Spiral In Nature And Art Essay

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. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.

In the 19th century, Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. Scottish biologist D'Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, British mathematician Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. Hungarian biologist Aristid Lindenmayer and French American mathematician Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns.

Mathematics, physics and chemistry can explain patterns in nature at different levels. Patterns in living things are explained by the biological processes of natural selection and sexual selection. Studies of pattern formation make use of computer models to simulate a wide range of patterns.


Early Greek philosophers attempted to explain order in nature, anticipating modern concepts. Plato (c. 427 – c. 347 BC) — looking only at his work on natural patterns — argued for the existence of universals. He considered these to consist of ideal forms (εἶδοςeidos: "form") of which physical objects are never more than imperfect copies. Thus, a flower may be roughly circular, but it is never a perfect mathematical circle.[2]Pythagoras explained patterns in nature like the harmonies of music as arising from number, which he took to be the basic constituent of existence.[3]Empedocles to an extent anticipated Darwin's evolutionary explanation for the structures of organisms.[4]

In 1202, Leonardo Fibonacci (c. 1170 – c. 1250) introduced the Fibonacci number sequence to the western world with his book Liber Abaci.[5] Fibonacci gave an (unrealistic) biological example, on the growth in numbers of a theoretical rabbit population.[6]

In 1658, the English physician and philosopher Sir Thomas Browne discussed "how Nature Geometrizeth" in The Garden of Cyrus, citing Pythagorean numerology involving the number 5, and the Platonic form of the quincunx pattern. The discourse's central chapter features examples and observations of the quincunx in botany.[7]

In 1917, D'Arcy Wentworth Thompson (1860–1948) 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.[8]

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, formulating 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.[10][11][12]

Ernst Haeckel (1834–1919) painted beautiful illustrations of marine organisms, in particular Radiolaria, emphasising 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.[14]

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.[15] He predicted oscillatingchemical 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.[17] 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, Wacław Sierpiński and others, Benoît Mandelbrot wrote a famous paper, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, crystallising mathematical thought into the concept of the fractal.[18]


Living things like orchids, hummingbirds, and the peacock's tail have abstract designs with a beauty of form, pattern and colour that artists struggle to match.[19] 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.[21][22] 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.[17]

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.[23] 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,[24]sexual selection,[24] and different kinds of signalling, including mimicry[25] and cleaning symbiosis.[26] In plants, the shapes, colours, and patterns of insect-pollinatedflowers like the lily have evolved to attract insects such as bees. Radial patterns of colours and stripes, some visible only in ultraviolet light serve as nectar guides that can be seen at a distance.[27]

Types of pattern[edit]


Further information: Symmetry in biology, Floral symmetry, and Crystal 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 echinoderms, the group that includes starfish, sea urchins, and sea lilies.

Among non-living things, snowflakes have striking sixfold symmetry; each flake's structure forms 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). Rotational symmetry is 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.

Symmetry has a variety of causes. Radial symmetry suits organisms like sea anemones whose adults do not move: food and threats may arrive from any direction. But 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 specialised with a mouth and sense organs (cephalisation), and the body becomes bilaterally symmetric (though internal organs need not be).[34] More puzzling is the reason for the fivefold (pentaradiate) symmetry of the echinoderms. 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.[35]

Trees, fractals[edit]

Fractals are infinitely self-similar, iterated mathematical constructs having fractal dimension.[18][36][37] Infinite iteration is not possible in nature so all 'fractal' patterns are only approximate. For example, the leaves of ferns and umbellifers (Apiaceae) are only self-similar (pinnate) to 2, 3 or 4 levels. Fern-like growth patterns occur in plants and in animals including bryozoa, corals, hydrozoa like the air fern, Sertularia argentea, and in non-living things, notably electrical discharges. Lindenmayer system fractals can model different patterns of tree growth by varying a small number of parameters including branching angle, distance between nodes or branch points (internode length), and number of branches per branch point.[17]

Fractal-like patterns occur widely in nature, in phenomena as diverse as clouds, river networks, geologic fault lines, mountains, coastlines,[38]animal coloration, snow flakes,[39]crystals,[40]blood vessel branching,[41] and ocean waves.[42]


