Divine Pattern

How does this little fish create such a structure, close to golden ratio and Fibonacci sequence, with sands on the bottom of ocean?  We are eager to say it is just for mating, to me it is a superimposition of loves in a praise to God: "The seven heavens and the earth and everyone in them glorify God. There is not a single thing that does not celebrate His praise, though you do not understand their praise: God is most forbearing, most forgiving." (17:44). 

“Do you not realize [Prophet] that everything in the heavens and earth submits to God: the sun, the moon, the stars, the mountains, the trees, and the animals? So do many human beings, though for many others punishment is well deserved." (22:18)

"They also say, ‘Why has no sign been sent down to him from his Lord?’ Say, ‘God certainly has the power to send down a sign,’ though most of them do not know: all the creatures that crawl on the earth and those that fly with their wings are communities like yourselves. We have missed nothing out of the Record– in the end they will be gathered to their Lord.  Those who reject Our signs are deaf, dumb, and in total darkness. God leaves whoever He will to stray, and sets whoever He will on a straight path.” (6:37-39)

Strangely the shape of structure built by Japanese puffer fish is similar to Islamic geometric patterns. From Wikipedia:

https://en.wikipedia.org/wiki/Islamic_geometric_patterns

"Many Islamic designs are built on squares and circles, typically repeated, overlapped and interlaced to form intricate and complex patterns. A recurring motif is the 8-pointed star, often seen in Islamic tilework; it is made of two squares, one rotated 45 degrees with respect to the other. The fourth basic shape is the polygon, including pentagons and octagons. All of these can be combined and reworked to form complicated patterns with a variety of symmetries including reflections and rotations. Such patterns can be seen as mathematical tessellations, which can extend indefinitely and thus suggest infinity. They are constructed on grids that require only ruler and compass to draw. Artist and educator Roman Verostko argues that such constructions are in effect algorithms, making Islamic geometric patterns forerunners of modern algorithmic art.

The circle symbolizes unity and diversity in nature, and many Islamic patterns are drawn starting with a circle. For example, the decoration of the 15th-century mosque in Yazd, Iran is based on a circle, divided into six by six circles drawn around it, all touching at its centre and each touching its two neighbours' centres to form a regular hexagon. On this basis is constructed a six-pointed star surrounded by six smaller irregular hexagons to form a tessellating star pattern. This forms the basic design which is outlined in white on the wall of the mosque. That design, however, is overlaid with an intersecting tracery in blue around tiles of other colours, forming an elaborate pattern that partially conceals the original and underlying design. A similar design forms the logo of the Mohammed Ali Research Center."

"Everything which prostrates itself bears witness to its own root from which it is absent by being a branch. When a thing is diverted from being a root by being a branch, it is said to it, "Seek that which is absent from you, your root from which you have emerged." So the thing prostrates itself to the soil which is its root. The spirit prostrates itself to the Universal Spirit (ar-Ruh al-Kull) from which it emerged. The inmost secret (sirr) prostrates itself to its Lord by means of whom it has achieved its level."
From the Futuhat of
Shaykh-ul Akbar Muhyiddin Ibn Arabi
Trans. William Chittick

********************************************

Creating the Never Ending Bloom:  John Edmark

Blooms: Strobe-Animated Sculptures

BLOOMS 2: Strobe Animated Sculptures Invented by John Edmark

***************************

From Andy Goldsworthy art works:

http://visualmelt.com/Andy-Goldsworthy

The magic of Fibonacci numbers | Arthur Benjamin

Fibonacci Sequence Documentary - Golden Section Explained - Secret Teachings

Islamic patterns:

What Does It Mean?



If one asked me "what are these patterns about?" a while ago, I would resort to all sorts of 'evolutionary theories'. Materialism has not much to say but 'mating' and 'survival'. Now, it seems to me obvious that it is "inside" and "outside" us. It is dependent origination and connection to the source in attraction of love. A fish or a bird doesn't have any claim to power but praising God in the depth of their existence-instinct. When human beings arrive at understanding the poverty of their existence, they receive scriptures and make mandala and Islamic geometric. The symmetry in the sand structure above or in human intuition and arts resonate with the existent of the whole in the parts, without reducing them to each other, and the whole and part all tend to a center which attracts them and in this very attraction-love holds them together. It just tells me that my independent existence is all dependent-in absolute poverty of existence-to be drawn-love-praise to this center.

**********************************

M.C. Escher Maurits Cornelis Escher (1898-1972) is one of the world's most famous graphic artists. His art is enjoyed by millions of people all over the world, as can be seen on the many web sites on the internet.

