Spherical water drop pattern created due to physical forces associated with surface tension.
03 Introduction Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically.
08 Causes 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.
11 Types Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes.
35 Formations The Fibonacci numbers are Nature's numbering system. They appear everywhere in nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple.
Patterns created in antelope canyon due to the erosion of sandstone during flash flooding
Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry. richard p feynman
2
PATTERNS IN NATURE
Photography by Mostafa Ozturk
RICHARD P FEYNMAN, a scientist not given to flights of fancy, expressed it perfectly. The spiral of a shell, the fractals of peacock’s tail, tessellations of a honecomb and the symmetry of snowflakes all display visual structures that are strikingly remarkable. These are the patterns in nature that are visible all around us and can be explained using mathematics, physics and chemistry. Patterns in living things are explained by the biological processes of natural selection and sexual selection. The modern understanding of visible patterns developed gradually over time. PATTERNS IN NATURE
3
PHOTO LEFT TO RIGHT: Shapes created when water runs through the rocks in Antelope Canyon; Multiple pattern formations in butterfly wings; Natural tessellation-like shapes in drangonfly wings, Wings that never took flight by cheryl.rose83 on Flickr, http://waitingforteaagain.tumblr.com/post/71797612300
In 1202, Leonardo Fibonacci introduced the Fibonacci number sequence to the western world with his book Liber Abaci. Fibonacci gave an (unrealistic) biological example, on the growth in numbers of a theoretical rabbit population. In 1917, D'Arcy Wentworth Thompson published his book On Growth and Form. His description of phyllotaxis and the Fibonacci sequence, the mathematical relationships in the spiral growth patterns of plants, is classic. He showed that simple equations could describe all the apparently complex spiral growth patterns of animal horns and mollusc shells. The American photographer Wilson Bentley (1865–1931) took the first micrograph of a snowflake in 1885.
4
PATTERNS IN NATURE
PATTERNS IN NATURE
5
Regular but not Identical Traditionally, we think of patterns as something that just repeats again and again throughout space in an identical way, sort of like a wallpaper pattern. But many patterns that we see in nature aren’t quite like that. We sense that there is something regular or at least not random about them, but that doesn’t mean that all the elements are identical. 6
PATTERNS IN NATURE
A very familiar example of that would be the zebra’s stripes. Everyone can recognize that as a pattern, but no stripe is like any other stripe. Anything that isn’t purely random has a kind of pattern in it. There must be something in that system that has pulled it away from that pure randomness or at the other extreme, from pure uniformity.
PHOTOS CLOCKWISE: Bilateral Symmetry in Zebra stripes, http:// apcalcpd41213.blogspot.in/I; Microscopic view of the moss leaf chroloplasts showing hexagonal shapes attached together, Waldo Nell / Flickr, https://www.flickr.com/photos/pwnell/ with/10655617516/; Succulent showing Fibonacci pattern; Vibrating blue spot pattern on a fish; Micro-atomic spiral pattern of the shell of a snail showing similarity with the macro-atomic spiral patterns of a galaxy.
PATTERNS IN NATURE
7
Causes Living things like orchids, roses, butterflies, 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, notable in the mathematics that governs what patterns can physical form, and among living things
8
PATTERNS IN NATURE
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. The laws of physics apply the abstractions of math-
ematics 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 signaling, including mimicry and cleaning symbiosis. In plants, the shapes, colors and patterns of flowers like lily have evolved to optimize insect pollination (other plants
may be pollinated by wind, birds, or bats). European honey bees and other pollinating insects are attracted to flowers by a radial pattern of colors and stripes (some visible only in ultraviolet light) that server as nectar guides that can be seen at a distance; by scent; and by rewards of sugar-rich nectar and edible pollen.