Further information: phyllotaxis

Spirals are common in plants and in some animals, notably molluscs. For example, in the nautilus, a cephalopod mollusc, each chamber of its shell is an approximate copy of the next one, scaled by a constant factor and arranged in a logarithmic spiral.[43] 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[45]) of other parts as in compositeflower heads and seed heads like the sunflower or fruit structures like the pineapple[46]:337 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. Phyllotaxis spirals can be generated mathematically from Fibonacci ratios: the Fibonacci sequence runs 1, 1, 2, 3, 5, 8, 13... (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.[48] In disc phyllotaxis as in the sunflower and daisy, the florets are arranged in Fermat's spiral with Fibonacci numbering, at least when the flowerhead is mature so all the elements are the same size. Fibonacci ratios approximate the golden angle, 137.508°, which governs the curvature of Fermat's spiral.[49]

From the point of view of physics, spirals are lowest-energy configurations[50] which emerge spontaneously through self-organizing processes in dynamic systems.[51] From the point of view of chemistry, a spiral can be generated by a reaction-diffusion process, involving both activation and inhibition. Phyllotaxis is controlled by proteins that manipulate the concentration of the plant hormone auxin, which activates meristem growth, alongside other mechanisms to control the relative angle of buds around the stem. 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.[46]

Chaos, flow, meanders[edit]

In mathematics, a dynamical system is chaotic if it is (highly) sensitive to initial conditions (the so-called "butterfly effect"[53]), which requires the mathematical properties of topological mixing and denseperiodic orbits.[54]

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.[55] Some cellular automata, simple sets of mathematical rules that generate patterns, have chaotic behaviour, notably Stephen Wolfram's Rule 30.[56]

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.[57] 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.[58]

Waves, dunes[edit]

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.[59]Wind waves are sea surface waves that create the characteristic chaotic pattern of any large body of water, though their statistical behaviour can be predicted with wind wave models.[60] 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 a range of patterns including crescents, very long straight lines, stars, domes, parabolas, and longitudinal or seif ('sword') shapes.[61]

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 onto 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 behaviour: 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.[62] Apart from this nonlinearity, barchans behave rather like solitary waves.[63]

Bubbles, foam[edit]

A soap bubble forms a sphere, a surface with minimal area — the smallest possible surface area for the volume enclosed. Two bubbles together form a more complex shape: the outer surfaces of both bubbles are spherical; these surfaces are joined by a third spherical surface as the smaller bubble bulges slightly into the larger one.

A foam is a mass of bubbles; foams of different materials occur in nature. Foams composed of soap films obey Plateau's laws, which require three soap films to meet at each edge at 120° and four soap edges to meet at each vertex at the tetrahedral angle of about 109.5°. Plateau's laws further require films to be smooth and continuous, and to have a constant average curvature at every point. For example, a film may remain nearly flat on average by being curved up in one direction (say, left to right) while being curved downwards in another direction (say, front to back).[65] Structures with minimal surfaces can be used as tents. Lord Kelvin identified the problem of the most efficient way to pack cells of equal volume as a foam in 1887; his solution uses just one solid, the bitruncated cubic honeycomb with very slightly curved faces to meet Plateau's laws. No better solution was found until 1993 when Denis Weaire and Robert Phelan proposed the Weaire–Phelan structure; the Beijing National Aquatics Center adapted the structure for their outer wall in the 2008 Summer Olympics.

At the scale of living cells, foam patterns are common; radiolarians, spongespicules, silicoflagellateexoskeletons and the calcite skeleton of a sea urchin, Cidaris rugosa, all resemble mineral casts of Plateau foam boundaries.[68] The skeleton of the Radiolarian, Aulonia hexagona, a beautiful marine form drawn by Ernst Haeckel, looks as if it is a sphere composed wholly of hexagons, but this is mathematically impossible. The Euler characteristic states that for any convex polyhedron, the number of faces plus the number of vertices (corners) equals the number of edges plus two. A result of this formula is that any closed polyhedron of hexagons has to include exactly 12 pentagons, like a soccer ball, Buckminster Fullergeodesic dome, or fullerene molecule. This can be visualised by noting that a mesh of hexagons is flat like a sheet of chicken wire, but each pentagon that is added forces the mesh to bend (there are fewer corners, so the mesh is pulled in).