He is most famous for his so-called impossible constructions, such as Ascending and Descending, Relativity, his Transformation Prints, such as Metamorphosis I, Metamorphosis II and Metamorphosis III, Sky & Water I or Reptiles.

http://www.mcescher.com/gallery/

*************************************

A Penrose tiling is an example of non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of prototiles implies that a shifted copy of a tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right.

Periodic and aperiodic tilings

Figure 1. Part of a periodic tiling

Penrose tilings are simple examples of aperiodic tilings of the plane.[1] A tiling is a covering of the plane by tiles with no overlaps or gaps; the tiles normally have a finite number of shapes, called prototiles, and a set of prototiles is said to admit a tiling or tile the plane if there is a tiling of the plane using only tiles congruent to these prototiles.[2] The most familiar tilings (e.g., by squares or triangles) are periodic: a perfect copy of the tiling can be obtained by translating all of the tiles by a fixed distance in a given direction. Such a translation is called a period of the tiling; more informally, this means that a finite region of the tiling repeats itself in periodic intervals. If a tiling has no periods it is said to be non-periodic. A set of prototiles is said to be aperiodic if it tiles the plane but every such tiling is non-periodic; tilings by aperiodic sets of prototiles are called aperiodic tilings:  https://en.wikipedia.org/wiki/Penrose_tiling

**************************

Jay Hambidge

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Jay_Hambidge

"At the Tomb of Omar Khayyam", by Jay Hambidge

Jay Hambidge (1867–1924) was a Canadian born American artist. He was a pupil at the Art Students' League in New York and of William Chase, and a thorough student of classical art. He conceived the idea that the study of arithmetic with the aid of geometrical designs was the foundation of the proportion and symmetry in Greek architecture, sculpture and ceramics.^[1] Careful examination and measurements of classical buildings in Greece, among them the Parthenon, the temple of Apollo at Bassæ, of Zeus at Olympia and Athenæ at Ægina, prompted him to formulate the theory of "dynamic symmetry"^{[citation needed]} as demonstrated in his works Dynamic Symmetry: The Greek Vase (1920)^[2] and The Elements of Dynamic Symmetry (1926).^[3] It created a great deal of discussion.^[1] He found a disciple in Dr. Lacey D. Caskey, the author of Geometry of Greek Vases (1922).^[4]

In 1921, articles critical of Hambidge's theories were published by Edwin M. Blake in Art Bulletin, and by Rhys Carpenter in American Journal of Archaeology. Art historian Michael Quick says Blake and Carpenter "used different methods to expose the basic fallacy of Hambidge's use of his system on Greek art—that in its more complicated constructions, the system could describe any shape at all."^[5] In 1979 Lee Malone said Hambidge's theories were discredited, but that they had appealed to many American artists in the early 20th century because "he was teaching precisely the things that certain artists wanted to hear, especially those who had blazed so brief a trail in observing the American scene and now found themselves displaced by the force of contemporary European trends."^[4]

A number of notable American and Canadian artists have used dynamic symmetry in their painting, including George Bellows (1882–1925),^[6] Maxfield Parrish (1870–1966),^[7] The New Yorker cartoonist Helen Hokinson (1893–1949), Al Nestler (1900–1971),^[8][9] Kathleen Munn (1887–1974),^[10] the children's book illustrator and author Robert McCloskey (1914–2003),^[11] and Clay Wagstaff (b. 1964).^[12]

Dynamic symmetry[edit]

Dynamic symmetry is a proportioning system and natural design methodology described in Hambidge's books. The system uses dynamic rectangles, including root rectangles based on ratios such as √2, √3, √5, the golden ratio (φ = 1.618...), its square root (√φ = 1.272...), and its square (φ² = 2.618....), and the silver ratio (${\displaystyle \delta _{s}=2.414...}$).^[13][14]

From the study of phyllotaxis and the related Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...), Hambidge says that "a much closer representation would be obtained by a substitute series such as 118, 191, 309, 500, 809, 1309, 2118, 3427, 5545, 8972, 14517, etc. One term of this series divided into the other equals 1.6180, which is the ratio needed to explain the plant design system."^[15] This substitute sequence is a generalization of the Fibonacci sequence that chooses 118 and 191 as the beginning numbers to generate the rest. In fact, the standard Fibonacci sequence provides the best possible rational approximations to the golden ratio for numbers of a given size.

*********************************

The Modulor was a standard model of the human form which Le Corbusier devised to determine the correct amount of living space needed for residents in his buildings. It was also his rather original way of dealing with differences between the metric system and British or American system, since the Modulor was not attached to either one.

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. Many scholars see the Modulor as a humanistic expression but it is also argued that: "It's exactly the opposite (...) It's the mathematicization of the body, the standardization of the body, the rationalization of the body."[69]

He took Leonardo's suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system.