PHOTOS LEFT TO RIGHT: Image of leaves, one showing fractal branching in the leaf and another without a certain pattern; Center of a cut lemon showing radial symmetry, the color temparature has been changed to blue; Branches of a plant coming out in spiral form, alternating up the stem; Close-up of a rose showing spiral fibonacci pattern of the petals around the center; Water splash approximates radial symmetry
PATTERNS IN NATURE
9
Naturally, patterns emerge through repetition, and repetition yields up a type of discovery that reveals everything about itself, especially its sorry limits.
jan peacock
Spiral galaxies, http://www.spacetelescope.org/news/heic0602/
10
PATTERNS IN NATURE
Types of Patterns
PATTERNS IN NATURE
11
Symmetry 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. Plants often have radial or rotational symmetry, as do many flowers and some groups of animals such as sea anemones. Fivefold symmetry is found in the echinderms, the group that includes starfish, sea urchins, and sea lilies. Among non-living things, snowflakes have striking six-fold symmetry; each flake is unique, its structure forming a record of the varying conditions during 12
PATTERNS IN NATURE
its crystallization, with nearly the same pattern of growth on each of its six arms. All snowflakes have this sort of symmetry is due to the way water molecules arrange themselves when ice forms. Crystals have a variety of symmetries and crystal habits; they can be cubic or octahedral, but true crystals cannot have five-fold symmetry (unlike quasicrystals). Rotational symmetry is found among non-living things like 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.
PHOTOS CLOCKWISE FROM TOP LEFT: Echinoderms like this starfish have fivefold symmetry, Photo by Paul Shaffner - https://www. flickr.com/people/paulshaffner/; Butterfly wings showing bilateral symmetry; Tiger showing mirror or bilateral symmetry, Photo by Chiswick Chap saved from https://commons.wikimedia. org/wiki/File:Tiger-berlin-5_symmetry.jpg; Snowflake showing sixfold symmetry, http:// pcwallart.com/real-snowflake-under-microscope-wallpaper-2.html
PATTERNS IN NATURE
13
Bubbles & Foam If you blow a layer of bubbles on the surface of water (also called “bubble raft”), the bubbles become hexagonal, or almost so. You’ll never find a raft of square bubbles: If four bubble walls come together, they instantly rearrange into three-wall junctions with more or less equal angles of 120 degrees between them. Evidently there are no agents shaping these rafts as bees do with their combs. All that’s guiding the pattern are the laws of physics. Those laws evidently have definite preferences, such as the bias toward three-way junctions of bubble walls. The same is true of more complicated foams. If you pile up bubbles in three dimensions by blowing
14
PATTERNS IN NATURE
through a straw into a bowl of soapy water you’ll see that when bubble walls meet at a vertex, it’s always a four-way union with angles between the intersecting films roughly equal to about 109 degrees—an angle related to the four-faceted geometric tetrahedron. What determines these rules of soapfilm junctions and bubble shapes? Nature is even more concerned about economy than the bees are. Bubbles and soap films are made of water (with a skin of soap molecules) and surface tension pulls at the liquid surface to give it as small an area as possible. That’s why raindrops are spherical (more or less) as they fall: A sphere has less surface area than any other shape with the
same volume. On a waxy leaf, droplets of water retract into little beads for the same reason. This surface tension explains the patterns of bubble rafts and foams. The foam will seek to find the structure that has the lowest total surface tension, which means the least area of soap-film wall. But the configuration of bubble walls also has to be mechanically stable: The tugs in different directions at a junction have to balance perfectly, just as the forces must be balanced in the walls of a cathedral if the building is going to stand up. The three-way junction in a bubble raft, and the four-way junctions in foam, are the configurations that achieve this balance.
Soap bubbles showing hexagonal-like shapes, Photo Credit: Shebeko, Shutterstock.com PATTERNS IN NATURE PATTERNS IN NATURE
15 25
From sea shells and spiral galaxies to the structure of human lungs, the patterns of chaos are all around us.
Fractals
16
PATTERNS IN NATURE
Fractals are patterns formed from chaotic equations and contain selfsimilar patterns of complexity increasing with magnification. If you divide a fractal pattern into parts you get a nearly identical reduced-size copy of the whole.
