Making a Math of Art

Please explore the relationship between mathematics and art

Definitions

What are math and art? How might one make a math of art?

I find it useful to begin any discussion with a definition of terms. The following definitions were adapted from an early online version of the Webster dictionary that is no longer available.

how:
• a question about manner (which might be a method of artistic execution or mode of presentation) or method (which might be a systematic procedure, technique, or mode of inquiry employed by or proper to a particular discipline or art). Method implies an orderly logical effective arrangement usually in steps and manner implies a procedure or method that is individual or distinctive.
make:
• to cause to exist, occur, or appear.
• to fit, intend, or destine by or as if by creating, to bring into being by forming, shaping, or altering material.
• to put together from components.
math:
• the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.
• a branch of, operation in, or use of mathematics — i.e. the mathematics of physical chemistry, the mathematics of art.
art:
• skill acquired by experience, study, or observation — i.e. the art of making friends.
• an occupation requiring knowledge or skill — i.e. the art of organ building.
• the conscious use of skill and creative imagination especially in the production of aesthetic objects; also: works so produced.
• decorative or illustrative elements in printed matter.
• The faculty of executing well what one has devised, also implying a personal, unanalyzable creative power.

Word games

Knowing the definitions of individual words, how can the phrase "Math of Art" be defined? This is an important point, especially if we want to make one.

The phrase "Work of Art" is certainly better known — something which displays artistic qualities.

• Art of Work
• Math of Work
• Work of Math
• Art of Math

What sort of relationship is there between "Work of Art" and "Work of Math"? What is the difference between "Art of Work" and "Math of Work"?

Interpretation

Is math relevant?

As far as the laws of mathematics refer to reality, they are uncertain, and as far as they are certain, they do not refer to reality.

— Albert Einstein

Stand firm in your refusal to remain conscious during algebra. In real life, I assure you, there is no such thing as algebra.

— Fran Leibowitz

Galileo said that mathematics is the language of nature "and its characters are triangles, circles, and other geometrical figures"

Mathematics ... would certainly not have come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude.

— Friedrich Nietzsche

clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line

— Benoit B Mandelbrot

(and fractals, representative of "other geometrical figures", can describe these phenomena.)

Is art relevant?

Nature is inside art as its content, not outside as its model.

— Northrop Frye

Art is the imposing of a pattern on experience, and our aesthetic enjoyment is the recognition of that pattern.

Art does not reproduce the visible; rather, it makes visible.

— Paul Klee

— Jean Cocteau

Can math and art combine?

Pythagoreans believed that all things have form and all forms can be defined by numbers.

In the final analysis, every force finds expression in number; this is called numerical expression. In art at present, this remains a rather theoretic contention, but, nevertheless, it must not be left out of consideration. We today lack the possibilities of measurement which some day, sooner or later, will be found beyond the Utopian. From this moment on, it will be possible to give every composition its numerical expression, even though this may at first hold true only of its "basic plan" and its larger complexes. The balance is chiefly a matter of patience which will accomplish the breaking down of the larger complexes into ever smaller, more subordinate groups. Only after the conquest of numerical expression will an exact theory of composition be fully realized.

A new form of art redefines the boundary between 'invention' and 'discovery', as understood in the sciences and 'creativity' in the plastic arts. Can pure geometry be perceived by the 'man in the street' as beautiful? To be more specific, can a shape that is defined by a simple equation or a simple rule of construction be perceived by people other than geometers as having aesthetic value — namely, as being at least surprisingly decorative — or perhaps even being a work of art? When the geometric shape is a fractal, the answer is yes.... Therefore we shall argue that fractal geometry appears to have created a new category of art, next to art for art's sake and art for the sake of commerce: art for the sake of science (and of mathematics).

— Benoit B Mandelbrot

Can Numbers Describe These Images?

Complicated (fractal) shapes may have simple descriptions.

Are these shapes complicated? How can they be described? Is there a relation between them?

What makes an image worthy of the title "art"? Is it a democratic process, or an objective standard? Modern computer technology and algorithms allow people to have a voice.

The personal process of creation is difficult to convey, but there are signposts along the way. The images Fiery Dragon , Steps to Infinity , and Captivating Rhythm from the MATH-ART exhibit are annotated.

Can Numbers Describe Nature?

Finding numbers to describe nature can bring to it a sense of order. Some examples from history include:

• musical scales (doubling frequency increases octave, also related to length of string)
• development of sunflower heads (Fibonacci numbers and Golden Angle describe placement of seeds)
The natural world is also filled with dynamic processes:
• Predator/Prey relationships in the wild (modelled by Lotka-Volterra equations)
• The Butterfly Effect (describing the potential impact on global weather of the flapping of a single butterfly's wings)
• Symmetry in Chaos (the work of Golubitsky and Field: finding patterns in the averaging of millions of iterations)
• The Mandelbrot Set: complexity from a simple equation

Can numbers describe coastlines or clouds? Mandelbrot, the father of fractal geometry once said:

clouds are not spheres,
mountains are not cones,
coastlines are not circles,
and bark is not smooth,
nor does lightning travel in a straight line ...

• What is the meaning of this quotation? Do you agree?
• What do clouds, mountains, coastlines, bark and lightning have in common?
• What do spheres, cones, circles, and straight lines have in common?
• What are some concise ways to describe a circle?
• What are some concise ways to describe a mountain?
• How does one measure a coastline? (the answer depends on the length of the measuring stick!) Look at this progression of shapes. Can you see a pattern?

Think of the original line in 3 segments. Replace those 3 segments with 4 segments, as shown in the second curve. Repeat. The dimension of this curve: log 4 / log 3

Fractals

What is a fractal? (precisely)

It's hard to be precise! like a biologists definition of life, a single definition doesn't capture all the important qualities.

It's a new word (c. 1975) and even those who know the word may have a hard time explaining it.

Which words does fractal sound like, or look like?

Fractals provides a way to quantify the roughness of a surface and generally have:

• detailed structure
• self-similarity in some sense
• simple formula, usually recursive

These are the same properties possessed by natural objects. Have a look at Ken Musgrave's fractal-based creations and see if you agree that fractals can help to make forgeries of nature: