What sort of relationship, if any, exists between mathematics and art?
Please explore and let me know if you have any questions or comments.
I find it useful to begin any discussion with a definition of terms. The following definitions have been adapted from the online version of the Webster dictionary.
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.
What about the following variations:
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"?
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
-- B. B. Mandelbrot
(and fractals, representative of "other geometrical figures", can describe these phenomena.)
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.
-- Alfred North Whitehead
Art does not reproduce the visible; rather, it makes visible.
-- Paul Klee
Art is science made clear.
-- Jean Cocteau
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.
-- V. Kadinsky
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).
-- B. B. Mandelbrot
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.
Are there properties of fractals are like those of natural objects?
Can fractals be used to create forgeries of nature?
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 this quotation saying? Do you agree?
What do clouds, mountains, coastlines, bark and lightning have in common? Rough, natural
What do spheres, cones, circles, and straight lines have in common? Euclidean, man-made
What are some concise ways to describe a circle? Center and radius, formula in x, y, and r.
What are some concise ways to describe a mountain? Fractals (surprise!)
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?
fraction, fracture: the broken edge of a twig or rock is rough and irregular
Fractals provides a way to quantify the roughness of a surface and generally have:
What happens if we zoom into the Mandelbrot set? (Courtesy of Eric Schol)