The mathematical beauty of fractals is that infinite complexity is formed with relatively simple equations. By iterating or repeating fractal-generating equations many times, random outputs create beautiful patterns that are unique, yet recognizable. Fractals permeate our lives, appearing in places as tiny as the membrane of a
cell and as majestic as the solar system. These patterns occur widely in nature, in phenomena as diverse as clouds, river networks, geologic fault lines, mountains, coastlines, animal coloration, snowflakes, crystals, bloodvessels branching, and ocean waves. Fractals describe the real world better than traditional mathematics & physics.
PHOTOS LEFT TO RIGHT: Fractal-like patterns in leaves; dendritic copper crystals, Vangert, Shutterstock.com, http://thechromologist.com/patterns-nature-striking-book-print-lovers/; Fractal pattern on leaf; Fractals from a bud; Fractals in plant structures; burst pattern branching from the bottom, http://www.flowermeaning.com/ dandelion-flower-
PATTERNS IN NATURE
17
Crack formations in a dried waterbed
18
PATTERNS IN NATURE
Cracks Cracks are linear openings that form in materials to relieve stress. When an elastic material stretches or shrinks uniformly, it eventually reaches its breaking strength and then fails suddenly in all directions, creating cracks with 120 degree joints, so three cracks meet at a node. Conversely, when an inelastic material fails, straight cracks form to relieve the stress. Further stress in the same
direction would then simply open the existing cracks; stress at right angles can create new cracks, at 90 degrees to the old ones. Thus the pattern of cracks indicates whether the material is elastic or not. In a tough fibrous material like oak tree bark, cracks form to relieve stress as usual, but they do not grow long as their growth is interrupted by bundles of strong elastic fibers.
PATTERNS IN NATURE
19
PHOTOS RIGHT CLOCKWISE: Pineapple scales are patterned into spirals; Fermat’s spiral at the center of a flower; Spiral horns of bighorn sheep; Multiple fibonacci spirals, green cabbage in cross section; Logarithmic spiral of a shell
20
PATTERNS IN NATURE
Spirals Spirals are common in plants and in some animals, notable molluscs. For example, in the nautilus, a cephalopod mollusk, each chamber of its shell is an approximation copy of the next one, scaled by a constant factor and arranged in a logarithmic spiral. Given a modern understanding of fractals, a growth spiral can be seen as a special case of self-similarity. Plant spirals can be seen in phyllotaxis, the arrangement of leaves on a stem, and in the arrangement (parastichy) of other parts as in composite flower heads and seed heads like the sunflower or fruit structures like the pineapple and snake fruit, as well as in the pattern of scales in pine cones, where multiple spirals run both clockwise and
anticlockwise. These arrangements have explanations at different levels – mathematics, physics, chemistry, biology – each individually correct, but all necessary together.
at least when the flower head is mature so all the elements are the same size. Fibonacci ratios approximate the golden angle, 137.508 deg, which governs the curvature of Fermat’s spiral.