Main article: tessellation

Tessellations are patterns formed by repeating tiles all over a flat surface. There are 17 wallpaper groups of tilings.[70] While common in art and design, exactly repeating tilings are less easy to find in living things. The cells in the paper nests of social wasps, and the wax cells in honeycomb built by honey bees are well-known examples. Among animals, bony fish, reptiles or the pangolin, or fruits like the salak are protected by overlapping scales or osteoderms, these form more-or-less exactly repeating units, though often the scales in fact vary continuously in size. Among flowers, the snake's head fritillary, Fritillaria meleagris, have a tessellated chequerboard pattern on their petals. The structures of minerals provide good examples of regularly repeating three-dimensional arrays. Despite the hundreds of thousands of known minerals, there are rather few possible types of arrangement of atoms in a crystal, defined by crystal structure, crystal system, and point group; for example, there are exactly 14 Bravais lattices for the 7 lattice systems in three-dimensional space.[71]


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 fibres. Since each species of tree has its own structure at the levels of cell and of molecules, each has its own pattern of splitting in its bark.

Composite patterns: aphids and newly born young in arraylike clusters on sycamore leaf, divided into polygons by veins, which are avoided by the young aphids


The Fibonacci sequence exhibits a certain numerical pattern which originated as the answer to an exercise in the first ever high school algebra text. This pattern turned out to have an interest and importance far beyond what its creator imagined. It can be used to model or describe an amazing variety of phenomena, in mathematics and science, art and nature. The mathematical ideas the Fibonacci sequence leads to, such as the golden ratio, spirals and self- similar curves, have long been appreciated for their charm and beauty, but no one can really explain why they are echoed so clearly in the world of art and nature.

The story began in Pisa, Italy in the year 1202. Leonardo Pisano Bigollo was a young man in his twenties, a member of an important trading family of Pisa. In his travels throughout the Middle East, he was captivated by the mathematical ideas that had come west from India through the Arabic countries. When he returned to Pisa he published these ideas in a book on mathematics called Liber Abaci, which became a landmark in Europe. Leonardo, who has since come to be known as Fibonacci, became the most celebrated mathematician of the Middle Ages. His book was a discourse on mathematical methods in commerce, but is now remembered mainly for two contributions, one obviously important at the time and one seemingly insignificant.

The important one: he brought to the attention of Europe the Hindu system for writing numbers. European tradesmen and scholars were still clinging to the use of the old Roman numerals; modern mathematics would have been impossible without this change to the Hindu system, which we call now Arabic notation, since it came west through Arabic lands.

The other: hidden away in a list of brain-teasers , Fibonacci posed the following question:

If a pair of rabbits is placed in an enclosed area, how many rabbits will be born there if we assume that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their birth?
This apparently innocent little question has as an answer a certain sequence of numbers, known now as the Fibonacci sequence, which has turned out to be one of the most interesting ever written down. It has been rediscovered in an astonishing variety of forms, in branches of mathematics way beyond simple arithmetic. Its method of development has led to far-reaching applications in mathematics and computer science.

But even more fascinating is the surprising appearance of Fibonacci numbers, and their relative ratios, in arenas far removed from the logical structure of mathematics: in Nature and in Art, in classical theories of beauty and proportion.

Consider an elementary example of geometric growth - asexual reproduction, like that of the amoeba. Each organism splits into two after an interval of maturation time characteristic of the species. This interval varies randomly but within a certain range according to external conditions, like temperature, availability of nutrients and so on. We can imagine a simplified model where, under perfect conditions, all amoebae split after the same time period of growth.

So, one amoebas becomes two, two become 4, then 8, 16, 32, and so on.

We get a doubling sequence. Notice the recursive formula:

This of course leads to exponential growth, one characteristic pattern of population growth.

Now in the Fibonacci rabbit situation, there is a lag factor; each pair requires some time to mature. So we are assuming

  • maturation time = 1 month
  • gestation time = 1 month
If you were to try this in your backyard, here's what would happen:

Now let the computer draw a few more lines:

The pattern we see here is that each cohort or generation remains as part of the next, and in addition, each grown-up pair contributes a baby pair. The number of such baby pairs matches the total number of pairs in the previous generation. Symbolically

  • fn = number of pairs during month n
  • fn = fn-1 + fn-2

So we have a recursive formula where each generation is defined in terms of the previous two generations. Using this approach, we can successively calculate fn for as many generations as we like.

