Dr. Joseph Valks – Mathematics in Art and Psychology

Galileo Galilei wrote that the universe is written in the language of mathematics, and its characters are triangles, circles and other geometric figures. Artists who strive and seek to study nature must therefore first fully understand mathematics. On the other hand mathematicians have sought to interpret and analyze art through the lens of geometry and rationality.

Mathematics is an esoteric, yet universal language. To many artists and psychologists it may seem the preserve of cold Spock-like logical robots, but modern maths is a living art itself with passion and intrigue. It can be intuitive, creative and even beautiful and could be considered akin to composing a symphony.

Mathematics and art have a long historical relationship. Early philosophers noticed the occurrence of geometric patterns in nature and studied them to try to gain an understanding of their surroundings. In particular, Plato argued that mathematical entities have real existence and mathematical truth is independent of human thought. Pythagoras taught that the universe is essentially a manifestation of mathematical relationships and is most famous for his theorem showing the relationship between the sides of a right-angled triangle. He also demonstrated harmonics in music arising from number.

Later Darwin explained the structure of an organism was the shape best suitable for survival in a species specific niche. Many designs in nature are creations of beauty such as lilies, butterflies and birds of paradise. Mathematics seeks to explain these patterns as they often tend to approximate to mathematical shapes, not usually being perfect in form. Mathematical laws govern what patterns can physically form e.g. the earth moves in an elliptical orbit around the sun. These natural patterns can then be observed as visible regularities of form present in the natural world. They may include:

Symmetry, such as the spectacular bilateral symmetry of a butterfly, the radial symmetry of flowers and sea anemones, or the spectacular six-fold symmetry of snowflakes.

Fractals, which are self-similar, repeated constructs such as ferns, clouds and lightening.

Spirals, such as the nautilus mollusk shell where each chamber is an approximate copy of the next one, scaled by a constant factor and arranged in a logarithmic spiral. Many galaxies and the path of a gliding hawk are also spirals. From the point of view of physics spirals are, ‘refined’, or lowest energy level configurations.

Tessellations, such as the wax cells in a honeycomb built by honey bees.

Stripes, or spots, such as those seen in tigers or ladybirds are adaptations of natural selection and often form camouflage or warning coloration. Other colorful patterns may increase attractiveness to a mate thereby encouraging sexual selection.

Light waves create colors and their mathematical properties such as refraction and interference create natural phenomena such as rainbows or flickering shadows on forest floors. There are also many others such as domes and hollows in sand dunes and spherical bubbles in foams.

The origins of mathematical thought lie in shape, size and number. These concepts have been shown through studies of animals, not to be unique to humans, and they would have been readily found in the everyday natural world of primitive man. Ancient megaliths from thousands of years BC have been shown to incorporate basic geometrical shapes such as triangles, circles and ellipses. However, the oldest undisputable mathematical usage is from ancient Babylonian and Egyptian texts.

The golden ratio =1.618 is believed to be such that when incorporated into geometric shapes such as the golden rectangle, or golden triangle, it has aesthetic qualities that are part of an instinctive and primal human cognitive preference. It was ‘formally’ discovered by the Greek mathematician Pythagoras along with the 3-4-5 triangle. However, some people believe the ancient Greeks and ancient Egyptians incorporated mathematical relationships such as the golden ratio into the design of many monuments including the Great Pyramid, the Parthenon and the Colosseum.

The Greek sculptor Polykleitos the Elder applied the concepts of Greek geometry and ratio of proportion, to create a harmony of the human body which he believed embodied the proportions of both physical and mathematical perfection.

It was centuries later that the Renaissance sparked a renewed explosion in interest in mathematics and art driven by the belief that maths was the ‘cosmic key’ to unraveling God’s truth of creation. The artists believed that both the old and new testament of the Bible contained numbers that revealed hidden concepts and meanings. The whole universe was ordered, and everything in it including the arts, could be explained in geometric terms.

The first thing they had to do was work out how create 3D scenes on a 2D canvas, and it was Piero della Francesca who realized that the way aspects of a figure changed with point of view obeyed precise mathematical laws.