Phyllotaxis spirals can be generated mathematically from Fibonacci ratios: the Fibonacci sequence runs 1, 1, 2, 3, 5, 8, 13, 21,… (each subsequent number being the sum of the two preceding ones). For example, when leaves alternate up a stem, one rotation of the spiral touches two leaves, so the pattern or ratio is 1/2.In hazel the ratio is 1/3; in apricot it is 2/5; in pear it is 3/8; in almond it is 5/13. In disc phyllotaxis as in the sunflower and daisy, the florets are arranged in Fermat’s spiral with Fibonacci numbering,
From the point of view of physics, spirals are lowest-energy configurations which emerge spontaneously through self-organizing processes in dynamic systems. From the point of view of chemistry, a spiral can be generated by a reaction-diffusion process, involving both activation and inhibition. From a biological perspective, arranging leaves as far apart as possible in any given space is favoured by natural selection as it maximises access to resources, especially sunlight for photosynthesis. PATTERNS IN NATURE
21
L ook
deep into nature , and then you
will understand everything better . albert einstein
Microscopic view of fractals from the center
22
PATTERNS IN NATURE
PHOTOS CLOCKWISE: Cracks in tree trunk shown as it grows old; the repeated hanging lines of the spider web; Fractals shown in a leaf
PATTERNS IN NATURE
23
Chaos, Flow & Meanders In mathematics, a dynamical system is chaotic if it is (highly) sensitive to initial conditions (the so-called “butterfly effect”), which requires the mathematical properties of topological mixing and dense periodic orbits. Alongside fractals, chaos theory ranks as an essentially universal influence on patterns in nature. There is a relationship between chaos and fractals—the strange attractors in chaotic systems have a fractal dimension. Some cellular automata, simple sets of mathematical rules that generate patterns, have
24
PATTERNS IN NATURE
chaotic behavior, notably Stephen Wolfram’sRule 30. Vortex streets are zigzagging patterns of whirling vortices created by the unsteady separation of flow of a fluid, most often air or water, over obstructing objects. Smooth (laminar) flow starts to break up when the size of the obstruction or the velocity of the flow become large enough compared to the viscosity of the fluid. Meanders are sinuous bends in rivers or other channels, which form as a fluid, most often water, flows around
bends. As soon as the path is slightly curved, the size and curvature of each loop increases as helical flow drags material like sand and gravel across the river to the inside of the bend. The outside of the loop is left clean and unprotected, so erosion accelerates, further increasing the meandering in a powerful positive feedback loop. Meander formation is a result of natural factors and processes. The waveform configuration of a stream is constantly changing. There are a number of theories as to why streams of any size become sinuous.
PHOTOS TOP TO BOTTOM: The curvy bends in rivers which forms as water flows around them, Glen Canyon, https://goo.gl/ n3DZH3; Meanders: sinuous path near San Francisco Bay
PATTERNS IN NATURE
25
Wind waves on a sea surface showing patterns of ripples
26
PATTERNS IN NATURE
Waves & Dunes Waves are disturbances that carry energy as they move. Mechanical waves propagate through a medium – air or water, making it oscillate as they pass by.
a range of patterns including crescents, very long straight lines, stars,domes, parabolas, and longitudinal or Seif ('sword') that undulate like snakes.
Wind waves are sea surface waves that create the characteristic chaotic pattern of any large body of water, though their statistical behavior can be predicted with wind wave models. As waves in water or wind pass over sand, they create patterns of ripples. When winds blow over large bodies of sand, they create dunes, sometimes in extensive dune fields as in the Taklamakan desert. Dunes may form
Barchans or crescent dunes are produced by wind acting on desert sand; the two horns of the crescent and the slip face point downwind. Sand blows over the upwind face, which stands at about 15 degrees from the horizontal, and falls on to the slip face, where it accumulates up to the angle of repose of the sand, which is about 35 degrees. When the slip face exceeds
the angle of repose, the sand avalanches, which is a nonlinear behavior: the addition of many small amounts of sand causes nothing much to happen, but then the addition of a further small amount suddenly causes a large amount to avalanche. Apart from this nonlinearity, barchans behave rather like solitary waves. Dunes have been seen on Mars, which also has vast sandy deserts stirred by winds-but the different planetary conditions may product patterns found nowhere on Earth.
PATTERNS IN NATURE
27
Tessellations The compound eyes of insects are packed hexagonally, just like the bubbles of a bubble raft—although, in fact, each facet is a lens connected to a long, thin, retinal cell beneath. The structures that are formed by clusters of biological cells often have forms governed by much the same rules as foams and bubble rafts—for example,
28
PATTERNS IN NATURE
just three cell walls meet at any vertex. The microscopic structure of the facets of a wasp’s eye—beyond what is visible here—supplies one of the best examples. Each facet contains a cluster of four light-sensitive cells that have the same shape as a cluster of four ordinary bubbles. The face and body of a wasp also has same structure.