So this sequence of numbers 1,1,2,3,5,8,13,21,... and the recursive way of constructing it ad infinitum, is the solution to the Fibonacci puzzle. But what Fibonacci could not have foreseen was the myriad of applications that these numbers and this method would eventually have. His idea was more fertile than his rabbits. Just in terms of pure mathematics - number theory, geometry and so on - the scope of his idea was so great that an entire professional journal has been devoted to it - the Fibonacci Quarterly.

Now let's look at another reasonably natural situation where the same sequence "mysteriously" pops up. Go back 350 years to 17th century France. Blaise Pascal is a young Frenchman, scholar who is torn between his enjoyment of geometry and mathematics and his love for religion and theology. In one of his more worldly moments he is consulted by a friend, a professional gambler, the Chevalier de Mé ré , Antoine Gombaud. The Chevalier asks Pascal some questions about plays at dice and cards, and about the proper division of the stakes in an unfinished game. Pascal's response is to invent an entirely new branch of mathematics, the theory of probability. This theory has grown over the years into a vital 20th century tool for science and social science. Pascal's work leans heavily on a collection of numbers now called Pascal's Triangle, and represented like this:

This configuration has many interesting and important properties:

  • Notice the left-right symmetry - it is its own mirror image.
  • Notice that in each row, the second number counts the row.
  • Notice that in each row, the 2nd + the 3rd counts the number of numbers above that line.
There are endless variations on this theme.

Next, notice what happens when we add up the numbers in each row - we get our doubling sequence.

Now for visual convenience draw the triangle left-justified. Add up the numbers on the various diagonals ...

and we get 1, 1, 2, 3, 5, 8, 13, . . . the Fibonacci sequence!

Fibonacci could not have known about this connection between his rabbits and probability theory - the theory didn't exist until 400 years later.

What is really interesting about the Fibonacci sequence is that its pattern of growth in some mysterious way matches the forces controlling growth in a large variety of natural dynamical systems. Quite analogous to the reproduction of rabbits, let us consider the family tree of a bee - so we look at ancestors rather than descendants. In a simplified reproductive model, a male bee hatches from an unfertilized egg and so he has only one parent, whereas a female hatches from a fertilized egg, and has two parents. Here is the family tree of a typical male bee:

Notice that this looks like the bunny chart, but moving backwards in time. The male ancestors in each generation form a Fibonacci sequence, as do the female ancestors, as does the total. You can see from the tree that bee society is female dominated.

The most famous and beautiful examples of the occurrence of the Fibonacci sequence in nature are found in a variety of trees and flowers, generally asociated with some kind of spiral structure. For instance, leaves on the stem of a flower or a branch of a tree often grow in a helical pattern, spiraling aroung the branch as new leaves form further out. Picture this: You have a branch in your hand. Focus your attention on a given leaf and start counting around and outwards. Count the leaves, and also count the number of turns around the branch, until you return to a position matching the original leaf but further along the branch. Both numbers will be Fibonacci numbers.

For example, for a pear tree there will be 8 leaves and 3 turns. Here are some more examples:

Branches of the Fibonacci Family
Tree Leaves Turns
Elm 2 1
Cherry 3 2
Beech 3 1
Poplar 52
Weeping willow 8 3
Pear 8 3
Almond 13 8

You can take a walk in a park and find this pattern on plants and bushes quite easily.

Many flowers offer a beautiful confirmation of the Fibonacci mystique. A daisy has a central core consisting of tiny florets arranged in opposing spirals. There are usually 21 going to the left and 34 to the right. A mountain aster may have 13 spirals to the left and 21 to the right. Sunflowers are the most spectacular example, typically having 55 spirals one way and 89 in the other; or, in the finest varieties, 89 and 144.

Pine cones are also constructed in a spiral fashion, small ones having commonly with 8 spirals one way and 13 the other. The most interesting is the pineapple - built from adjacent hexagons, three kinds of spirals appear in three dimensions. There are 8 to the right, 13 to the left, and 21 vertically - a Fibonacci triple.

