "is a form of reasoning, in which perceiving and thinking are indivisibly intertwined. A person who paints, writes, composes, dances, I felt compelled to say, thinks with his senses" (1969)..

Arnheim springs from the original school of Gestalt philosophy. He believes that a certain degree of inbred responses to certain shapes, colours and movements exist, “Shape perception operates at a high cognitive level of concept formation” (29). He also spends a great deal of time describing the way that images are meaningful only in a context, “Things fit together by assimilation and contrast” (65). “To lift something out of its context means to neglect an important aspect of its nature” (71). His writings on figure and ground have been particularly influential (Schriver, Kostelnick, and Kress and Van Leeuwen).

Arnheim also analyzes metaphor and the notion of “routine metaphor” (112). Mark Johnson, who studies the way the body and bodily motion contributes to the metaphoric system underlying all language, makes direct reference to Arnheim.

Human Beauty

2007-04-05T08:55:00.000-07:00

"Beauty is in the phi of the beholder."

It has long been said that beauty is in the eye of the beholder and thought that beauty varies by race, culture or era. The evidence, however, shows that our perception of physical beauty is hard wired into our being and based entirely on how closely one's features reflect phi in their proportions. Take another look at beauty through the eyes of medical science.

A template for human beauty is found in phi and the pentagon

Dr. Stephen Marquardt has studied human beauty for years in his practice of oral and maxillofacial surgery. Dr. Marquardt performed cross-cultural surveys on beauty and found that all groups had the same perceptions of facial beauty. He also analyzed the human face from ancient times to the modern day. Through his research, he discovered that beauty is not only related to phi, but can be defined for both genders and for all races, cultures and eras with the beauty mask which he developed and patented. This mask uses the pentagon and decagon as its foundation, which embody phi in all their dimensions. For more information and other examples, see his site at Marquardt Beauty Analysis.

The Marquardt Beauty Mask


Asian	Black	Caucasian


1350 B.C. Egypt	500 B.C. Greece	164 A.D. Rome	1794 A.D.

Even with a perfectly proportioned face though, there are endless variations in coloring and the shapes of each facial feature (eyes, eyebrows, lips, nose, etc.) that give rise to the distinctive appearance of each race and provide for endless variations in beauty that are as unique as each individual.

The human face communicates an incredible array of emotions which are an integral element of one's total beauty. The human face conforms most closely to phi proportions when we smile. You'll be perceived as more beautiful with a warm smile than with a cold-hearted look of anger, arrogance or contempt.

See also sites at BeautyRank.com and BeautyCheck.de

Fibonacci Numbers in Nature

2007-04-05T08:53:00.000-07:00

Here are some not-too technical papers about the maths which justifies the occurrence of the Fibonacci numbers in nature:

A H Church On the relation of Phyllotaxis to Mechanical Laws, published by Williams and Norgat, London 1904.
E E Leppik, Phyllotaxis, anthotaxis and semataxis Acta Biotheoretica Vol 14, 1961, pages 1-28.
F J Richards Phyllotaxis: Its Quantitative Expression and Relation to growth in the Apex Phil. Trans. Series B Vol 235, 1951, pages 509-564.
D'Arcy W Thompson On Growth and Form Dover Press 1992.
This is the complete edition! (Click on the title-link for more information and to order it now.)
There is also an abridged version from Cambridge University press (more information and order it on line via the title-link.)
T A Davis, Fibonacci Numbers for Palm Foliar Spirals Acta Botanica Neelandica, Vol 19, 1970, pages 236-243.
T A Davis Why Fibonacci Sequence for Palm Leaf Spirals?, Fibonacci Quarterly, Vol 9, 1971, pages 237-244.
The Algorithmic Beauty of Plants by P Prusinkiewicz, and A Lindenmayer, published by Springer-Verlag (Second printing 1996) is an astounding book of wonderful images and patterns in plant shapes as well as algorithms for modelling and simulation by computer. (For more information and how to order it online use the title-link).
Related to this book is:
The Algorithmic Beauty of Sea Shells (Virtual Laboratory) in hardback by Hans Meinhardt, Przemyslaw Prusinkiewicz, Deborah R. Fowler (more information and order it online via this title-link).
The Curves of Life: Being an Account of Spiral Formations and Their Application to Growth in Nature, to Science, and to Art Sir Theodore A Cook, Dover books, 1979, ISBN 0 486 23701 X.
A Dover reprint of a classic 1914 book. (More information and you can order it online via the title-link.)
Also see H S M Coxeter's Introduction to Geometry, published by Wiley, in its Wiley Classics Library series, 1989, ISBN 0471504580, especially chapter 11 on Phyllotaxis. (More information and order it online via the title-link.)

The Human Face

2007-04-05T08:51:00.000-07:00

The human face abounds with examples of the Golden Section or Divine Proportion. We'll use our building blocks again to understand design in the face:

The head forms a golden rectangle with the eyes at its midpoint. The mouth and nose are each placed at golden sections of the distance between the eyes and the bottom of the chin. The beauty unfolds as you look further.

Human beauty is based on the Divine Proportion

The 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.

Click on the photo to the left to see another photo which illustrates the following golden sections in the human face:

Center of pupil : Bottom of teeth : Bottom of chin
Outer & inner edge of eye: Center of nose
Outer edges of lips : Upper ridges of lips
Width of center tooth : Width of second tooth
Width of eye : Width of iris

Phi defines the dimensions of the human profile

Even when viewed from the side, the human head illustrates the Divine Proportion.

The first golden section (blue) from the front of the head defines the position of the ear opening. The successive golden sections define the neck (yellow), the back of the eye (green) and the front of the eye and back of the nose and mouth (magenta). The dimensions of the face from top to bottom also exhibit the Divine Proportion, in the positions of the eye brow (blue), nose (yellow) and mouth (green and magenta).

The ear reflects the shape of a Fibonacci spiral.

Even the dimensions of our teeth are based on phi

The front two incisor teeth form a golden rectangle, with a phi ratio in the heighth to the width.

The ratio of the width of the first tooth to the second tooth from the center is also phi.

The ratio of the width of the smile to the third tooth from the center is phi as well.

Visit the site of Dr. Eddy Levin for more on the Golden Section and Dentistry.

Phi ( Ø )

2007-04-05T08:49:00.000-07:00

Phi for Neo-phi-tes
What is Phi?

Phi ( Ø ), is simply an irrational number like pi (p), but one with many unusual mathematical properties. Phi to the first 15 places is 1.618033988749895. What makes it more unusual is that it can be derived in many ways and shows up in relationships throughout the universe.
Phi Ø can be derived through:

*A numerical series discovered by Leonardo Fibonacci

*Mathematics

*Geometry

Phi Ø appears in:

*The proportions of the human body
*The proportions of many other animals
*Plants
*DNA
*The solar system
*Art and architecture
*Music
*Population growth
*The stock market
*The Bible and in theology

The ratio, or proportion, determined by Phi (1.618...) was known to the Greeks as the "Golden Section" and to Renaissance artists as the "Divine Proportion"

Phi with an upper case "P" is 1.618033987..., while phi with a lower case "p" is 0.6180339887, the reciprocal of Phi and also Phi minus 1.