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2805:. 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
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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.
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2852:(troughs) have been demonstrated in models of the brain starting from smooth, layered gels, with the patterns caused by compressive mechanical forces resulting from the expansion of the outer layer (representing the cortex) after the addition of a solvent. Numerical models in computer simulations support natural and experimental observations that the surface folding patterns increase in larger brains.
2783:'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 guineafowl
1950:
7052:
865:), and the body becomes bilaterally symmetric (though internal organs need not be). 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. In the case of
1084:
A more general version states that when a parent branch splits into two or more child branches, the surface areas of the child branches add up to that of the parent branch. An equivalent formulation is that if a parent branch splits into two child branches, then the cross-sectional diameters of the
860:
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
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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
2758:, 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
668:
have abstract designs with a beauty of form, pattern and colour 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
2616:
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 undisturbed. The young leopards and ladybirds, inheriting
2778:
Later research has managed to create convincing models of patterns as diverse as zebra stripes, giraffe blotches, jaguar spots (medium-dark patches surrounded by dark broken rings) and ladybird shell patterns (different geometrical layouts of spots and stripes, see illustrations).
1879:
that create the characteristic chaotic pattern of any large body of water, though their statistical behaviour 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
2005:) — 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.
1473:
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
3109:
Aristotle reports
Empedocles arguing that, "herever, then, everything turned out as it would have if it were happening for a purpose, there the creatures survived, being accidentally compounded in a suitable way; but where this did not happen, the creatures perished."
1407:, 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 from
2028:
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). Structures with minimal surfaces can be used as tents.
1411:: 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
1910:
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. Apart from this nonlinearity, barchans behave rather like
2766:
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
2789:
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.
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565:
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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).
6851:
935:
2084:, 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
478:; his description of phyllotaxis and the Fibonacci sequence, the mathematical relationships in the spiral growth patterns of plants showed that simple equations could describe the spiral growth patterns of
2832:
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.
600:
2840:
that develops on vertebrate brains is caused by a physical process of constrained expansion dependent on two geometric parameters: relative tangential cortical expansion and relative thickness of the
698:
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
1713:
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
869:, the gentle churn of water, blown by a suitably stiff breeze makes concentric layers of ice form on a seed particle that then grows into a floating ball as it rolls through the freezing currents.
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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
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that somehow create spottedness, survive. But while these evolutionary and functional arguments explain why these animals need their patterns, they do not explain how the patterns are formed.
1731:
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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
2297:
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
1296:
579:
4275:
Takeda, T; Ishikawa, A; Ohtomo, K; Kobayashi, Y; Matsuoka, T (February 1992). "Fractal dimension of dendritic tree of cerebellar
Purkinje cell during onto- and phylogenetic development".
4260:
Hahn, Horst K.; Georg, Manfred; Peitgen, Heinz-Otto (2005). "Fractal aspects of three-dimensional vascular constructive optimization". In Losa, Gabriele A.; Nonnenmacher, Theo F. (eds.).
1781:
3658:
3092:, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.
3985:
Sumrall, Colin D.; Wray, Gregory A. (January 2007). "Ontogeny in the fossil record: diversification of body plans and the evolution of "aberrant" symmetry in
Paleozoic echinoderms".
2195:
2106:
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174:
pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, the
British mathematician
2168:
2544:
2505:
248:(c. 570–c. 495 BC) explained patterns in nature like the harmonies of music as arising from number, which he took to be the basic constituent of existence.
2384:
1461:. 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
2437:
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1898:
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
2717:
1837:
2152:
1143:
fractals can model different patterns of tree growth by varying a small number of parameters including branching angle, distance between nodes or branch points (
1254:
2698:
2489:
1888:
desert. Dunes may form a range of patterns including crescents, very long straight lines, stars, domes, parabolas, and longitudinal or seif ('sword') shapes.
829:; 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.
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1581:
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2287:, 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,
3010:
2528:
1320:
1965:
1200:
2564:
1614:
1597:
3524:
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of tilings. 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
4211:
Progress in wavelet analysis and applications: proceedings of the
International Conference "Wavelets and Applications," Toulouse, France – June 1992
895:
875:
277:: "form") of which physical objects are never more than imperfect copies. Thus, a flower may be roughly circular, but it is never a perfect circle.
2817:
of
Namibia appear to be created by the interaction of competing groups of sand termites, along with competition for water among the desert plants.
515:") of stripes and spots in animals, and contribute to the spiral patterns seen in plant phyllotaxis. In 1968, the Hungarian theoretical biologist
4425:
425:
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
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616:
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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
1538:
1450:); when the flowerhead is mature so all the elements are the same size, this spacing creates a Fibonacci number of more obvious spirals.
5386:
421:
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
4036:
2876:
1517:
1089:. One explanation is that this allows trees to better withstand high winds. Simulations of biomechanical models agree with the rule.
2992:
2588:
which increase the chances that the offspring of the patterned animal will survive to reproduce. One function of animal patterns is
3949:
1115:(Apiaceae) are only self-similar (pinnate) to 2, 3 or 4 levels. Fern-like growth patterns occur in plants and in animals including
2809:
of the
Northwestern United States and some other areas, which appear to be created over many years by the burrowing activities of
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17:
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992:
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seeks to discover and explain abstract patterns or regularities of all kinds. Visual patterns in nature find explanations in
289:(1452–1519) noted the spiral arrangement of leaf patterns, that tree trunks gain successive rings as they age, and proposed
6113:
4209:
2516:
1703:) flow starts to break up when the size of the obstruction or the velocity of the flow become large enough compared to the
953:
1659:
ranks as an essentially universal influence on patterns in nature. There is a relationship between chaos and fractals—the
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and his brother Louis connected phyllotaxis ratios to the
Fibonacci sequence in 1837, also noting its appearance in
6070:
5739:
4156:
3260:
2234:
1809:
496:, an analysis of the mechanisms that would be needed to create patterns in living organisms, in the process called
430:
143:
attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.
6732:
6425:
2953:
5363:
4131:
Minamino, Ryoko; Tateno, Masaki (2014). "Tree
Branching: Leonardo da Vinci's Rule versus Biomechanical Models".
2943:
2024:
angle of about 109.5°. Plateau's laws further require films to be smooth and continuous, and to have a constant
2020:, which require three soap films to meet at each edge at 120° and four soap edges to meet at each vertex at the
1921:
350:
pattern. The discourse's central chapter features examples and observations of the quincunx in botany. In 1754,
5719:
4510:
3044:
2709:
1453:
From the point of view of physics, spirals are lowest-energy configurations which emerge spontaneously through
1080:
All the branches of a tree at every stage of its height when put together are equal in thickness to the trunk .
1026:
1365:. Given a modern understanding of fractals, a growth spiral can be seen as a special case of self-similarity.
212:
can explain patterns in nature at different levels and scales. Patterns in living things are explained by the
6410:
6335:
4693:
4668:
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2644:
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281:(c. 372–c. 287 BC) noted that plants "that have flat leaves have them in a regular series";
7082:
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607:
426:
422:
5200:
841:). Rotational symmetry is found at different scales among non-living things, including the crown-shaped
7092:
6560:
6390:
5548:
2085:
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6267:
5836:
4801:
2076:, looks as if it is a sphere composed wholly of hexagons, but this is mathematically impossible. The
1311:
1107:
is not possible in nature so all 'fractal' patterns are only approximate. For example, the leaves of
6545:
5273:
1949:
1894:
or crescent dunes are produced by wind acting on desert sand; the two horns of the crescent and the
934:
413:
with a given boundary, which is now named after him. He studied soap films intensively, formulating
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5643:
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1361:
of its shell is an approximate copy of the next one, scaled by a constant factor and arranged in a
979:
842:
261:
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5846:
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1800:
324:(1571–1630) pointed out the presence of the Fibonacci sequence in nature, using it to explain the
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1902:
of the sand, which is about 35 degrees. When the slip face exceeds the angle of repose, the sand
367:
92:
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4609:
Douady, S.; Couder, Y. (March 1992). "Phyllotaxis as a physical self-organized growth process".
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Leopards and ladybirds are spotted; angelfish and zebras are striped. These patterns have an
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1605:
1240:
691:
665:
4798:
Lecture Notes in Incompressible Fluid Dynamics: Phenomenology, Concepts and Analytical Tools
1673:, simple sets of mathematical rules that generate patterns, have chaotic behaviour, notably
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6752:
6655:
6176:
5436:
4973:
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Schwämmle, V.; Herrman, H. J. (11 December 2003). "Solitary wave behaviour of sand dunes".
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2754:. The cells of a young organism have genes that can be switched on by a chemical signal, a
2467:
2443:
2334:
2174:
2077:
1136:
909:
837:; they can be cubic or octahedral, but true crystals cannot have fivefold symmetry (unlike
434:
406:
6435:
5302:
3960:
752:. Radial patterns of colours and stripes, some visible only in ultraviolet light serve as
560:
191:
8:
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7011:
6986:
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6774:
6769:
6583:
6540:
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3729:
Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe
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1086:
1063:
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795:
787:
770:
516:
511:. These activator-inhibitor mechanisms can, Turing suggested, generate patterns (dubbed "
465:
334:
187:
6500:
5440:
4977:
4910:
4684:
4622:
4579:
4345:
3998:
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679:, fractals, logarithmic spirals, topology and other mathematical patterns. For example,
7006:
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6136:
6018:
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built by honey bees are well-known examples. Among animals, bony fish, reptiles or the
2089:
1754:
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313:
301:
171:
128:
6052:
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
5720:
Phyllotaxis: an Interactive Site for the Mathematical Study of Plant Pattern Formation
4985:
3232:
Singh, Parmanand (1986). "Acharya Hemachandra and the (so called) Fibonacci Numbers".
2012:
is a mass of bubbles; foams of different materials occur in nature. Foams composed of
721:
can cause the development of patterns in living things for several reasons, including
566:
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
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2017:
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329:
286:
225:
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2119:: four edges meet at each vertex, at angles close to 109.5°, as in two C-H bonds in
6520:
6470:
6415:
6330:
6315:
6262:
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5614:
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4002:
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778:
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221:
73:
57:
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3553:
2820:
In permafrost soils with an active upper layer subject to annual freeze and thaw,
2750:, described a mechanism that spontaneously creates spotted or striped patterns: a
7087:
6852:
A Philosophical Enquiry into the Origin of Our Ideas of the Sublime and Beautiful
6405:
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6222:
5819:
5599:
5298:
4141:
3684:
3495:
3459:
3335:
3305:
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2915:, feathers transition from barred to spotted, both in-feather and across the bird
2747:
2668:
2409:
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2182:
2002:
1899:
1864:
1692:
1674:
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Fractal-like patterns occur widely in nature, in phenomena as diverse as clouds,
1096:
983:
774:
410:
375:
371:
321:
282:
265:
229:
183:
155:
4630:
4320:"Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork"
1442:, but this is disguised because successive florets are spaced far apart, by the
32:
6991:
6971:
6841:
6598:
6550:
6510:
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6395:
6375:
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Proceedings of the National Academy of Sciences of the United States of America
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2025:
1972:
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1458:
707:
552:
540:
524:
512:
501:
454:
450:
441:(1834–1919) painted beautiful illustrations of marine organisms, in particular
402:
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of animals and the branching patterns of their veins and nerves, as well as in
351:
253:
163:
147:
45:
5542:
4353:
3356:
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6242:
5997:
5992:
5689:
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5510:
5013:"Geometry and Pattern in Nature 3: The holes in radiolarian and diatom tests"
3245:
3054:
2957:
2810:
2763:
2496:
2375:
2293:, have a tessellated chequerboard pattern on their petals. The structures of
2139:
2092:
2073:
2064:, all resemble mineral casts of Plateau foam boundaries. The skeleton of the
1769:
1746:
1684:
1637:
is chaotic if it is (highly) sensitive to initial conditions (the so-called "
1505:
1183:
963:
862:
838:
834:
753:
690:
apply the abstractions of mathematics to the real world, often as if it were
649:
497:
483:
438:
387:
343:
285:(23–79 AD) noted their patterned circular arrangement. Centuries later,
179:
159:
120:
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52:
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Tallinen, Tuoma; Chung, Jun Young; Biggins, John S.; Mahadevan, L. (2014).
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3545:
3465:
3193:
2998:
2837:
2814:
2780:
2354:
2256:
2251:
1876:
1788:
1714:
1700:
1656:
1447:
1443:
1268:
1179:
699:
676:
548:
390:
explored the golden ratio expressed in the arrangement of plant parts, the
363:
278:
37:
4296:
3488:
6794:
6747:
6603:
6555:
6515:
6465:
6257:
6026:
5984:
5903:
5799:
5794:
5555:
5494:
4189:
Batty, Michael (4 April 1985). "Fractals – Geometry Between Dimensions".
3744:. "Perfection in the Sciences. II. Perfection in Physics and Chemistry".
3519:
3455:
3183:
2983:
2806:
2609:
2310:
2209:
2201:
2159:
2116:
2065:
2051:
2037:
2021:
1994:
1976:
1868:
1844:
1816:
1796:
1524:
1369:
1341:
1013:
901:
803:
799:
672:
661:
487:
479:
418:
355:
307:
201:
175:
69:
4845:. Howard University, USA, 13–18 May 2008. World Scientific Publications.
4235:
Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (2000).
924:
Fivefold symmetry can be seen in many flowers and some fruits like this
622:
D'Arcy Thompson pioneered the study of growth and form in his 1917 book.
6764:
6742:
6635:
6420:
6380:
6340:
6207:
6122:
5973:
5968:
5886:
5789:
5071:(2nd Edition). Manchester Physics Series, John Wiley & Sons, 2010.
4014:
3836:
3654:
3461:
The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
3189:
The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
2928:
2798:
2724:
2589:
2284:
2135:
2055:
1940:
1885:
1792:
1404:
1377:
1373:
1307:
1155:
1112:
846:
815:
811:
722:
442:
249:
245:
140:
136:
96:
65:
2608:
birds that hunt by sight, if it has bold warning colours, and is also
260:(c. 427–c. 347 BC) argued for the existence of natural
6675:
6645:
6440:
6385:
6305:
6237:
6075:
5978:
5724:
5489:
5201:"Reaction–diffusion models of within-feather pigmentation patterning"
4566:
Levitov, L. S. (15 March 1991). "Energetic Approach to Phyllotaxis".
3947:
3093:
3039:
2979:
2949:
2825:
2755:
2581:
2570:
Palm trunk with branching vertical cracks (and horizontal leaf scars)
2350:
2272:
2213:
2186:
2096:
2013:
1903:
1895:
1872:
1738:
1704:
1646:
1569:
1550:
1431:
1396:
1388:
1191:
1171:
1163:
1104:
940:
822:
458:
383:
359:
328:
form of some flowers. In 1658, the English physician and philosopher
297:
293:
purportedly satisfied by the cross-sectional areas of tree-branches.
209:
151:
4918:
4006:
3827:
3810:
606:
Beijing's National Aquatics Center for the 2008 Olympic games has a
6976:
6710:
6660:
6640:
6505:
5804:
5769:
5431:
4336:
2802:
2759:
2705:
2605:
2601:
2459:
2294:
2276:
1545:
1470:
1354:
1350:
1283:
1207:
1159:
1140:
1128:
1124:
1052:
gain spherical symmetry by being rolled about by wind and currents.
1049:
959:
905:
866:
807:
783:
680:
641:
536:
520:
446:
391:
379:
347:
325:
195:
167:
124:
108:
88:
5329:"The secret of Namibia's 'fairy circles' may be explained at last"
6789:
6737:
6695:
6680:
6650:
5930:
5925:
5913:
5891:
5876:
5784:
4843:
CBMS Conference Proceedings on Water Waves: Theory and Experiment
4527:
3002:
2931:
2686:
2613:
2593:
2368:
2298:
2120:
1956:
1912:
1891:
1751:
1741:
1718:
1710:
1678:
1588:
1462:
1416:
1346:
1260:
1245:
1175:
1116:
1092:
830:
741:
730:
714:
703:
695:
687:
645:
570:
532:
395:
213:
205:
104:
41:
5670:
Design by Nature: Using Universal Forms and Principles in Design
4884:
Physical Geography: Science and Systems of the Human Environment
3948:
Hickman, Cleveland P.; Roberts, Larry S.; Larson, Allan (2002).
786:
is pervasive in living things. Animals mainly have bilateral or
48:
on their surfaces repeat wherever there are suitable conditions.
6722:
6700:
6630:
6608:
5253:
2551:
2535:
2512:
2330:
2041:
1998:
1980:
1688:
1424:
1380:
1372:, the arrangement of leaves on a stem, and in the arrangement (
1287:
1223:
1033:
998:
854:
850:
791:
745:
657:
637:
468:
studied the patterns of phyllotaxis in his 1904 book. In 1917,
366:
series. Mathematical observations of phyllotaxis followed with
317:
241:
84:
5256:
Banded vegetation patterning in arid and semiarid environments
6495:
6091:
5963:
5544:
Mathematics in Nature: Modeling Patterns in the Natural World
5412:
4645:
4274:
3560:
3384:
3382:
3017:
2975:
2935:
2845:
2596:
that is harder to see catches more prey. Another function is
2425:
2280:
1721:
accelerates, further increasing the meandering in a powerful
1699:, most often air or water, over obstructing objects. Smooth (
1696:
1466:
1412:
1400:
1392:
1151:
1120:
886:
683:
form convincing models of different patterns of tree growth.
569:, crystallising mathematical thought into the concept of the
269:
257:
256:'s evolutionary explanation for the structures of organisms.
132:
116:
100:
36:
Natural patterns form as wind blows sand in the dunes of the
4964:(July 1976). "The geometry of soap films and soap bubbles".
4037:"Image of the Week – Goodness gracious, great balls of ice!"
6716:
6690:
5841:
5779:
5774:
5387:"Human Brain's Bizarre Folding Pattern Re-Created in a Vat"
5174:
5152:
5150:
5133:. Springer Science & Business Media. pp. 436–450.
5110:
5108:
4161:
Fractal Geometry: Mathematical Foundations and Applications
2618:
2520:
2268:
2260:
2112:
2009:
1881:
1860:
1420:
1384:
1108:
845:
pattern formed when a drop falls into a pond, and both the
252:(c. 494–c. 434 BC) to an extent anticipated
112:
4234:
3379:
6685:
5578:
Nature's Patterns: a tapestry in three parts. 3. Branches
4800:. Syracuse, New York: Syracuse University. Archived from
4207:
3811:"Cleaning symbioses: proximate and adaptive explanations"
2162:; the skeletons of these Radiolaria have foam-like forms.
588:
patterns occur widely in plants such as this queen sago,
5274:"Fir Waves: Regeneration in New England Conifer Forests"
5162:
5147:
5105:
5061:
4694:
10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
3905:
3881:
3869:
3373:
Neue Lehre van den Proportionen des meschlischen Körpers
2390:
Tilings: tessellated flower of snake's head fritillary,
798:, as do many flowers and some groups of animals such as
417:
which describe the structures formed by films in foams.
5560:
Nature's Patterns: a tapestry in three parts. 1: Shapes
4992:
2483:
Old pottery surface, white glaze with mainly 90° cracks
1465:
that manipulate the concentration of the plant hormone
240:
Early Greek philosophers attempted to explain order in
5093:
5081:
5030:
4406:
3635:
3632:. The Guardian. Review: Non-fiction. 11 February 2012.
3394:
2771:
is a non-biological example of this kind of scheme, a
2558:. Vertical mainly 120° cracks giving hexagonal columns
2462:
are linear openings that form in materials to relieve
2317:
for the 7 lattice systems in three-dimensional space.
790:, as do the leaves of plants and some flowers such as
6902:
The Work of Art in the Age of Mechanical Reproduction
5611:
Survival of the Beautiful: Art, Science and Evolution
5569:
Nature's Patterns: a tapestry in three parts. 2: Flow
3929:
3917:
3893:
1867:
propagate through a medium – air or water, making it
27:
Visible regularity of form found in the natural world
5677:
Pattern Sourcebook: 250 Patterns Inspired by Nature.
5640:
What Shape is a Snowflake? Magical Numbers in Nature
1206:
The growth patterns of certain trees resemble these
1062:
The branching pattern of trees was described in the
5378:
5254:Tongway, D. J.; Valentin, C.; Seghieri, J. (2001).
4941:
4491:
3406:
3127:
3016:Human brain (superior view) exhibiting patterns of
861:
becomes specialised with a mouth and sense organs (
5415:"Gyrification from constrained cortical expansion"
5050:
3791:
2746:Alan Turing, and later the mathematical biologist
1641:"), which requires the mathematical properties of
1349:are common in plants and in some animals, notably
1147:length), and number of branches per branch point.
737:. In plants, the shapes, colours, and patterns of
433:adapted the structure for their outer wall in the
5010:
4881:
4791:"Flow Separation and Secondary Flow: Section 9.1"
4237:Pattern formation in biology, vision and dynamics
4106:"The Wisdom of Trees (Leonardo Da Vinci Knew It)"
3525:Philosophical Transactions of the Royal Society B
3217:Da Vinci, Leonardo (1971). Taylor, Pamela (ed.).
1863:are disturbances that carry energy as they move.
453:theories of evolution. The American photographer
91:recur in different contexts and can sometimes be
7069:
5271:
4896:
4737:
4707:
4259:
3304:
3522:(1952). "The Chemical Basis of Morphogenesis".
3340:"Proportion: Science, Philosophy, Architecture"
1884:, sometimes in extensive dune fields as in the
1620:Water droplets fly off a wet, spinning ball in
729:, and different kinds of signalling, including
5195:
4130:
3622:
3428:Encyclopedia of Nineteenth-Century Photography
3280:
3145:
640:and newly born young in arraylike clusters on
186:of spots and stripes. The Hungarian biologist
83:are visible regularities of form found in the
6107:
5740:
4959:
4722:
4457:
4403:. Princeton University Press, 2009. Page 135.
3959:(Third ed.). p. 139. Archived from
3808:
3802:
3609:
3607:
3311:Proportion: Science, Philosophy, Architecture
2424:Tilings: overlapping scales of snakefruit or
2407:Tilings: overlapping scales of common roach,
2228:Equal spheres (gas bubbles) in a surface foam
6832:The Literary Mind and the Carving of Dragons
5303:"'Digital gophers' solve Mima mound mystery"
5249:
5247:
5208:Proceedings of the Royal Society of London B
4608:
4214:. Atlantica Séguier Frontières. p. 25.
3763:
3761:
3759:
3734:
3731:(Scientific American Paperback Library) 1996
3268:Faculty of Engineering and Physical Sciences
3192:(First trade paperback ed.). New York:
833:in general have a variety of symmetries and
669:selection, that govern how patterns evolve.
358:of plants were frequently expressed in both
290:
154:, leading him to formulate the concept of a
4453:
4451:
4135:. Vol. 9, no. 4. p. e93535.
3984:
3450:
3448:
2793:Patterns can form for other reasons in the
2240:Circus tent approximates a minimal surface.
1001:showing rhombic dodecahedral crystal habit.
401:In the 19th century, the Belgian physicist
305:
146:In the 19th century, the Belgian physicist
6114:
6100:
5747:
5733:
5684:Surfaces and Textures: A Visual Sourcebook
5605:
5267:
5265:
5180:
5120:
4754:
4086:: CS1 maint: location missing publisher (
3782:
3780:
3613:
3604:
3314:. Taylor & Francis. pp. 305–306.
3178:
3176:
3174:
3172:
3170:
2337:, isometric hexoctahedral crystal symmetry
1099:, iterated mathematical constructs having
5458:
5448:
5430:
5244:
5227:
5048:
5042:
4692:
4387:Fractals and chaos: an illustrated course
4361:
4335:
4201:
4140:
3826:
3756:
3514:
3512:
3355:
1403:, as well as in the pattern of scales in
1085:parent and the two child branches form a
5715:Fibonacci Numbers and the Golden Section
4886:(4th ed.). John Wiley. p. 442.
4448:
4167:
4155:
4149:
3740:
3445:
3216:
1628:
748:have evolved to attract insects such as
631:
386:. In his 1854 book, German psychologist
51:
31:
5634:
5621:
5501:———— translated by Sigler, Laurence E.
5297:
5262:
5191:
5189:
5099:
5087:
4882:Strahler, A.; Archibold, O. W. (2008).
4788:
4565:
4509:
4384:
4228:
4059:
3957:Chapter 8: Acoelomate Bilateral Animals
3935:
3923:
3911:
3899:
3887:
3875:
3786:
3777:
3641:
3431:. Vol. 1. CRC Press. p. 149.
3424:
3388:
3370:
3334:
3167:
3133:
2882:Detail of giant pufferfish skin pattern
2283:are protected by overlapping scales or
1446:, 137.508° (dividing the circle in the
652:, which are avoided by the young aphids
332:discussed "how Nature Geometrizeth" in
14:
7070:
5754:
5326:
5126:
4833:
4818:
4712:. Chapman & Hall/CRC. p. 117.
4663:
4378:
4317:
4173:
4099:
4097:
3852:"Plant Physiology Information Website"
3849:
3718:
3518:
3509:
2263:all over a flat surface. There are 17
1927:Waves: breaking wave in a ship's wake
539:of symbols that can be combined using
232:to simulate a wide range of patterns.
190:and the French American mathematician
6095:
5728:
5554:
5168:
5156:
5114:
5036:
4998:
4947:
4651:
4497:
4460:"Growth in Plants: A Study in Number"
4412:
4188:
4182:
4103:
3653:
3647:
3566:
3454:
3412:
3400:
3231:
3182:
374:'s 1830 and 1830 work, respectively;
5406:
5186:
4208:Meyer, Yves; Roques, Sylvie (1993).
4124:
3588:The Mathematical Theory of L Systems
3579:
3577:
3575:
3221:. New American Library. p. 121.
2735:
2313:; for example, there are exactly 14
1687:are zigzagging patterns of whirling
1135:, and in non-living things, notably
802:. Fivefold symmetry is found in the
264:. He considered these to consist of
198:could create plant growth patterns.
6033:The Chemical Basis of Morphogenesis
5291:
5011:Brodie, Christina (February 2005).
4673:Journal of the Atmospheric Sciences
4094:
4053:
3153:Stanford Encyclopedia of Philosophy
3114:, B8, 198b29 in Kirk, et al., 304).
2072:, a beautiful marine form drawn by
759:
493:The Chemical Basis of Morphogenesis
304:to the western world with his book
24:
4426:"Spiral Lattices & Parastichy"
4063:The Notebooks of Leonardo da Vinci
3809:Poulin, R.; Grutter, A.S. (1996).
3219:The Notebooks of Leonardo da Vinci
2824:can form, creating circles, nets,
2329:Crystals: cube-shaped crystals of
1057:
486:. In 1952, the computer scientist
158:. The German biologist and artist
25:
7114:
5708:
5519:(Art Forms in Nature), 1899–1904.
5384:
5272:D'Avanzo, C. (22 February 2004).
5127:Murray, James D. (9 March 2013).
4986:10.1038/scientificamerican0776-82
4178:. Thames and Hudson. p. 148.
4060:Richter, Jean Paul, ed. (1970) .
3572:
3258:
2610:distastefully bitter or poisonous
2604:is less likely to be attacked by
2575:
2259:are patterns formed by repeating
1438:, the florets are arranged along
7050:
5855:
5679:. Books 1 and 2. Rockport, 2009.
4669:"Deterministic Nonperiodic Flow"
4538:The Algorithmic Beauty of Plants
4262:Fractals in biology and medicine
4239:. World Scientific. p. 78.
4104:Palca, Joe (December 26, 2011).
3146:Balaguer, Mark (7 April 2009) .
3009:
2991:
2964:
2942:
2920:
2902:
2887:
2875:
2857:
2716:
2697:
2679:
2661:
2643:
2626:
2563:
2543:
2527:
2504:
2488:
2476:
2436:
2417:
2400:
2383:
2361:
2342:
2322:
2245:
2233:
2221:
2194:
2167:
2151:
2128:
2105:
1988:
1964:
1948:
1932:
1920:
1836:
1831:Meanders: sinuous snake crawling
1824:
1808:
1780:
1761:
1730:
1613:
1596:
1587:Multiple Fibonacci spirals: red
1580:
1558:
1537:
1516:
1498:
1483:
1319:
1295:
1276:
1253:
1231:
1215:
1199:
1042:
1025:
1006:
991:
971:
952:
933:
917:
894:
874:
756:that can be seen at a distance.
615:
599:
578:
431:Beijing National Aquatics Center
342:involving the number 5, and the
244:, anticipating modern concepts.
5478:
5348:
5327:Sample, Ian (18 January 2017).
5320:
5004:
4953:
4890:
4875:
4849:
4827:
4812:
4782:
4748:
4731:
4716:
4701:
4657:
4602:
4559:
4521:
4503:
4418:
4393:
4311:
4268:
4253:
4029:
3978:
3941:
3843:
3482:
3418:
3364:
3328:
3298:
2446:: a rare rock formation on the
2185:and colleagues synthesised the
2001:, a surface with minimal area (
1855:
1330:copper crystals (in microscope)
457:took the first micrograph of a
6121:
5356:"Permafrost: Patterned Ground"
4836:"Practical wind wave modeling"
4176:Fractals:The Patterns of Chaos
3616:The fractal geometry of nature
3614:Mandelbrot, Benoît B. (1983).
3274:
3252:
3225:
3210:
3139:
3103:
3082:
3045:Evolutionary history of plants
3001:in the Marienflusstal area in
2828:polygons, steps, and stripes.
2054:and the calcite skeleton of a
1430:In disc phyllotaxis as in the
1290:made of spheres (self-similar)
794:. Plants often have radial or
316:on the growth of an idealized
194:showed how the mathematics of
13:
1:
5652:Patterns from nature (as art)
4755:von Kármán, Theodore (1963).
4727:. Princeton University Press.
3794:Mimicry in plants and animals
3070:
2896:Belousov–Zhabotinsky reaction
2769:Belousov–Zhabotinsky reaction
1368:Plant spirals can be seen in
1357:, a cephalopod mollusc, each
881:Animals often show mirror or
509:Belousov–Zhabotinsky reaction
6962:Aestheticization of politics
5258:. New York: Springer-Verlag.
5199:; Williamson, Scott (2002).
5057:. New York: Springer-Verlag.
4541:. Springer-Verlag. pp.
4389:. CRC Press. pp. 44–46.
4289:10.1016/0168-0102(92)90031-7
4142:10.1371/journal.pone.0093535
3685:10.1126/science.240.4852.611
3506:. Retrieved 16 October 2012.
2495:Drying inelastic mud in the
2036:, foam patterns are common;
7:
4960:Almgren, Frederick J. Jr.;
4631:10.1103/PhysRevLett.68.2098
4264:. Springer. pp. 55–66.
3234:Mathematics Education Siwan
3028:
2374:illustrating the stairstep
764:
527:which can be used to model
423:bitruncated cubic honeycomb
405:(1801–1883) formulated the
95:. Natural patterns include
10:
7119:
5622:Stevens, Peter S. (1974).
5580:. Oxford University Press.
5571:. Oxford University Press.
5562:. Oxford University Press.
5549:Princeton University Press
5523:Thompson, D'Arcy Wentworth
5360:US Army Corps of Engineers
4841:. In Mahmood, M.F. (ed.).
4588:10.1209/0295-5075/14/6/006
3148:"Platonism in Metaphysics"
2894:Snapshot of simulation of
2739:
2249:
1815:Meanders: sinuous path of
1665:in chaotic systems have a
1339:
1335:
806:, the group that includes
768:
519:(1925–1989) developed the
270:
235:
7030:
6954:
6803:
6576:
6283:
6195:
6129:
6063:
6013:D'Arcy Wentworth Thompson
5956:
5864:
5853:
5762:
5644:Weidenfeld & Nicolson
5067:Hook, J. R.; Hall, H. E.
4789:Lewalle, Jacques (2006).
4738:Wolfram, Stephen (2002).
4708:Elaydi, Saber N. (1999).
4529:Prusinkiewicz, Przemyslaw
4473:: 335–354. Archived from
4385:Addison, Paul S. (1997).
4354:10.1103/PhysRevX.7.011031
3659:"The Science of Patterns"
3357:10.1007/s00004-001-0008-7
2752:reaction–diffusion system
2454:
821:Among non-living things,
627:
470:D'Arcy Wentworth Thompson
354:observed that the spiral
7103:Recreational mathematics
5049:Armstrong, M.A. (1988).
4654:, pp. 163, 249–250.
4515:Introduction to geometry
4401:e: The Story of a Number
3798:. New York: McGraw-Hill.
3772:On the Origin of Species
3569:, pp. 163, 247–250.
3504:University of St Andrews
2633:Dirce beauty butterfly,
1744:the cloth of gold cone,
1691:created by the unsteady
312:. Fibonacci presented a
178:predicted mechanisms of
6982:Evolutionary aesthetics
6932:The Aesthetic Dimension
5503:Fibonacci's Liber Abaci
5450:10.1073/pnas.1406015111
4611:Physical Review Letters
3856:Pollination Adaptations
3752:(2 (spring 1980)): 139.
3746:Dialectics and Humanism
3742:Tatarkiewicz, Władysław
3630:All that useless beauty
3371:Zeising, Adolf (1854).
2584:explanation: they have
2538:with mainly 120° cracks
2271:, and the wax cells in
2032:At the scale of living
1376:) of other parts as in
1222:Branching pattern of a
1037:has spherical symmetry.
706:can be explained using
608:Weaire–Phelan structure
427:Weaire–Phelan structure
368:Karl Friedrich Schimper
93:modelled mathematically
18:Da Vinci branching rule
6912:Avant-Garde and Kitsch
6862:Lectures on Aesthetics
5628:Little, Brown & Co
5576:Ball, Philip (2009c).
5567:Ball, Philip (2009b).
5220:10.1098/rspb.2001.1896
4834:Tolman, H. L. (2008).
4819:French, A. P. (1971).
4723:Ruelle, David (1991).
4458:Kappraff, Jay (2004).
4318:Sadegh, Sanaz (2017).
3546:10.1098/rstb.1952.0012
3425:Hannavy, John (2007).
2844:. Similar patterns of
2813:, while the so-called
2691:Panthera pardus pardus
2534:Drying elastic mud in
2515:with 90° cracks, near
2499:with mainly 90° cracks
1843:Meanders: symmetrical
1774:Juan Fernandez Islands
1723:positive feedback loop
1353:. For example, in the
1082:
1073:A Treatise on Painting
653:
563:wrote a famous paper,
409:of the existence of a
362:and counter-clockwise
340:Pythagorean numerology
306:
131:studied pattern, with
77:
49:
7057:Philosophy portal
5847:Widmanstätten pattern
5659:Patterns of the Earth
5516:Kunstformen der Natur
4807:on 29 September 2011.
4741:A New Kind of Science
4517:. Wiley. p. 169.
4174:Briggs, John (1992).
3850:Koning, Ross (1994).
3344:Nexus Network Journal
3261:"Fibonacci's Rabbits"
3065:Widmanstätten pattern
3035:Developmental biology
2909:Helmeted guineafowl,
2673:Pygoplites diacanthus
2393:Fritillaria meleagris
2290:Fritillaria meleagris
2279:, or fruits like the
2144:Kunstformen der Natur
1939:Dunes: sand dunes in
1629:Chaos, flow, meanders
1606:Trochoidea liebetruti
1415:the ratio is 1/3; in
1340:Further information:
1241:Anthriscus sylvestris
1238:Leaf of cow parsley,
1137:electrical discharges
1087:right-angled triangle
1078:
849:shape and rings of a
769:Further information:
635:
529:plant growth patterns
464:In the 20th century,
170:. Scottish biologist
62:Chamaeleo calyptratus
55:
42:crescent shaped dunes
35:
7002:Philosophy of design
6882:In Praise of Shadows
6872:The Critic as Artist
5594:and Neill, William.
5130:Mathematical Biology
4821:Vibrations and Waves
4533:Lindenmayer, Aristid
4041:Cryospheric Sciences
3500:University of Dundee
3266:University of Surrey
2723:Breeding pattern of
2444:Tessellated pavement
2335:cubic crystal system
2212:) often approximate
2175:Buckminsterfullerene
2080:states that for any
2078:Euler characteristic
1655:Alongside fractals,
1603:Spiralling shell of
1395:structures like the
636:Composite patterns:
535:. L-systems have an
507:, in particular the
449:to support his faux-
445:, emphasising their
435:2008 Summer Olympics
407:mathematical problem
162:painted hundreds of
7078:Applied mathematics
7012:Philosophy of music
6987:Mathematical beauty
6081:Mathematics and art
6071:Pattern recognition
6041:Aristid Lindenmayer
5675:Nakamura, Shigeki.
5672:. New Riders, 2012.
5490:Fibonacci, Leonardo
5441:2014PNAS..11112667T
5425:(35): 12667–12672.
5392:Scientific American
5171:, pp. 168–180.
5159:, pp. 159–167.
5117:, pp. 156–158.
5069:Solid State Physics
5053:Groups and Symmetry
4978:1976SciAm.235a..82A
4966:Scientific American
4911:2003Natur.426..619S
4685:1963JAtS...20..130L
4623:1992PhRvL..68.2098D
4580:1991EL.....14..533L
4568:Europhysics Letters
4346:2017PhRvX...7a1031S
3999:2007Pbio...33..149S
3677:1988Sci...240..611S
3584:Rozenberg, Grzegorz
3538:1952RSPTB.237...37T
3391:, pp. 108–109.
3291:The Garden of Cyrus
3050:Mathematics and art
2830:Thermal contraction
2795:vegetated landscape
2773:chemical oscillator
1787:Meanders: dramatic
1622:equiangular spirals
1188:actin cytoskeletons
1133:Sertularia argentea
1064:Italian Renaissance
1018:rotational symmetry
796:rotational symmetry
771:Symmetry in biology
656:Living things like
644:leaf, divided into
517:Aristid Lindenmayer
335:The Garden of Cyrus
188:Aristid Lindenmayer
182:which give rise to
166:to emphasise their
127:and stripes. Early
7083:History of science
7007:Philosophy of film
6997:Patterns in nature
6967:Applied aesthetics
6942:Why Beauty Matters
6728:Life imitating art
6589:Art for art's sake
6019:On Growth and Form
5919:Logarithmic spiral
5756:Patterns in nature
5657:Edmaier, Bernard.
5624:Patterns in Nature
5596:By Nature's Design
5586:Patterns in Nature
5531:. Cambridge, 1917.
5528:On Growth and Form
5484:Pioneering authors
5001:, pp. 96–101.
3950:"Animal Diversity"
3774:. 1859, chapter 4.
3594:, New York, 1980.
3494:2017-07-01 at the
2600:— for instance, a
2592:; for instance, a
2090:Buckminster Fuller
1959:crescent sand dune
1943:desert, from space
1755:cellular automaton
1693:separation of flow
1662:strange attractors
1643:topological mixing
1633:In mathematics, a
1469:, which activates
1363:logarithmic spiral
1304:Lichtenberg figure
1265:Romanesco broccoli
1208:Lindenmayer system
1141:Lindenmayer system
883:bilateral symmetry
735:cleaning symbiosis
654:
505:chemical reactions
490:(1912–1954) wrote
475:On Growth and Form
314:thought experiment
302:Fibonacci sequence
298:Leonardo Fibonacci
129:Greek philosophers
81:Patterns in nature
78:
74:breeding condition
50:
7093:Pattern formation
7065:
7064:
7017:Psychology of art
6892:Art as Experience
6089:
6088:
6046:Benoît Mandelbrot
5946:Self-organization
5882:Natural selection
5872:Pattern formation
5607:Rothenberg, David
5505:. Springer, 2002.
5214:(1493): 781–792.
5183:, pp. 93–95.
5140:978-3-662-08539-4
5077:978-0-471-92804-1
5039:, pp. 51–54.
4905:(6967): 619–620.
4863:. 29 October 1997
4823:. Nelson Thornes.
4665:Lorenz, Edward N.
4617:(13): 2098–2101.
4552:978-0-387-97297-8
4511:Coxeter, H. S. M.
4415:, pp. 29–32.
4324:Physical Review X
4277:Neurosci Research
4157:Falconer, Kenneth
4073:978-0-486-22572-2
3914:, pp. 82–84.
3890:, pp. 64–65.
3878:, pp. 48–49.
3788:Wickler, Wolfgang
3671:(4852): 611–616.
3586:; Salomaa, Arto.
3475:978-0-7679-0816-0
3438:978-0-415-97235-2
3403:, pp. 73–76.
3321:978-0-419-22780-9
3203:978-0-7679-0816-0
3060:Pattern formation
2978:with surrounding
2927:Aerial view of a
2742:Pattern formation
2736:Pattern formation
2729:Sepia officinalis
2303:crystal structure
2189:molecule in 1985.
2082:convex polyhedron
2026:average curvature
1871:as they pass by.
1849:Diploria strigosa
1803:, seen from space
1671:cellular automata
1667:fractal dimension
1574:Helianthus annuus
1168:animal coloration
1101:fractal dimension
1068:Leonardo da Vinci
910:fivefold symmetry
739:insect-pollinated
719:natural selection
694:. For example, a
561:Benoît Mandelbrot
557:Wacław Sierpiński
545:Gottfried Leibniz
330:Sir Thomas Browne
287:Leonardo da Vinci
226:pattern formation
218:natural selection
192:Benoît Mandelbrot
16:(Redirected from
7110:
7055:
7054:
7053:
6947:
6937:
6927:
6917:
6907:
6897:
6887:
6877:
6867:
6857:
6847:
6837:
6827:
6817:
6116:
6109:
6102:
6093:
6092:
5897:Sexual selection
5859:
5749:
5742:
5735:
5726:
5725:
5682:O'Neill, Polly.
5668:Macnab, Maggie.
5647:
5631:
5618:
5615:Bloomsbury Press
5588:. Chicago, 2016.
5581:
5572:
5563:
5473:
5472:
5462:
5452:
5434:
5410:
5404:
5403:
5401:
5399:
5382:
5376:
5375:
5373:
5371:
5362:. Archived from
5352:
5346:
5345:
5343:
5341:
5324:
5318:
5317:
5315:
5313:
5299:Morelle, Rebecca
5295:
5289:
5288:
5286:
5284:
5269:
5260:
5259:
5251:
5242:
5241:
5231:
5205:
5197:Prum, Richard O.
5193:
5184:
5178:
5172:
5166:
5160:
5154:
5145:
5144:
5124:
5118:
5112:
5103:
5097:
5091:
5085:
5079:
5065:
5059:
5058:
5056:
5046:
5040:
5034:
5028:
5027:
5025:
5023:
5008:
5002:
4996:
4990:
4989:
4957:
4951:
4945:
4939:
4938:
4894:
4888:
4887:
4879:
4873:
4872:
4870:
4868:
4857:"Types of Dunes"
4853:
4847:
4846:
4840:
4831:
4825:
4824:
4816:
4810:
4808:
4806:
4795:
4786:
4780:
4771:. Dover (1994):
4770:
4752:
4746:
4745:
4744:. Wolfram Media.
4735:
4729:
4728:
4725:Chance and Chaos
4720:
4714:
4713:
4705:
4699:
4698:
4696:
4661:
4655:
4649:
4643:
4642:
4606:
4600:
4599:
4563:
4557:
4556:
4525:
4519:
4518:
4507:
4501:
4495:
4489:
4488:
4486:
4485:
4479:
4464:
4455:
4446:
4445:
4443:
4441:
4432:. Archived from
4422:
4416:
4410:
4404:
4397:
4391:
4390:
4382:
4376:
4375:
4365:
4339:
4315:
4309:
4308:
4272:
4266:
4265:
4257:
4251:
4250:
4232:
4226:
4225:
4205:
4199:
4198:
4186:
4180:
4179:
4171:
4165:
4164:
4153:
4147:
4146:
4144:
4128:
4122:
4121:
4119:
4117:
4101:
4092:
4091:
4085:
4077:
4057:
4051:
4050:
4048:
4047:
4033:
4027:
4026:
3982:
3976:
3975:
3973:
3971:
3965:
3954:
3945:
3939:
3933:
3927:
3921:
3915:
3909:
3903:
3897:
3891:
3885:
3879:
3873:
3867:
3866:
3864:
3862:
3847:
3841:
3840:
3830:
3806:
3800:
3799:
3797:
3784:
3775:
3765:
3754:
3753:
3738:
3732:
3722:
3716:
3715:
3713:
3712:
3703:. Archived from
3651:
3645:
3639:
3633:
3626:
3620:
3619:
3611:
3602:
3581:
3570:
3564:
3558:
3557:
3516:
3507:
3486:
3480:
3479:
3452:
3443:
3442:
3422:
3416:
3410:
3404:
3398:
3392:
3386:
3377:
3376:
3368:
3362:
3361:
3359:
3336:Padovan, Richard
3332:
3326:
3325:
3306:Padovan, Richard
3302:
3296:
3295:
3278:
3272:
3271:
3256:
3250:
3249:
3229:
3223:
3222:
3214:
3208:
3207:
3180:
3165:
3164:
3162:
3160:
3143:
3137:
3131:
3115:
3107:
3101:
3098:Metaphysics 1–5
3086:
3013:
2995:
2972:Patterned ground
2968:
2946:
2924:
2912:Numida meleagris
2906:
2891:
2879:
2865:Giant pufferfish
2861:
2838:fissured pattern
2822:patterned ground
2786:Numida meleagris
2720:
2701:
2683:
2665:
2647:
2630:
2567:
2556:Giant's Causeway
2547:
2531:
2508:
2492:
2480:
2448:Tasman Peninsula
2440:
2421:
2404:
2387:
2365:
2346:
2326:
2315:Bravais lattices
2265:wallpaper groups
2237:
2225:
2198:
2171:
2155:
2132:
2109:
2070:Aulonia hexagona
2049:silicoflagellate
1968:
1952:
1936:
1924:
1865:Mechanical waves
1840:
1828:
1812:
1784:
1765:
1737:Chaos: shell of
1734:
1639:butterfly effect
1635:dynamical system
1617:
1600:
1591:in cross section
1584:
1562:
1541:
1527:of spiral aloe,
1520:
1502:
1487:
1409:Fibonacci ratios
1323:
1310:breakdown in an
1299:
1280:
1257:
1235:
1219:
1203:
1076:he stated that:
1046:
1029:
1010:
995:
975:
956:
945:sixfold symmetry
937:
921:
898:
878:
827:sixfold symmetry
779:Crystal symmetry
760:Types of pattern
727:sexual selection
619:
603:
591:Cycas circinalis
586:Fibonacci number
582:
541:production rules
531:in the style of
311:
273:
272:
222:sexual selection
164:marine organisms
58:veiled chameleon
56:Patterns of the
21:
7118:
7117:
7113:
7112:
7111:
7109:
7108:
7107:
7068:
7067:
7066:
7061:
7051:
7049:
7026:
6950:
6945:
6935:
6925:
6922:Critical Essays
6915:
6905:
6895:
6885:
6875:
6865:
6855:
6845:
6835:
6825:
6815:
6799:
6572:
6486:Ortega y Gasset
6279:
6191:
6125:
6120:
6090:
6085:
6059:
5952:
5860:
5851:
5758:
5753:
5711:
5600:Chronicle Books
5575:
5566:
5481:
5476:
5411:
5407:
5397:
5395:
5383:
5379:
5369:
5367:
5366:on 7 March 2015
5354:
5353:
5349:
5339:
5337:
5325:
5321:
5311:
5309:
5296:
5292:
5282:
5280:
5270:
5263:
5252:
5245:
5203:
5194:
5187:
5181:Rothenberg 2011
5179:
5175:
5167:
5163:
5155:
5148:
5141:
5125:
5121:
5113:
5106:
5098:
5094:
5086:
5082:
5066:
5062:
5047:
5043:
5035:
5031:
5021:
5019:
5009:
5005:
4997:
4993:
4962:Taylor, Jean E.
4958:
4954:
4946:
4942:
4919:10.1038/426619a
4895:
4891:
4880:
4876:
4866:
4864:
4855:
4854:
4850:
4838:
4832:
4828:
4817:
4813:
4804:
4793:
4787:
4783:
4767:
4759:. McGraw-Hill.
4753:
4749:
4736:
4732:
4721:
4717:
4706:
4702:
4662:
4658:
4650:
4646:
4607:
4603:
4564:
4560:
4553:
4526:
4522:
4508:
4504:
4496:
4492:
4483:
4481:
4477:
4462:
4456:
4449:
4439:
4437:
4424:
4423:
4419:
4411:
4407:
4398:
4394:
4383:
4379:
4316:
4312:
4273:
4269:
4258:
4254:
4247:
4233:
4229:
4222:
4206:
4202:
4187:
4183:
4172:
4168:
4154:
4150:
4129:
4125:
4115:
4113:
4110:Morning Edition
4102:
4095:
4079:
4078:
4074:
4058:
4054:
4045:
4043:
4035:
4034:
4030:
4007:10.1666/06053.1
3983:
3979:
3969:
3967:
3963:
3952:
3946:
3942:
3934:
3930:
3922:
3918:
3910:
3906:
3898:
3894:
3886:
3882:
3874:
3870:
3860:
3858:
3848:
3844:
3828:10.2307/1312929
3807:
3803:
3785:
3778:
3768:Darwin, Charles
3766:
3757:
3739:
3735:
3723:
3719:
3710:
3708:
3652:
3648:
3640:
3636:
3628:Forbes, Peter.
3627:
3623:
3612:
3605:
3582:
3573:
3565:
3561:
3517:
3510:
3498:. D' Arcy 150.
3496:Wayback Machine
3487:
3483:
3476:
3468:. p. 111.
3453:
3446:
3439:
3423:
3419:
3411:
3407:
3399:
3395:
3387:
3380:
3369:
3365:
3333:
3329:
3322:
3303:
3299:
3279:
3275:
3257:
3253:
3230:
3226:
3215:
3211:
3204:
3196:. p. 110.
3181:
3168:
3158:
3156:
3144:
3140:
3132:
3128:
3119:
3118:
3108:
3104:
3087:
3083:
3073:
3031:
3024:
3014:
3005:
2996:
2987:
2969:
2960:
2954:White Mountains
2947:
2938:
2925:
2916:
2907:
2898:
2892:
2883:
2880:
2871:
2862:
2744:
2738:
2731:
2721:
2712:
2702:
2693:
2684:
2675:
2669:Royal angelfish
2666:
2657:
2648:
2639:
2631:
2578:
2571:
2568:
2559:
2548:
2539:
2532:
2523:
2509:
2500:
2493:
2484:
2481:
2457:
2450:
2441:
2432:
2430:Salacca zalacca
2422:
2413:
2410:Rutilus rutilus
2405:
2396:
2388:
2379:
2366:
2357:
2347:
2338:
2327:
2254:
2248:
2241:
2238:
2229:
2226:
2217:
2199:
2190:
2183:Richard Smalley
2180:
2172:
2163:
2156:
2147:
2133:
2124:
2110:
2003:minimal surface
1991:
1984:
1969:
1960:
1953:
1944:
1937:
1928:
1925:
1900:angle of repose
1858:
1851:
1841:
1832:
1829:
1820:
1813:
1804:
1785:
1776:
1766:
1757:
1735:
1675:Stephen Wolfram
1650:periodic orbits
1631:
1624:
1618:
1609:
1601:
1592:
1585:
1576:
1568:: seed head of
1566:Fermat's spiral
1563:
1554:
1542:
1533:
1530:Aloe polyphylla
1521:
1512:
1510:Ovis canadensis
1503:
1494:
1488:
1459:dynamic systems
1455:self-organizing
1440:Fermat's spiral
1344:
1338:
1331:
1324:
1315:
1312:acrylic polymer
1306:: high voltage
1300:
1291:
1281:
1272:
1258:
1249:
1236:
1227:
1220:
1211:
1204:
1095:are infinitely
1060:
1058:Trees, fractals
1053:
1047:
1038:
1030:
1021:
1011:
1002:
996:
987:
984:radial symmetry
976:
967:
957:
948:
938:
929:
922:
913:
899:
890:
879:
788:mirror symmetry
781:
775:Floral symmetry
767:
762:
702:; for example,
630:
623:
620:
611:
604:
595:
583:
513:Turing patterns
500:. He predicted
411:minimal surface
376:Auguste Bravais
372:Alexander Braun
370:and his friend
322:Johannes Kepler
300:introduced the
283:Pliny the Elder
238:
230:computer models
172:D'Arcy Thompson
156:minimal surface
28:
23:
22:
15:
12:
11:
5:
7116:
7106:
7105:
7100:
7095:
7090:
7085:
7080:
7063:
7062:
7060:
7059:
7047:
7042:
7037:
7031:
7028:
7027:
7025:
7024:
7019:
7014:
7009:
7004:
6999:
6994:
6992:Neuroesthetics
6989:
6984:
6979:
6974:
6972:Arts criticism
6969:
6964:
6958:
6956:
6952:
6951:
6949:
6948:
6938:
6928:
6918:
6908:
6898:
6888:
6878:
6868:
6858:
6848:
6842:On the Sublime
6838:
6828:
6818:
6807:
6805:
6801:
6800:
6798:
6797:
6792:
6787:
6782:
6777:
6772:
6767:
6762:
6755:
6750:
6745:
6740:
6735:
6730:
6725:
6720:
6713:
6708:
6706:Interpretation
6703:
6698:
6693:
6688:
6683:
6678:
6673:
6668:
6663:
6658:
6653:
6648:
6643:
6638:
6633:
6628:
6623:
6622:
6621:
6616:
6606:
6601:
6599:Artistic merit
6596:
6591:
6586:
6580:
6578:
6574:
6573:
6571:
6570:
6563:
6558:
6553:
6548:
6543:
6538:
6533:
6528:
6523:
6518:
6513:
6508:
6503:
6498:
6493:
6488:
6483:
6478:
6473:
6468:
6463:
6458:
6453:
6448:
6443:
6438:
6433:
6428:
6423:
6418:
6413:
6408:
6403:
6398:
6393:
6388:
6383:
6378:
6373:
6368:
6363:
6358:
6353:
6348:
6343:
6338:
6333:
6328:
6323:
6318:
6313:
6308:
6303:
6298:
6293:
6287:
6285:
6281:
6280:
6278:
6277:
6270:
6265:
6260:
6255:
6250:
6248:Psychoanalysis
6245:
6240:
6235:
6230:
6225:
6220:
6215:
6210:
6205:
6199:
6197:
6193:
6192:
6190:
6189:
6184:
6179:
6174:
6169:
6164:
6159:
6154:
6149:
6144:
6139:
6133:
6131:
6127:
6126:
6119:
6118:
6111:
6104:
6096:
6087:
6086:
6084:
6083:
6078:
6073:
6067:
6065:
6061:
6060:
6058:
6057:
6056:
6055:
6043:
6038:
6037:
6036:
6024:
6023:
6022:
6010:
6008:Wilson Bentley
6005:
6003:Joseph Plateau
6000:
5995:
5990:
5989:
5988:
5976:
5971:
5966:
5960:
5958:
5954:
5953:
5951:
5950:
5949:
5948:
5943:
5941:Plateau's laws
5938:
5936:Fluid dynamics
5933:
5923:
5922:
5921:
5916:
5911:
5901:
5900:
5899:
5894:
5889:
5884:
5874:
5868:
5866:
5862:
5861:
5854:
5852:
5850:
5849:
5844:
5839:
5834:
5829:
5828:
5827:
5822:
5817:
5812:
5802:
5797:
5792:
5787:
5782:
5777:
5772:
5766:
5764:
5760:
5759:
5752:
5751:
5744:
5737:
5729:
5723:
5722:
5717:
5710:
5709:External links
5707:
5706:
5705:
5702:Viking Penguin
5698:Nature's Chaos
5687:
5686:. Black, 2008.
5680:
5673:
5666:
5649:
5648:
5632:
5619:
5603:
5589:
5584:Ball, Philip.
5582:
5573:
5564:
5552:
5541:Adam, John A.
5533:
5532:
5520:
5511:Haeckel, Ernst
5508:
5507:
5506:
5480:
5477:
5475:
5474:
5405:
5377:
5347:
5319:
5301:(2013-12-09).
5290:
5261:
5243:
5185:
5173:
5161:
5146:
5139:
5119:
5104:
5102:, p. 208.
5092:
5090:, p. 207.
5080:
5060:
5041:
5029:
5003:
4991:
4972:(235): 82–93.
4952:
4940:
4889:
4874:
4848:
4826:
4811:
4781:
4777:978-0486434858
4766:978-0070676022
4765:
4747:
4730:
4715:
4710:Discrete Chaos
4700:
4679:(2): 130–141.
4667:(March 1963).
4656:
4644:
4601:
4574:(6): 533–539.
4558:
4551:
4520:
4502:
4490:
4447:
4436:on 26 May 2010
4417:
4405:
4392:
4377:
4310:
4267:
4252:
4246:978-9810237929
4245:
4227:
4220:
4200:
4181:
4166:
4148:
4123:
4093:
4072:
4052:
4028:
3993:(1): 149–163.
3977:
3966:on 17 May 2016
3940:
3928:
3916:
3904:
3892:
3880:
3868:
3842:
3821:(7): 512–517.
3801:
3776:
3755:
3733:
3717:
3646:
3644:, p. 222.
3634:
3621:
3603:
3592:Academic Press
3571:
3559:
3532:(641): 37–72.
3508:
3481:
3474:
3466:Broadway Books
3444:
3437:
3417:
3405:
3393:
3378:
3363:
3350:(1): 113–122.
3327:
3320:
3297:
3282:Browne, Thomas
3273:
3251:
3224:
3209:
3202:
3194:Broadway Books
3166:
3138:
3125:
3117:
3116:
3102:
3088:The so-called
3080:
3079:
3072:
3069:
3068:
3067:
3062:
3057:
3052:
3047:
3042:
3037:
3030:
3027:
3026:
3025:
3015:
3008:
3006:
2997:
2990:
2988:
2982:polygons near
2970:
2963:
2961:
2948:
2941:
2939:
2926:
2919:
2917:
2908:
2901:
2899:
2893:
2886:
2884:
2881:
2874:
2872:
2863:
2856:
2811:pocket gophers
2740:Main article:
2737:
2734:
2733:
2732:
2722:
2715:
2713:
2703:
2696:
2694:
2685:
2678:
2676:
2667:
2660:
2658:
2649:
2642:
2640:
2636:Colobura dirce
2632:
2625:
2577:
2576:Spots, stripes
2574:
2573:
2572:
2569:
2562:
2560:
2549:
2542:
2540:
2533:
2526:
2524:
2510:
2503:
2501:
2494:
2487:
2485:
2482:
2475:
2456:
2453:
2452:
2451:
2442:
2435:
2433:
2423:
2416:
2414:
2406:
2399:
2397:
2389:
2382:
2380:
2372:hopper crystal
2367:
2360:
2358:
2348:
2341:
2339:
2328:
2321:
2307:crystal system
2250:Main article:
2247:
2244:
2243:
2242:
2239:
2232:
2230:
2227:
2220:
2218:
2206:microparticles
2200:
2193:
2191:
2178:
2173:
2166:
2164:
2157:
2150:
2148:
2134:
2127:
2125:
2111:
2104:
2061:Cidaris rugosa
2018:Plateau's laws
1990:
1987:
1986:
1985:
1970:
1963:
1961:
1954:
1947:
1945:
1938:
1931:
1929:
1926:
1919:
1913:solitary waves
1857:
1854:
1853:
1852:
1842:
1835:
1833:
1830:
1823:
1821:
1814:
1807:
1805:
1786:
1779:
1777:
1767:
1760:
1758:
1736:
1729:
1707:of the fluid.
1685:Vortex streets
1630:
1627:
1626:
1625:
1619:
1612:
1610:
1602:
1595:
1593:
1586:
1579:
1577:
1564:
1557:
1555:
1543:
1536:
1534:
1522:
1515:
1513:
1504:
1497:
1495:
1489:
1482:
1476:photosynthesis
1423:it is 3/8; in
1419:it is 2/5; in
1337:
1334:
1333:
1332:
1325:
1318:
1316:
1301:
1294:
1292:
1286:flowerhead, a
1282:
1275:
1273:
1259:
1252:
1250:
1248:, not infinite
1237:
1230:
1228:
1221:
1214:
1212:
1205:
1198:
1184:Purkinje cells
1152:river networks
1059:
1056:
1055:
1054:
1048:
1041:
1039:
1031:
1024:
1022:
1012:
1005:
1003:
997:
990:
988:
977:
970:
968:
962:showing cubic
958:
951:
949:
939:
932:
930:
923:
916:
914:
900:
893:
891:
880:
873:
835:crystal habits
825:have striking
766:
763:
761:
758:
708:fluid dynamics
666:peacock's tail
629:
626:
625:
624:
621:
614:
612:
605:
598:
596:
584:
577:
553:Helge von Koch
525:formal grammar
484:mollusc shells
455:Wilson Bentley
415:Plateau's laws
403:Joseph Plateau
352:Charles Bonnet
237:
234:
148:Joseph Plateau
26:
9:
6:
4:
3:
2:
7115:
7104:
7101:
7099:
7096:
7094:
7091:
7089:
7086:
7084:
7081:
7079:
7076:
7075:
7073:
7058:
7048:
7046:
7043:
7041:
7038:
7036:
7033:
7032:
7029:
7023:
7022:Theory of art
7020:
7018:
7015:
7013:
7010:
7008:
7005:
7003:
7000:
6998:
6995:
6993:
6990:
6988:
6985:
6983:
6980:
6978:
6975:
6973:
6970:
6968:
6965:
6963:
6960:
6959:
6957:
6953:
6944:
6943:
6939:
6934:
6933:
6929:
6924:
6923:
6919:
6913:
6909:
6903:
6899:
6894:
6893:
6889:
6884:
6883:
6879:
6873:
6869:
6864:
6863:
6859:
6854:
6853:
6849:
6844:
6843:
6839:
6834:
6833:
6829:
6824:
6823:
6819:
6814:
6813:
6812:Hippias Major
6809:
6808:
6806:
6802:
6796:
6793:
6791:
6788:
6786:
6783:
6781:
6778:
6776:
6773:
6771:
6768:
6766:
6763:
6761:
6760:
6756:
6754:
6751:
6749:
6746:
6744:
6741:
6739:
6736:
6734:
6731:
6729:
6726:
6724:
6721:
6719:
6718:
6714:
6712:
6709:
6707:
6704:
6702:
6699:
6697:
6694:
6692:
6689:
6687:
6684:
6682:
6679:
6677:
6674:
6672:
6671:Entertainment
6669:
6667:
6664:
6662:
6659:
6657:
6654:
6652:
6649:
6647:
6644:
6642:
6639:
6637:
6634:
6632:
6629:
6627:
6624:
6620:
6617:
6615:
6612:
6611:
6610:
6607:
6605:
6602:
6600:
6597:
6595:
6594:Art manifesto
6592:
6590:
6587:
6585:
6584:Appropriation
6582:
6581:
6579:
6575:
6569:
6568:
6564:
6562:
6559:
6557:
6554:
6552:
6549:
6547:
6544:
6542:
6539:
6537:
6534:
6532:
6529:
6527:
6524:
6522:
6519:
6517:
6514:
6512:
6509:
6507:
6504:
6502:
6499:
6497:
6494:
6492:
6489:
6487:
6484:
6482:
6479:
6477:
6476:Merleau-Ponty
6474:
6472:
6469:
6467:
6464:
6462:
6459:
6457:
6454:
6452:
6449:
6447:
6444:
6442:
6439:
6437:
6434:
6432:
6429:
6427:
6424:
6422:
6419:
6417:
6414:
6412:
6409:
6407:
6404:
6402:
6399:
6397:
6394:
6392:
6389:
6387:
6384:
6382:
6379:
6377:
6374:
6372:
6369:
6367:
6364:
6362:
6359:
6357:
6354:
6352:
6349:
6347:
6344:
6342:
6339:
6337:
6334:
6332:
6329:
6327:
6324:
6322:
6319:
6317:
6314:
6312:
6309:
6307:
6304:
6302:
6299:
6297:
6294:
6292:
6291:Abhinavagupta
6289:
6288:
6286:
6282:
6276:
6275:
6271:
6269:
6266:
6264:
6261:
6259:
6256:
6254:
6251:
6249:
6246:
6244:
6243:Postmodernism
6241:
6239:
6236:
6234:
6231:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6209:
6206:
6204:
6201:
6200:
6198:
6194:
6188:
6185:
6183:
6180:
6178:
6175:
6173:
6170:
6168:
6165:
6163:
6160:
6158:
6155:
6153:
6150:
6148:
6145:
6143:
6140:
6138:
6135:
6134:
6132:
6128:
6124:
6117:
6112:
6110:
6105:
6103:
6098:
6097:
6094:
6082:
6079:
6077:
6074:
6072:
6069:
6068:
6066:
6062:
6054:
6053:
6049:
6048:
6047:
6044:
6042:
6039:
6035:
6034:
6030:
6029:
6028:
6025:
6021:
6020:
6016:
6015:
6014:
6011:
6009:
6006:
6004:
6001:
5999:
5998:Ernst Haeckel
5996:
5994:
5993:Adolf Zeising
5991:
5987:
5986:
5982:
5981:
5980:
5977:
5975:
5972:
5970:
5967:
5965:
5962:
5961:
5959:
5955:
5947:
5944:
5942:
5939:
5937:
5934:
5932:
5929:
5928:
5927:
5924:
5920:
5917:
5915:
5912:
5910:
5907:
5906:
5905:
5902:
5898:
5895:
5893:
5890:
5888:
5885:
5883:
5880:
5879:
5878:
5875:
5873:
5870:
5869:
5867:
5863:
5858:
5848:
5845:
5843:
5840:
5838:
5837:Vortex street
5835:
5833:
5830:
5826:
5823:
5821:
5818:
5816:
5815:Quasicrystals
5813:
5811:
5808:
5807:
5806:
5803:
5801:
5798:
5796:
5793:
5791:
5788:
5786:
5783:
5781:
5778:
5776:
5773:
5771:
5768:
5767:
5765:
5761:
5757:
5750:
5745:
5743:
5738:
5736:
5731:
5730:
5727:
5721:
5718:
5716:
5713:
5712:
5703:
5699:
5695:
5694:Gleick, James
5691:
5690:Porter, Eliot
5688:
5685:
5681:
5678:
5674:
5671:
5667:
5664:
5663:Phaidon Press
5660:
5656:
5655:
5654:
5653:
5645:
5641:
5637:
5633:
5629:
5625:
5620:
5616:
5612:
5608:
5604:
5601:
5597:
5593:
5590:
5587:
5583:
5579:
5574:
5570:
5565:
5561:
5557:
5553:
5550:
5546:
5545:
5540:
5539:
5538:
5537:
5536:General books
5530:
5529:
5524:
5521:
5518:
5517:
5512:
5509:
5504:
5500:
5499:
5497:
5496:
5491:
5488:
5487:
5486:
5485:
5470:
5466:
5461:
5456:
5451:
5446:
5442:
5438:
5433:
5428:
5424:
5420:
5416:
5409:
5394:
5393:
5388:
5381:
5365:
5361:
5357:
5351:
5336:
5335:
5330:
5323:
5308:
5304:
5300:
5294:
5279:
5275:
5268:
5266:
5257:
5250:
5248:
5239:
5235:
5230:
5225:
5221:
5217:
5213:
5209:
5202:
5198:
5192:
5190:
5182:
5177:
5170:
5165:
5158:
5153:
5151:
5142:
5136:
5132:
5131:
5123:
5116:
5111:
5109:
5101:
5096:
5089:
5084:
5078:
5074:
5070:
5064:
5055:
5054:
5045:
5038:
5033:
5018:
5017:Microscopy-UK
5014:
5007:
5000:
4995:
4987:
4983:
4979:
4975:
4971:
4967:
4963:
4956:
4950:, p. 68.
4949:
4944:
4936:
4932:
4928:
4924:
4920:
4916:
4912:
4908:
4904:
4900:
4893:
4885:
4878:
4862:
4858:
4852:
4844:
4837:
4830:
4822:
4815:
4803:
4799:
4792:
4785:
4778:
4774:
4768:
4762:
4758:
4751:
4743:
4742:
4734:
4726:
4719:
4711:
4704:
4695:
4690:
4686:
4682:
4678:
4674:
4670:
4666:
4660:
4653:
4648:
4640:
4636:
4632:
4628:
4624:
4620:
4616:
4612:
4605:
4597:
4593:
4589:
4585:
4581:
4577:
4573:
4569:
4562:
4554:
4548:
4544:
4540:
4539:
4534:
4530:
4524:
4516:
4512:
4506:
4500:, p. 13.
4499:
4494:
4480:on 2016-03-04
4476:
4472:
4468:
4461:
4454:
4452:
4435:
4431:
4430:Smith College
4427:
4421:
4414:
4409:
4402:
4396:
4388:
4381:
4373:
4369:
4364:
4359:
4355:
4351:
4347:
4343:
4338:
4333:
4330:(1): 011031.
4329:
4325:
4321:
4314:
4306:
4302:
4298:
4294:
4290:
4286:
4282:
4278:
4271:
4263:
4256:
4248:
4242:
4238:
4231:
4223:
4221:9782863321300
4217:
4213:
4212:
4204:
4196:
4192:
4191:New Scientist
4185:
4177:
4170:
4163:. John Wiley.
4162:
4158:
4152:
4143:
4138:
4134:
4127:
4111:
4107:
4100:
4098:
4089:
4083:
4075:
4069:
4065:
4064:
4056:
4042:
4038:
4032:
4024:
4020:
4016:
4012:
4008:
4004:
4000:
3996:
3992:
3988:
3981:
3962:
3958:
3951:
3944:
3938:, p. 71.
3937:
3932:
3926:, p. 60.
3925:
3920:
3913:
3908:
3902:, p. 52.
3901:
3896:
3889:
3884:
3877:
3872:
3857:
3853:
3846:
3838:
3834:
3829:
3824:
3820:
3816:
3812:
3805:
3796:
3795:
3789:
3783:
3781:
3773:
3769:
3764:
3762:
3760:
3751:
3747:
3743:
3737:
3730:
3726:
3725:Devlin, Keith
3721:
3707:on 2010-10-28
3706:
3702:
3698:
3694:
3690:
3686:
3682:
3678:
3674:
3670:
3666:
3665:
3660:
3656:
3650:
3643:
3638:
3631:
3625:
3617:
3610:
3608:
3601:
3600:0-12-597140-0
3597:
3593:
3589:
3585:
3580:
3578:
3576:
3568:
3563:
3555:
3551:
3547:
3543:
3539:
3535:
3531:
3527:
3526:
3521:
3520:Turing, A. M.
3515:
3513:
3505:
3501:
3497:
3493:
3490:
3485:
3477:
3471:
3467:
3463:
3462:
3457:
3451:
3449:
3440:
3434:
3430:
3429:
3421:
3415:, p. 41.