From Wikipedia , Le Corbusier: https://en.wikipedia.org/wiki/Le_Corbusier

***************************************

The Sri Yantra is a form of mystical diagram, known as a yantra, found in the Shri Vidya school of Hindu tantra. The diagram is formed by nine interlocking triangles that surround and radiate out from the central (bindu) point. The two dimensional Sri Chakra, when it is projected into three dimensions is called a Maha Meru. Mount Meru derives its name from this Meru like shape.

https://en.wikipedia.org/wiki/Sri_Yantra

****************************************************

On Growth and Form

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/On_Growth_and_Form

On Growth and Form

Title page of first edition
Author	D'Arcy Wentworth Thompson
Illustrator	Thompson
Country	United Kingdom
Subject	Mathematical biology
Genre	Descriptive science
Publisher	Cambridge University Press
Publication date	1917
Pages	793 1942 edition, 1116
Awards	Daniel Giraud Elliot Medal

On Growth and Form is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942.

The book covers many topics including the effects of scale on the shape of animals and plants, large ones necessarily being relatively thick in shape; the effects of surface tension in shaping soap films and similar structures such as cells; the logarithmic spiral as seen in mollusc shells and ruminant horns; the arrangement of leaves and other plant parts (phyllotaxis); and Thompson's own method of transformations, showing the changes in shape of animal skulls and other structures on a Cartesian grid.

The work is widely admired by biologists, anthropologists and architects among others, but less often read than cited.^[1] Peter Medawar explains this as being because it clearly pioneered the use of mathematics in biology, and helped to defeat mystical ideas of vitalism; but that the book is weakened by Thompson's failure to understand the role of evolution and evolutionary history in shaping living structures. Philip Ball on the other hand suspects that while Thompson argued for physical mechanisms, his rejection of natural selection bordered on vitalism.

Contents[edit]

The contents of the chapters in the first edition are summarized below. All but Chapter 11 have the same titles in the second edition, but many are longer, as indicated by the page numbering of the start of each chapter. Bonner's abridgment shortened all the chapters, and removed some completely, again as indicated at the start of each chapter's entry below.

1. Introductory[edit]

(1st edition p1 - 2nd edition p1 - Bonner p1)

Thompson names the progress of chemistry towards Kant's goal of a mathematical science able to explain reactions by molecular mechanics, and points out that zoology has been slow to look to mathematics. He agrees that zoologists rightly seek for reasons in animals' adaptations, and reminds readers of the related but far older philosophical search for teleology, explanation by some Aristotelian final cause. His analysis of "growth and form" will try to show how these can be explained with ordinary physical laws.

2. On Magnitude[edit]

Models used (by William Froude) to show that the drag on a hull varies with square root of waterline length^[11]

(1st p16 - 2nd p22 - Bonner p15)

Thompson begins by showing that an animal's surface and volume (or weight) increase with the square and cube of its length, respectively, and deducing simple rules for how bodies will change with size. He shows in a few short equations that the speed of a fish or ship rises with the square root of its length. He then derives the slightly more complex scaling laws for birds or aircraft in flight. He shows that an organism thousands of times smaller than a bacterium is essentially impossible.

3. The Rate of Growth[edit]

(1st p50 - 2nd p78 - Bonner removed)

Thompson points out that all changes of form are phenomena of growth. He analyses growth curves for man, noting rapid growth before birth and again in the teens; and then curves for other animals. In plants, growth is often in pulses, as in Spirogyra, peaks at a specific temperature, and below that value roughly doubles every 10 degrees Celsius. Tree growth varies cyclically with season (less strongly in evergreens), preserving a record of historic climates. Tadpole tails regenerate rapidly at first, slowing exponentially.

4. On the Internal Form and Structure of the Cell[edit]

(1st p156 - 2nd p286 - Bonner removed)

Thompson argues for the need to study cells with physical methods, as morphology alone had little explanatory value. He notes that in mitosis the dividing cells look like iron filings between the poles of a magnet, in other words like a force field.

5. The Forms of Cells[edit]

Vorticella campanula (stalked cup shaped organisms) attached to a green plant

(1st p201 - 2nd p346 - Bonner p49)

He considers the forces such as surface tension acting on cells, and Plateau's experiments on soap films. He illustrates the way a splash breaks into droplets and compares this to the shapes of Campanularian zoophytes (Hydrozoa). He looks at the flask-like shapes of single-celled organisms such as species of Vorticella, considering teleological and physical explanations of their having minimal areas; and at the hanging drop shapes of some Foraminifera such as Lagena. He argues that the cells of trypanosomes are similarly shaped by surface tension.

