Patterns in Nature
Patterns in nature are visible regular forms found in the natural world. The patterns can sometimes be modeled mathematically and they include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes.
Mathematics, physics and chemistry can explain patterns in nature at different levels. Patterns in living things express the underlying biological processes. Studies of pattern formation make use of computer models to simulate a wide range of patterns.
In 1202, Leonardo Fibonacci introduced the Fibonacci number sequence. It turns out that simple equations involving the Fibonacci numbers can describe most of the complex spiral growth patterns found in nature.
The Belgian physicist Joseph Plateau (1801–1883) formulated the mathematical problem of the existence of a minimal surface with a given boundary, which is now named after him. He studied soap films intensively and formulated Plateau’s laws, which describe the structures formed by films in foams.
The German psychologist Adolf Zeising (1810–1876) claimed that the golden ratio was expressed in the arrangement of plant parts, in the skeletons of animals and the branching patterns of their veins and nerves, as well as in the geometry of crystals.
Ernst Haeckel (1834–1919) painted beautiful illustrations of marine organisms, in particular Radiolaria, emphasizing their symmetry to support his faux-Darwinian theories of evolution.
The American photographer Wilson Bentley (1865–1931) took the first micrograph of a snowflake in 1885.
D’Arcy Thompson pioneered the study of growth and form in his 1917 book.
In 1952, Alan Turing (1912–1954), better known for his work on computing and codebreaking, wrote The Chemical Basis of Morphogenesis, an analysis of the mechanisms that would be needed to create patterns in living organisms, in the process called morphogenesis. He predicted oscillating chemical reactions, in particular the Belousov–Zhabotinsky reaction. These activator-inhibitor mechanisms can, Turing suggested, generate patterns of stripes and spots in animals, and contribute to the spiral patterns seen in plant phyllotaxis.
In 1968, the Hungarian theoretical biologist Aristid Lindenmayer (1925–1989) developed the L-system, a formal grammar which can be used to model plant growth patterns in the style of fractals. L-systems have an alphabet of symbols that can be combined using production rules to build larger strings of symbols, and a mechanism for translating the generated strings into geometric structures. In 1975, after centuries of slow development of the mathematics of patterns by Gottfried Leibniz, Georg Cantor, Helge von Koch (the Koch snowflake), Wacław Sierpiński and others, Benoît Mandelbrot wrote a famous paper, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, crystallizing mathematical thought into the concept of the fractal and the Mandelbrot set.
Living things like orchids, hummingbirds, and the peacock’s tail have abstract designs with a beauty of form, pattern and color that artists struggle to match. The beauty that people perceive in nature has causes at different levels, notably in the mathematics that governs what patterns can physically form, and among living things in the effects of natural selection, that govern how patterns evolve.
Mathematics seeks to discover and explain abstract patterns or regularities of all kinds. Visual patterns in nature find explanations in chaos theory, fractals, logarithmic spirals, topology and other mathematical patterns. For example, L-systems form convincing models of different patterns of tree growth.
The laws of physics apply the abstractions of mathematics to the real world, often as if it were perfect. For example, a crystal is perfect when it has no structural defects such as dislocations and is fully symmetric. Exact mathematical perfection can only approximate real objects. Visible patterns in nature are governed by physical laws; for example, meanders can be explained using fluid dynamics.
In biology, natural selection can cause the development of patterns in living things for several reasons, including camouflage, sexual selection, and different kinds of signalling, including mimicry and cleaning symbiosis. In plants, the shapes, colors, and patterns of insect-pollinated flowers like the lily have evolved to attract insects such as bees. Radial patterns of colors and stripes, some visible only in ultraviolet light serve as nectar guides that can be seen at a distance.
Types of pattern
Symmetry is pervasive in living things. Animals mainly have bilateral or mirror symmetry, as do the leaves of plants and some flowers such as orchids. Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The head becomes specialized with a mouth and sense organs (cephalization), and the body becomes bilaterally symmetric (though internal organs need not be).
Plants often have radial or rotational symmetry, as do many flowers and some groups of animals such as sea anemones.
Rotational symmetry is also found at different scales among non-living things including the crown-shaped splash pattern formed when a drop falls into a pond, and both the spheroidal shape and rings of a planet like Saturn.
Radial symmetry suits organisms like sea anemones whose adults do not move: food and threats may arrive from any direction.
Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies. The reason for the fivefold (penta-radiate) symmetry of the echinoderms is puzzling. Early echinoderms were bilaterally symmetrical, as their larvae still are. Sumrall and Wray argue that the loss of the old symmetry had both developmental and ecological causes.
Among non-living things, snowflakes have striking six-fold symmetry: each flake’s structure forming a record of the varying conditions during its crystallization, with nearly the same pattern of growth on each of its six arms.
Crystals in general have a variety of symmetries and crystal habits; they can be cubic or octahedral, but true crystals cannot have fivefold symmetry (unlike quasicrystals).