PHOTOS CLOCKWISE: Wasp showing hexagonal shapes in eyes, face and body; Sugar apple skin showing tessellations; Pomegranate seeds placed in the form of a giraffe head and neck; Wings of drangonfly showing tessellations
Tessellations are patterns formed by repeating tiles all over a flat surface. The cells in the paper nests of social wasps, and the wax cells in honeycomb built by honey bees are well-known examples. The honeycombs in which bees store their amber nectar are marvels of precision engineering, an array of prismshaped cells with a perfectly hexagonal cross-section. The wax walls are made with a very precise thickness, the cells are gently tilted from the horizontal to prevent the viscous honey from running out, and the entire comb is aligned with the Earth’s magnetic field. Yet this structure is made without any blueprint or foresight, by many bees working
simultaneously and somehow coordinating their efforts to avoid mismatched cells. The ancient Greek philosopher Pappus of Alexandria thought that the bees must be endowed with “a certain geometrical forethought.” And who could have given them this wisdom, but God? According to William Kirby in 1852, bees are “Heaven-instructed mathematicians.” Charles Darwin wasn’t so sure, and he conducted experiments to establish whether bees are able to build perfect honeycombs using nothing but evolved and inherited instincts, as his theory of evolution would imply.
PATTERNS IN NATURE
29
Wasps building their nest using hexagonally shaped paper-like material attached together
30
PATTERNS IN NATURE
Nature prefer's hexagons; It’s a simple matter of geometry. If you want to pack together cells that are identical in shape and size so that they fill all of a flat plane, only three regular shapes (with all sides and angles identical) will work: equilateral triangles, squares, and hexagons. Of these, hexagonal cells require the least total length of wall, compared with triangles or squares of the same area. So it makes sense that bees would choose hexagons, since making wax costs them energy, and they will want to use up as little as possible—just as builders might want to save on the cost of
bricks. This was understood in the 18th century, and Darwin declared that the hexagonal honeycomb is “absolutely perfect in economizing labor and wax.” Darwin thought that natural selection had endowed bees with instincts for making these wax chambers, which had the advantage of requiring less energy and time than those with other shapes. But even though bees do seem to possess specialized abilities to measure angles and wall thickness, not everyone agrees about how much they have to rely on them. That’s because making hexagonal arrays of cells is something that nature does anyway.
PATTERNS IN NATURE
31
32
PATTERNS IN NATURE
Spots & Stripes
PHOTOS CLOCKWISE FROM TOP LEFT: Spotted jellyfish; Running deer with white spots; Ladybug showing zig-zag stripe pattern; Yellow fish with blue eyes and blue stripes;
Leopards and ladybirds are spotted; angelfish and zebras are striped. These patterns have an evolutionary explanation: they have functions which increase the chances that the offspring of the patterned animal will survive to reproduce. One function of animal patterns is camouflage; for instance, a leopard that is harder to see catches more prey. Another function is signaling—for instance, a ladybird is less likely to be attacked by predatory birds that hunt by sight, if it has bold warning colors, and is also distastefully bitter or poisonous, or mimics
other distasteful insects. A young bird may see a warning patterned insect like a ladybird and try to eat it, but it will only do this once; very soon it will spit out the bitter insect; the other ladybirds in the area will remain unmolested. The young leopards and ladybirds, inheriting genes that somehow create spottiness, survive. But while these evolutionary and functional arguments explain why these animals need their patterns for multiple reasons discussed here, they do not explain how the patterns are formed until now.