3414:
3409:
3402:
3397:
3390:
3385:
3383:
3374:
3367:
3358:
3353:
3349:
3345:
3341:
3337:
3331:
3323:
3317:
3313:
3312:
3307:
3301:
3293:
3292:
3287:
3286:"Chapter III"
3283:
3277:
3269:
3267:
3262:
3255:
3247:
3243:
3239:
3235:
3228:
3220:
3213:
3205:
3199:
3195:
3191:
3190:
3185:
3179:
3177:
3175:
3173:
3171:
3155:
3154:
3149:
3142:
3135:
3130:
3126:
3124:
3123:
3113:
3106:
3099:
3095:
3091:
3085:
3081:
3078:
3077:
3066:
3063:
3061:
3058:
3056:
3055:Morphogenesis
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3036:
3033:
3032:
3023:
3019:
3012:
3007:
3004:
3000:
2999:Fairy circles
2994:
2989:
2985:
2981:
2977:
2973:
2967:
2962:
2959:
2958:New Hampshire
2955:
2951:
2945:
2940:
2937:
2933:
2930:
2923:
2918:
2914:
2913:
2905:
2900:
2897:
2890:
2885:
2878:
2873:
2870:
2869:Tetraodon mbu
2866:
2860:
2855:
2854:
2853:
2851:
2847:
2843:
2839:
2834:
2831:
2827:
2823:
2818:
2816:
2815:fairy circles
2812:
2808:
2804:
2800:
2796:
2791:
2788:
2787:
2782:
2776:
2774:
2770:
2765:
2764:standing wave
2761:
2757:
2753:
2749:
2743:
2730:
2726:
2719:
2714:
2711:
2710:G.G. Jacobson
2707:
2700:
2695:
2692:
2688:
2682:
2677:
2674:
2670:
2664:
2659:
2656:
2652:
2651:Grevy's zebra
2646:
2641:
2638:
2637:
2629:
2624:
2623:
2622:
2620:
2615:
2611:
2607:
2603:
2599:
2595:
2591:
2587:
2583:
2566:
2561:
2557:
2553:
2546:
2541:
2537:
2530:
2525:
2522:
2518:
2517:Sgurr na Stri
2514:
2507:
2502:
2498:
2497:Rann of Kutch
2491:
2486:
2479:
2474:
2473:
2472:
2469:
2465:
2461:
2449:
2445:
2439:
2434:
2431:
2427:
2420:
2415:
2412:
2411:
2403:
2398:
2395:
2394:
2386:
2381:
2377:
2376:crystal habit
2373:
2370:
2364:
2359:
2356:
2353:is a natural
2352:
2345:
2340:
2336:
2333:(rock salt);
2332:
2325:
2320:
2319:
2318:
2316:
2312:
2308:
2304:
2301:, defined by
2300:
2296:
2292:
2291:
2286:
2282:
2278:
2274:
2270:
2266:
2262:
2258:
2257:Tessellations
2253:
2246:Tessellations
2236:
2231:
2224:
2219:
2215:
2211:
2207:
2203:
2197:
2192:
2188:
2184:
2176:
2170:
2165:
2161:
2154:
2149:
2145:
2141:
2137:
2131:
2126:
2122:
2118:
2114:
2108:
2103:
2102:
2101:
2098:
2094:
2093:geodesic dome
2091:
2087:
2083:
2079:
2075:
2074:Ernst Haeckel
2071:
2067:
2063:
2062:
2057:
2053:
2050:
2046:
2043:
2039:
2035:
2030:
2027:
2023:
2019:
2015:
2011:
2006:
2004:
2000:
1996:
1989:Bubbles, foam
1983:, Afghanistan
1982:
1978:
1974:
1967:
1962:
1958:
1951:
1946:
1942:
1935:
1930:
1923:
1918:
1917:
1916:
1914:
1909:
1906:, which is a
1905:
1901:
1897:
1893:
1889:
1887:
1883:
1878:
1877:surface waves
1874:
1870:
1866:
1862:
1850:
1846:
1839:
1834:
1827:
1822:
1818:
1811:
1806:
1802:
1798:
1795:in the broad
1794:
1790:
1789:meander scars
1783:
1778:
1775:
1772:of clouds at
1771:
1770:vortex street
1764:
1759:
1756:
1753:
1749:
1748:
1747:Conus textile
1743:
1740:
1733:
1728:
1727:
1726:
1724:
1720:
1716:
1712:
1708:
1706:
1702:
1698:
1694:
1690:
1686:
1682:
1680:
1676:
1672:
1668:
1664:
1663:
1658:
1653:
1651:
1648:
1644:
1640:
1636:
1623:
1616:
1611:
1608:
1607:
1599:
1594:
1590:
1583:
1578:
1575:
1571:
1567:
1561:
1556:
1553:growth spiral
1552:
1548:
1547:
1540:
1535:
1532:
1531:
1526:
1519:
1514:
1511:
1507:
1506:Bighorn sheep
1501:
1496:
1492:
1486:
1481:
1480:
1479:
1477:
1472:
1468:
1464:
1460:
1457:processes in
1456:
1451:
1449:
1445:
1441:
1437:
1433:
1428:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1379:
1375:
1371:
1366:
1364:
1360:
1356:
1352:
1348:
1343:
1329:
1322:
1317:
1313:
1309:
1305:
1298:
1293:
1289:
1285:
1279:
1274:
1270:
1266:
1262:
1256:
1251:
1247:
1244:, is 2- or 3-
1243:
1242:
1234:
1229:
1225:
1218:
1213:
1209:
1202:
1197:
1196:
1195:
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1157:
1153:
1148:
1146:
1142:
1138:
1134:
1130:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1090:
1088:
1081:
1077:
1075:
1074:
1069:
1065:
1051:
1045:
1040:
1036:
1035:
1028:
1023:
1019:
1015:
1009:
1004:
1000:
994:
989:
985:
982:approximates
981:
974:
969:
965:
964:crystal habit
961:
955:
950:
946:
942:
936:
931:
927:
920:
915:
911:
907:
903:
897:
892:
888:
884:
877:
872:
871:
870:
868:
864:
863:cephalisation
858:
856:
852:
848:
844:
840:
839:quasicrystals
836:
832:
828:
824:
819:
817:
813:
809:
805:
801:
797:
793:
789:
785:
780:
776:
772:
757:
755:
754:nectar guides
751:
747:
743:
740:
736:
732:
728:
724:
720:
716:
711:
709:
705:
701:
700:physical laws
697:
693:
689:
684:
682:
678:
674:
670:
667:
663:
659:
651:
647:
643:
639:
634:
618:
613:
609:
602:
597:
593:
592:
587:
581:
576:
575:
574:
572:
568:
567:
562:
558:
554:
550:
546:
542:
538:
534:
530:
526:
522:
518:
514:
510:
506:
503:
499:
498:morphogenesis
495:
494:
489:
485:
481:
477:
476:
471:
467:
462:
460:
456:
452:
448:
444:
440:
439:Ernst Haeckel
436:
432:
428:
424:
420:
416:
412:
408:
404:
399:
397:
393:
389:
388:Adolf Zeising
385:
381:
377:
373:
369:
365:
361:
357:
353:
349:
345:
344:Platonic form
341:
337:
336:
331:
327:
323:
319:
315:
310:
309:
303:
299:
294:
292:
288:
284:
280:
276:
267:
263:
259:
255:
251:
247:
243:
233:
231:
227:
224:. Studies of
223:
219:
216:processes of
215:
211:
207:
203:
199:
197:
193:
189:
185:
181:
180:morphogenesis
177:
173:
169:
165:
161:
160:Ernst Haeckel
157:
153:
149:
144:
142:
138:
134:
130:
126:
122:
121:tessellations
118:
114:
110:
106:
102:
98:
94:
90:
86:
85:natural world
82:
75:
71:
67:
63:
59:
54:
47:
43:
39:
34:
30:
19:
6996:
6940:
6930:
6920:
6890:
6880:
6860:
6850:
6840:
6830:
6820:
6810:
6757:
6733:Magnificence
6715:
6565:
6531:Schopenhauer
6366:Coomaraswamy
6284:Philosophers
6272:
6203:Aestheticism
6050:
6031:
6017:
5983:
5909:Chaos theory
5832:Tessellation
5755:
5697:
5683:
5676:
5669:
5658:
5651:
5650:
5639:
5636:Stewart, Ian
5623:
5610:
5595:
5585:
5577:
5568:
5559:
5556:Ball, Philip
5543:
5535:
5534:
5526:
5514:
5502:
5493:
5483:
5482:
5479:Bibliography
5422:
5418:
5408:
5396:. Retrieved
5390:
5385:Ghose, Tia.
5380:
5368:. Retrieved
5364:the original
5359:
5350:
5338:. Retrieved
5334:The Guardian
5332:
5322:
5310:. Retrieved
5306:
5293:
5281:. Retrieved
5277:
5255:
5211:
5207:
5176:
5164:
5129:
5122:
5100:Stevens 1974
5095:
5088:Stevens 1974
5083:
5068:
5063:
5052:
5044:
5032:
5020:. Retrieved
5016:
5006:
4994:
4969:
4965:
4955:
4943:
4902:
4898:
4892:
4883:
4877:
4865:. Retrieved
4860:
4851:
4842:
4829:
4820:
4814:
4802:the original
4797:
4784:
4757:Aerodynamics
4756:
4750:
4739:
4733:
4724:
4718:
4709:
4703:
4676:
4672:
4659:
4647:
4614:
4610:
4604:
4571:
4567:
4561:
4537:
4523:
4514:
4505:
4493:
4482:. Retrieved
4475:the original
4470:
4466:
4440:24 September
4438:. Retrieved
4434:the original
4429:
4420:
4408:
4400:
4395:
4386:
4380:
4327:
4323:
4313:
4283:(1): 19–31.
4280:
4276:
4270:
4261:
4255:
4236:
4230:
4210:
4203:
4194:
4190:
4184:
4175:
4169:
4160:
4151:
4132:
4126:
4114:. Retrieved
4109:
4062:
4055:
4044:. Retrieved
4040:
4031:
3990:
3987:Paleobiology
3986:
3980:
3968:. Retrieved
3961:the original
3956:
3943:
3936:Stewart 2001
3931:
3924:Stewart 2001
3919:
3912:Stewart 2001
3907:
3900:Stewart 2001
3895:
3888:Stewart 2001
3883:
3876:Stewart 2001
3871:
3859:. Retrieved
3855:
3845:
3818:
3814:
3804:
3793:
3771:
3749:
3745:
3736:
3728:
3720:
3709:. Retrieved
3705:the original
3668:
3662:
3649:
3642:Stevens 1974
3637:
3629:
3624:
3618:. Macmillan.
3615:
3587:
3562:
3529:
3523:
3489:About D'Arcy
3484:
3464:. New York:
3460:
3456:Livio, Mario
3427:
3420:
3408:
3396:
3389:Stewart 2001
3372:
3366:
3347:
3343:
3330:
3310:
3300:
3289:
3276:
3264:
3259:Knott, Ron.
3254:
3240:(1): 28–30.
3237:
3233:
3227:
3218:
3212:
3188:
3184:Livio, Mario
3157:. Retrieved
3151:
3141:
3136:, p. 3.
3134:Stevens 1974
3129:
3121:
3120:
3111:
3105:
3097:
3090:Pythagoreans
3084:
3075:
3074:
2974:: a melting
2910:
2868:
2848:(peaks) and
2835:
2819:
2792:
2784:
2781:Richard Prum
2777:
2748:James Murray
2745:
2728:
2690:
2672:
2655:Equus grevyi
2654:
2634:
2582:evolutionary
2579:
2458:
2429:
2408:
2391:
2355:tessellation
2288:
2255:
2252:Tessellation
2208:produced by
2143:
2117:soap bubbles
2069:
2059:
2052:exoskeletons
2038:radiolarians
2031:
2007:
1992:
1977:dislocations
1890:
1859:
1856:Waves, dunes
1848:
1750:, resembles
1745:
1715:helical flow
1709:
1683:
1660:
1657:chaos theory
1654:
1632:
1604:
1573:
1544:
1528:
1509:
1452:
1448:golden ratio
1444:golden angle
1429:
1427:it is 5/13.
1381:flower heads
1367:
1345:
1269:self-similar
1239:
1180:blood vessel
1149:
1132:
1097:self-similar
1091:
1083:
1079:
1071:
1061:
1032:
1014:Sea anemones
885:, like this
859:
820:
800:sea anemones
782:
712:
686:The laws of
685:
677:chaos theory
671:
662:hummingbirds
655:
589:
564:
559:and others,
549:Georg Cantor
491:
480:animal horns
473:
466:A. H. Church
463:
400:
364:golden ratio
333:
320:population.
295:
279:Theophrastus
274:
239:
228:make use of
200:
145:
80:
79:
61:
38:Namib Desert
29:
6826:(c. 335 BC)
6816:(c. 390 BC)
6795:Work of art
6748:Picturesque
6604:Avant-garde
6561:Winckelmann
6436:Kierkegaard
6361:Collingwood
6331:Baudrillard
6258:Romanticism
6228:Historicism
6162:Mathematics
6027:Alan Turing
5985:Liber Abaci
5904:Mathematics
5810:in crystals
5800:Soap bubble
5795:Phyllotaxis
5592:Murphy, Pat
5495:Liber Abaci
5370:17 February
4399:Maor, Eli.
4197:(1450): 31.
3655:Steen, L.A.