Leonardo da Vinci studied the new found laws of art and maths and he realized that by making the lines in a painting converge onto a single invisible point on the horizon he could solve this problem of creating a 3D image by adding depth to a painting. The disappearing point adds realism to a landscape so that you almost feel you can walk out of the room into it. In the Mona Lisa he deliberately creates a mismatch between the left and right scenery behind the figure to give her an illusion of perspective and depth. Her arms form the base and sides of a triangle whose top vertex finishes at her head. Thus, attention is drawn towards the face. It is also claimed that the picture contains a golden rectangle that can be subdivided into smaller rectangles intersecting at various focal points and forming part of a golden spiral.

More recently M. C. Escher pioneered tessellations, polyhedrons and shaping of space to create geometrical objects and impossible constructions that cannot exist yet are favorable to the human eye.

Fractal art copies nature, except at each level the self similarities are absolute identical copies, which can only be approximate within the natural world e.g. trees. The Mandlebrot set is one of the better-known algorithmic fractal art forms.

So how might some of the ancient beliefs of mathematics in art and psychology apply to a modern nature photographer’s portfolio?

Bluebell. The flower stem forms part of a distinctive arch.
Bluebell. The flower stem forms part of a distinctive arch.
Wind on barley. This is classic da Vinci. The two parallel lines converge onto a single invisible point on the horizon adding depth and taking the observer into the picture.
Wind on barley. This is classic da Vinci. The two parallel lines converge onto a single invisible point on the horizon adding depth and taking the observer into the picture.
In this picture, the light rays all meet at a single invisible point in the sky.
In this picture, the light rays all meet at a single invisible point in the sky.
Heron in treetop. Again there is a mismatch between the foreground and background. The heron's beak and foreground both contain strong straight lines. The heron's body almost splits the photo into two equal triangles. The bottom left of a rectangle begins at the point level with the heron's right foot formed by continuing the diagonal of the triangle mentioned previously down through the tail. The rectangle extends above the heron's head forming a golden spiral in which some of the smaller rectangles intersect at the focal points of the beak and eye.
Heron in treetop. Again there is a mismatch between the foreground and background. The heron’s beak and foreground both contain strong straight lines. The heron’s body almost splits the photo into two equal triangles. The bottom left of a rectangle begins at the point level with the heron’s right foot formed by continuing the diagonal of the triangle mentioned previously down through the tail. The rectangle extends above the heron’s head forming a golden spiral in which some of the smaller rectangles intersect at the focal points of the beak and eye.
There is a mismatch between the foreground and background. The foreground snow is in the shape of a right angled traingle. The jay is aligned with the sloping edge of the triangle and the snowy edge beneath the jay, the jay itself, and the snowy edge above the jay are all in the golden ratio respectively.
There is a mismatch between the foreground and background. The foreground snow is in the shape of a right angled traingle. The jay is aligned with the sloping edge of the triangle and the snowy edge beneath the jay, the jay itself, and the snowy edge above the jay are all in the golden ratio respectively.

What do you think?

If we were to communicate with aliens, it is widely accepted that mathematics is likely to be the only mutually comprehended language. The Arecibo message was broadcast into space via modulated radio waves to mark the remodeling of the Arecibo radio telescope in Puerto Rico on 16th November 1974 and aimed at a globular star cluster 25,000 light years away. It included the numbers 1 to 10 in binary format.

For those who may think the origin of mathematics was a men-only vocation: take heart in Hypatia of Alexandria, who succeeded her father as librarian of the Great Library and authored many mathematical works. She was reportedly stripped naked and her skin scraped off with clamshells after a political dispute with Christians. Who says the devil has all the fun?

Modern mathematics is now using computers to model patterns in nature with neural networks, a kind of artificial intelligence. (See using neurological processes and computer art to model global warming).

John A Adam does an excellent book on Mathematics in Nature. If anyone ever engineers a virtual forest from his stuff looking at tree structure, flexibility, light filtering, transpiration, Aeolian tones etc. please let me know what it looks like!

Dr. Joe Valks

Joe recently helped develop the British Woodlands food webs educational simulation for Newbyte and is donating his share of The Last Tiger (available on Amazon kindle) children’s fantasy novel profits to the Animals on the Edge conservation project.

For a Divine Architect Theory of Evolution click here.

For a Matlab program to create a fractal fern click here.

For mathematical models of seashells click here.

 

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