PATTERNS IN NATURE
33
Mathematics is the science of patterns, and nature exploits just about every pattern that there is.
ian stewart
Butterfly wings showing white spots, tessellation like patterns and scales along the edge
34
PATTERNS IN NATURE
Pattern Formations Alan Turing and later the mathematical biologist James Murray, described a mechanism that spontaneously creates spotted or striped patterns: a reaction-diffusion system. The cells of a young organism have genes that can be switched on by a chemical signal, a morphogen, resulting in the growth of a certain type of structure, say a darkly pigmented patch of skin. If the morphogen is present everywhere, the result is an even pigmentation, as in a black leopard. But if it is unevenly distributed, spots or stripes can result. Turing suggested that there could be
feedback control of the production of the morphogen itself. This could cause continuous fluctuations in the amount of morphogen as it diffused around the body. A second mechanism is needed to create standing wave patterns (to result in spots or stripes): an inhibitor chemical that switches off production of the morphogen, and that itself diffuses through the body more quickly than the morphogen, resulting in an activator inhibitor scheme. The Belousov–Zhabotinsky reaction is a non-biological example of this kind of scheme, a chemical oscillator. PATTERNS IN NATURE
35
Later research has managed to create convincing models of patterns as diverse as zebra stripes, giraffe blotches, jaguar spots and ladybird shell patterns. Richard Prum's activation-inhibition models, developed from Turing's work, use six variables to account for the observed range of nine basic within-feather pigmentation patterns, from the simplest, a central pigment patch, via concentric patches, bars, chevrons, eye spot, pair of central spots, rows of paired spots and an array of dots. More elaborate models simulate complex feather patterns in the Guinea fowl, Numida meleagris, in which the individual feathers feature transitions from bars at the base to an array of dots at the far (distal) end. These require an oscillation created by two inhibiting signals, with interactions in both space and time. 36
PATTERNS IN NATURE
PHOTOS LEFT TO RIGHT: Spines coming out of a stem showing fractal patterns; Flower center showing fibonacci pattern; Fractal-like formations of rock in Badlands national park
PATTERNS IN NATURE
37
Butterfly in motion, showing large white spots and wings in symmetry
30 38
PATTERNS PATTERNS IN IN NATURE NATURE
Patterns can form for other reasons in the vegetated landscape of tiger bush and fir waves. Tiger bush stripes occur on arid slopes where plant growth is limited by rainfall. Each roughly horizontal stripe of vegetation effectively collects the rainwater from the bare zone immediately above it. Fir waves occur in forests on mountain slopes after wind disturbance, during regeneration. When trees fall, the trees that they had sheltered become exposed and are in turn more likely to be damaged, so gaps tend to expand downwind. Meanwhile, on the windward side, young trees grow, protected by the wind shadow of the remaining tall trees. Natural patterns are
sometimes formed by animals, as in the Mima mounds of the Northwestern United States and some other areas, which appear to be created over many years by the burrowing activities of pocket gophers. In permafrost soils with an active upper layer subject to annual freeze and thaw, patterned ground can form, creating circles, nets, ice wedge polygons, steps, and stripes. Thermal contraction causes shrinkage cracks to form; in a thaw, water fills the cracks, expanding to form ice when next frozen, and widening the cracks into wedges. These cracks may join up to form polygons and other shapes.
PATTERNS IN NATURE
39
References Earth’s Most Stunning Natural Fractal Patterns (2010, October 9). Retrieved from https://www.wired.com/2010/09/ fractal-patterns-in-nature/
Selection (2001, August).Retrieved from https://answersingenesis.org/natural-selection/peacock-tail-beauty-and-problems-theory-of-sexual-selection/
Patterns – The Art, Soul, and Science of Beholding Nature (2013, November 17). Retrieved from http://www.patternsinnature.org/Book/PhysicalLaws.html
The Science Behind Nature’s Patterns (2016, May 10). Retrieved from http:// www.smithsonianmag.com/science-nature/science-behind-natures-patterns-180959033/?no-ist
Patterns in Nature (2016, October 10). Retrieved from https://en.wikipedia.org/ wiki/Patterns_in_nature The Beauty of the Peacock Tail and the Problems with the Theory of Sexual
40
PATTERNS IN NATURE
Why Nature Prefers Hexagons (2016, April 7). Retrieved from http://nautil.us/ issue/35/boundaries/why-nature-prefers-hexagons