3112:The Physics
3100:, c. 350 BC
2984:Tuktoyaktuk
2807:Mima mounds
2311:point group
2210:leafhoppers
2204:(secretory
2202:Brochosomes
2160:Spumellaria
2086:soccer ball
2066:Radiolarian
2022:tetrahedral
1995:soap bubble
1845:brain coral
1797:flood plain
1793:oxbow lakes
1551:logarithmic
1525:phyllotaxis
1401:snake fruit
1370:phyllotaxis
1342:Phyllotaxis
1192:ocean waves
1182:branching,
1172:snow flakes
1156:fault lines
1154:, geologic
1113:umbellifers
1103:. Infinite
902:Echinoderms
812:sea urchins
804:echinoderms
673:Mathematics
502:oscillating
488:Alan Turing
419:Lord Kelvin
356:phyllotaxis
308:Liber Abaci
266:ideal forms
202:Mathematics
176:Alan Turing
72:as well as
68:and signal
7072:Categories
6765:Recreation
6743:Perception
6636:Creativity
6336:Baumgarten
6326:Baudelaire
6208:Classicism
6123:Aesthetics
5974:Empedocles
5969:Pythagoras
5887:Camouflage
5825:in biology
5820:in flowers
5790:Parastichy
5432:1503.03853
5340:18 January
5312:9 December
5169:Ball 2009a
5157:Ball 2009a
5115:Ball 2009a
5037:Ball 2009a
4999:Ball 2009a
4948:Ball 2009a
4652:Ball 2009a
4498:Ball 2009a
4484:2012-05-02
4413:Ball 2009a
4337:1702.03997
4046:2022-04-23
3970:25 October
3815:BioScience
3711:2012-05-02
3567:Ball 2009a
3413:Ball 2009a
3401:Ball 2009a
3375:. preface.
3071:References
2929:tiger bush
2799:tiger bush
2725:cuttlefish
2598:signalling
2590:camouflage
2466:. When an
2285:osteoderms
2158:Haeckel's
2136:Radiolaria
2056:sea urchin
2014:soap films
1941:Taklamakan
1904:avalanches
1886:Taklamakan
1873:Wind waves
1405:pine cones
1385:seed heads
1374:parastichy
1308:dielectric
1164:coastlines
941:Snowflakes
904:like this
847:spheroidal
823:snowflakes
816:sea lilies
723:camouflage
664:, and the
472:published
443:Radiolaria
384:pineapples
326:pentagonal
262:universals
250:Empedocles
246:Pythagoras
214:biological
152:soap films
141:Empedocles
137:Pythagoras
97:symmetries
66:camouflage
64:, provide
6770:Reverence
6676:Eroticism
6646:Depiction
6619:Masculine
6521:Santayana
6481:Nietzsche
6426:Hutcheson
6416:Heidegger
6401:Greenberg
6356:Coleridge
6321:Balthasar
6306:Aristotle
6268:Theosophy
6263:Symbolism
6238:Modernism
6223:Formalism
6076:Emergence
5979:Fibonacci
5558:(2009a).
4596:250864634
4082:cite book
4066:. Dover.
3458:(2003) .
3246:0047-6269
3186:(2003) .
3122:Citations
3094:Aristotle
3076:Footnotes
3040:Emergence
2980:ice wedge
2950:Fir waves
2826:ice wedge
2803:fir waves
2756:morphogen
2706:ladybirds
2704:Array of
2606:predatory
2586:functions
2351:honeycomb
2273:honeycomb
2216:geometry.
2214:fullerene
2187:fullerene
2138:drawn by
2097:fullerene
1908:nonlinear
1896:slip face
1869:oscillate
1817:Rio Cauto
1801:Rio Negro
1739:gastropod
1705:viscosity
1570:sunflower
1523:Spirals:
1491:Fibonacci
1432:sunflower
1397:pineapple
1389:sunflower
1387:like the
1378:composite
1328:dendritic
1263:spirals:
1210:fractals.
1160:mountains
1145:internode
1127:like the
1105:iteration
744:like the
681:L-systems
461:in 1885.
459:snowflake
451:Darwinian
392:skeletons
380:pinecones
360:clockwise
338:, citing
296:In 1202,
210:chemistry
150:examined
7098:Patterns
7045:Category
6977:Axiology
6846:(c. 500)
6836:(c. 100)
6711:Judgment
6666:Emotions
6661:Elegance
6641:Cuteness
6614:Feminine
6577:Concepts
6546:Tanizaki
6526:Schiller
6511:Richards
6501:Rancière
6471:Maritain
6406:Hanslick
6346:Benjamin
6218:Feminism
6187:Theology
6167:Medieval
6157:Japanese
6152:Internet
5805:Symmetry
5763:Patterns
5638:(2001).
5609:(2011).
5498:, 1202.
5469:25136099
5307:BBC News
5238:11958709
4927:14668849
4639:10045303
4535:(1990).
4513:(1961).
4372:28690919
4159:(2003).
4133:PLoS One
4023:84195721
3790:(1968).
3693:17840903
3657:(1988).
3502:and the
3492:Archived
3338:(2002).
3308:(1999).
3284:(1658).
3029:See also
2986:, Canada
2760:feedback
2602:ladybird
2349:Arrays:
2295:minerals
2277:pangolin
2045:spicules
1997:forms a
1892:Barchans
1875:are sea
1711:Meanders
1689:vortices
1549:shell's
1546:Nautilus
1471:meristem
1463:proteins
1355:nautilus
1351:molluscs
1284:Angelica
1267:showing
1176:crystals
1129:air fern
1125:hydrozoa
1093:Fractals
1050:Ice eggs
960:Fluorite
906:starfish
867:ice eggs
831:Crystals
808:starfish
784:Symmetry
765:Symmetry
704:meanders
646:polygons
642:sycamore
537:alphabet
533:fractals
521:L-system
447:symmetry
396:crystals
348:quincunx
196:fractals
184:patterns
168:symmetry
109:meanders
89:patterns
87:. These
44:and the
7040:Outline
6955:Related
6822:Poetics
6790:Tragedy
6780:Sublime
6753:Quality
6738:Mimesis
6696:Harmony
6681:Fashion
6656:Ecstasy
6651:Disgust
6567:more...
6536:Scruton
6461:Lyotard
6396:Goodman
6376:Deleuze
6311:Aquinas
6301:Alberti
6274:more...
6253:Realism
6233:Marxism
6213:Fascism
6196:Schools
6182:Science
6137:Ancient
6064:Related
5931:Crystal
5926:Physics
5914:Fractal
5892:Mimicry
5877:Biology
5785:Meander
5704:, 1990.
5665:, 2007.
5602:, 1993.
5551:, 2006.
5460:4156754
5437:Bibcode
5398:5 April
5229:1690965
4974:Bibcode
4907:Bibcode
4681:Bibcode
4619:Bibcode
4576:Bibcode
4543:101–107
4363:5500227
4342:Bibcode
4305:4158401
4297:1314350
4116:16 July
4015:4500143
3995:Bibcode
3837:1312929
3701:4849363
3673:Bibcode
3664:Science
3534:Bibcode
3003:Namibia
2932:plateau
2687:Leopard
2594:leopard
2550:Cooled
2511:Veined
2468:elastic
2369:Bismuth
2299:crystal
2146:(1904).
2142:in his
2140:Haeckel
2121:methane
1973:ripples
1957:barchan
1955:Dunes:
1799:of the
1752:Rule 30
1742:mollusc
1719:erosion
1701:laminar
1679:Rule 30
1669:. Some
1589:cabbage
1417:apricot
1359:chamber
1347:Spirals
1336:Spirals
1326:Trees:
1302:Trees:
1261:Fractal
1246:pinnate
1117:bryozoa
792:orchids
742:flowers
731:mimicry
715:biology
696:crystal
692:perfect
688:physics
658:orchids
571:fractal
346:of the
236:History
206:physics
105:spirals
46:ripples
7088:Nature
6946:(2009)
6936:(1977)
6926:(1946)
6916:(1939)
6906:(1935)
6896:(1934)
6886:(1933)
6876:(1891)
6866:(1835)
6856:(1757)
6723:Kitsch
6701:Humour
6631:Comedy
6609:Beauty
6551:Vasari
6541:Tagore
6516:Ruskin
6456:Lukács
6446:Langer
6391:Goethe
6316:Balázs
6296:Adorno
6177:Nature
6142:Africa
5957:People
5865:Causes
5692:, and
5467:
5457:
5283:26 May
5236:
5226:
5137:
5075:
5022:28 May
4935:688445
4933:
4925:
4899:Nature
4867:May 2,
4775:
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4637:
4594:
4549:
4370:
4360:
4303:
4295:
4243:
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3554:937133
3552:
3472:
3435:
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3244:
3200:
2842:cortex
2614:mimics
2552:basalt
2536:Sicily
2513:gabbro
2464:stress
2460:Cracks
2455:Cracks
2331:halite
2309:, and
2042:sponge
1999:sphere
1981:Sistan
1819:, Cuba
1768:Flow:
1493:spiral
1425:almond
1288:sphere
1224:baobab
1190:, and
1121:corals
1034:Volvox
999:Garnet
980:splash
978:Water
926:medlar
855:Saturn
851:planet
843:splash
814:, and
777:, and
638:aphids
628:Causes
429:; the
318:rabbit
291:a rule
254:Darwin
242:nature
125:cracks
40:. The
7035:Index
6804:Works
6785:Taste
6775:Style
6556:Wilde
6496:Plato
6491:Pater
6451:Lipps
6411:Hegel
6381:Dewey
6371:Danto
6351:Burke
6172:Music
6147:India
6130:Areas
5964:Plato
5770:Crack
5427:arXiv
5204:(PDF)
4931:S2CID
4839:(PDF)
4805:(PDF)
4794:(PDF)
4592:S2CID
4478:(PDF)
4467:Forma
4463:(PDF)
4332:arXiv
4301:S2CID
4112:. NPR
4019:S2CID
4011:JSTOR
3964:(PDF)
3953:(PDF)
3861:2 May
3833:JSTOR
3697:S2CID
3550:S2CID
3159:4 May
3022:sulci
2976:pingo
2936:Niger
2850:sulci
2619:genes
2612:, or
2426:salak
2281:salak
2269:wasps
2261:tiles
2095:, or
2034:cells
2016:obey
1975:with
1971:Wind
1882:dunes
1861:Waves
1697:fluid
1695:of a
1647:dense
1467:auxin
1436:daisy
1413:hazel
1393:fruit
1314:block
1109:ferns
1070:. In
1016:have
943:have
908:have
887:tiger
853:like
650:veins
275:eidos
271:εἶδος
258:Plato
133:Plato
117:foams
113:waves
101:trees
6759:Rasa
6717:Kama
6691:Gaze
6626:Camp
6506:Rand
6441:Klee
6431:Kant
6421:Hume
6341:Bell
5842:Wave
5780:Foam
5775:Dune
5465:PMID
5400:2018
5372:2015
5342:2017
5314:2013
5285:2012
5278:TIEE
5234:PMID
5135:ISBN
5073:ISBN
5024:2012
4923:PMID
4869:2012
4861:USGS
4773:ISBN
4761:ISBN
4635:PMID
4547:ISBN
4442:2013
4368:PMID
4293:PMID
4241:ISBN
4216:ISBN
4118:2019
4088:link
4068:ISBN
3972:2012
3863:2012
3689:PMID
3596:ISBN
3470:ISBN
3433:ISBN
3316:ISBN
3242:ISSN
3198:ISBN
3161:2012
3020:and
3018:gyri
2846:gyri
2836:The
2801:and
2521:Skye
2113:Foam
2010:foam
1791:and
1645:and
1434:and
1421:pear
1399:and
1383:and
1271:form
1226:tree
1111:and
750:bees
746:lily
733:and
523:, a
482:and
382:and
220:and
208:and
139:and
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6686:Fun
6466:Man
6386:Fry
5455:PMC
5445:doi
5423:111
5224:PMC
5216:doi
5212:269
4982:doi
4970:235
4915:doi
4903:426
4689:doi
4627:doi
4584:doi
4358:PMC
4350:doi
4285:doi
4195:105
4137:doi
4003:doi
3823:doi
3681:doi
3669:240
3542:doi
3530:237
3352:doi
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