Why should this be? Why has Mother Nature found an evolutionary advantage in arranging plant structures in spiral shapes exhibiting the Fibonacci sequence?

We have no certain answer. In 1875, a mathematician named Wiesner provided a mathematical demonstration that the helical arrangement of leaves on a branch in Fibonacci proportions was an efficient way to gather a maximum amount of sunlight with a few leaves - he claimed, the best way. But recently, a Cornell University botanist named Karl Niklas decided to test this hypothesis in his laboratory; he discovered that almost any reasonable arrangement of leaves has the same sunlight-gathering capability. So we are still in the dark about light.

But if we think in terms of natural growth patterns I think we can begin to understand the presence of spirals and the connection between spirals and the Fibonacci sequence.

Spirals arise from a property of growth called self-similarity or scaling - the tendency to grow in size but to maintain the same shape. Not all organisms grow in this self-similar manner. We have seen that adult people, for example, are not just scaled up babies: babies have larger heads, shorter legs, and a longer torso relative to their size. But if we look for example at the shell of the chambered nautilus we see a differnet growth pattern. As the nautilus outgrows each chamber, it builds new chambers for itself, always the same shape - if you imagine a very long-lived nautilus, its shell would spiral around and around, growing ever larger but always looking exactly the same at every scale.

Here is where Fibonacci comes in - we can build a squarish sort of nautilus by starting with a square of size 1 and successively building on new rooms whose sizes correspond to the Fibonacci sequence:

Running through the centers of the squares in order with a smooth curve we obtain the nautilus spiral = the sunflower spiral.

This is a special spiral, a self-similar curve which keeps its shape at all scales (if you imagine it spiraling out forever). It is called equiangular because a radial line from the center makes always the same angle to the curve. This curve was known to Archimedes of ancient Greece, the greatest geometer of ancient times, and maybe of all time.

We should really think of this curve as spiraling inward forever as well as outward. It is hard to draw; you can visualize water swirling around a tiny drainhole, being drawn in closer as it spirals but never falling in. This effect is illustrated by another classical brain-teaser:

Four bugs are standing at the four corners of a square. They are hungry (or lonely) and at the same moment they each see the bug at the next corner over and start crawling toward it. What happens?
The picture tells the story. As they crawl towards each other they spiral into the center, always forming an ever smaller square, turning around and around forever. Yet they reach each other! This is not a paradox because the length of this spiral is finite. They trace out the same equiangular spiral.

Now since all these spirals are self-similar they look the same at every scale - the scale does not matter. What matters is the proportion - these spirals have a fixed proportion determining their shape. It turns out that this proportion is the same as the proportions generated by successive entries in the Fibonacci sequence: 5:3, 8:5,13:8, and so on. Here is the calculation:

Fibonacci Proportions

As we go further out in the sequence, the proportions of adjacent terms begins to approach a fixed limiting value of 1.618034 . . . This is a very famous ratio with a long and honored history; the Golden Mean of Euclid and Aristotle, the divine proportion of Leonardo daVinci, considered the most beautiful and important of quantities. This number has more tantalizing properties than you can imagine.

By simple calculation, we see that if we subtract 1 we get .618 . . which is its reciprocal. If we add 1 we get 2.618 . . . which is its square.

Using the traditional name for this number, the Greek letter ("phi") we can write symbolically:

Solving this quadratic equation we obtain

Here are some other strange but fascinating expressions that can be derived:

, an infinite cascade of square roots.

, an infinite cascade of fractions.

Using this golden ratio as a foundation, we can build an explicit formula for the Fibonacci numbers:

Formula for the Fibonacci numbers:

But the Greeks had a more visual point of view about the golden mean. They asked: what is the most natural and well-proportioned way to divide a line into 2 pieces? They called this a section. The Greeks felt strongly that the ideal should match the proportion between the parts with that of the parts to the whole. This results in a proportion of exactly .

Forming a rectangle with the sections of the line as sides results in a visually pleasing shape that was the basis of their art and architecture. This esthetic was adopted by the great Renaissance artists in their painting, and is still with us today.

Dan Reich
Department of Mathematics, Temple University

Translations of this page include:
A Ukrainian translation provided by Anna Matesh, a student at Kyiv International University.
A Finnish translation posted on Elsa Jansson's blog.

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