6. A Note on Adsorption[edit]

(1st p277 - 2nd p444 - Bonner removed)

Thompson notes that surface tension in living cells is reduced by substances resembling oils and soaps; where the concentrations of these vary locally, the shapes of cells are affected. In the green alga Pleurocarpus (Zygnematales), potassium is concentrated near growing points in the cell.

7. The Forms of Tissues, or Cell-aggregates[edit]

(1st p293 - 2nd p465 - Bonner p88)

Thompson observes that in multicellular organisms, cells influence each other's shapes with triangles of forces. He analyses parenchyma and the cells in a frog's egg as soap films, and considers the symmetries bubbles meeting at points and edges. He compares the shapes of living and fossil corals such as Cyathophyllum and Comoseris, and the hexagonal structure of honeycomb, to such soap bubble structures.

8. The same (continued)[edit]

(1st p346 - 2nd p566 - Bonner merged with previous chapter)

Thompson considers the laws governing the shapes of cells, at least in simple cases such as the fine hairs (a cell thick) in the rhizoids of mosses. He analyses the geometry of cells in a frog's egg when it has divided into 4, 8 and even 64 cells. He shows that uniform growth can lead to unequal cell sizes, and argues that the way cells divide is driven by the shape of the dividing structure (and not vice versa).

9. On Concretions, Spicules, and Spicular Skeletons[edit]

A selection of spicules in the Demospongiae

(1st p411 - 2nd p645 - Bonner p132)

Thompson considers the skeletal structures of diatoms, radiolarians, foraminifera and sponges, many of which contain hard spicules with geometric shapes. He notes that these structures form outside living cells, so that physical forces must be involved.

10. A Parenthetic Note on Geodetics[edit]

(1st p488 - 2nd p741 - Bonner removed)

Thompson applies the use of the geodetic line, "the shortest distance between two points on the surface of a solid of revolution", to the spiral thickening of plant cell walls and other cases.

11. The Logarithmic Spiral ['The Equiangular Spiral' in 2nd Ed.][edit]

Halved shell of Nautilus showing the chambers (camerae) in a logarithmic spiral

(1st p493 - 2nd p748 - Bonner p172)

Thompson observes that there are many spirals in nature, from the horns of ruminants to the shells of molluscs; other spirals are found among the florets of the sunflower. He notes that the mathematics of these are similar but the biology differs. He describes the spiral of Archimedes before moving on to the logarithmic spiral, which has the property of never changing its shape: it is equiangular and is continually self-similar. Shells as diverse as Haliotis, Triton, Terebra and Nautilus (illustrated with a halved shell and a radiograph) have this property; different shapes are generated by sweeping out curves (or arbitrary shapes) by rotation, and if desired also by moving downwards. Thompson analyses both living molluscs and fossils such as ammonites.

12. The Spiral Shells of the Foraminifera[edit]

(1st p587 - 2nd p850 - Bonner merged with previous chapter)

Thompson analyses diverse forms of minute spiral shells of the foraminifera, many of which are logarithmic, others irregular, in a manner similar to the previous chapter.

13. The Shapes of Horns, and of Teeth or Tusks: with A Note on Torsion[edit]

The spiral horns of the male bighorn sheep, Ovis canadensis

(1st p612 - 2nd p874 - Bonner p202)

Thompson considers the three types of horn that occur in quadrupeds: the keratin horn of the rhinoceros; the paired horns of sheep or goats; and the bony antlers of deer.

In a note on torsion, Thompson mentions Charles Darwin's treatment of climbing plants which often spiral around a support, noting that Darwin also observed that the spiralling stems were themselves twisted. Thompson disagrees with Darwin's teleological explanation, that the twisting makes the stems stiffer in the same way as the twisting of a rope; Thompson's view is that the mechanical adhesion of the climbing stem to the support sets up a system of forces which act as a 'couple' offset from the centre of the stem, making it twist.

14. On Leaf-arrangement, or Phyllotaxis[edit]

Phyllotaxis of sunflower florets

(1st p635 - 2nd p912 - Bonner removed)

Thompson analyses phyllotaxis, the arrangement of plant parts around an axis. He notes that such parts include leaves around a stem; fir cones made of scales; sunflower florets forming an elaborate crisscrossing pattern of different spirals (parastichies). He recognises their beauty but dismisses any mystical notions; instead he remarks that

When the bricklayer builds a factory chimney, he lays his bricks in a certain steady, orderly way, with no thought of the spiral patterns to which this orderly sequence inevitably leads, and which spiral patterns are by no means "subjective".

— Thompson, 1917, page 641

The numbers that result from such spiral arrangements are the Fibonacci sequence of ratios 1/2, 2/3, 3.5 ... converging on 0.61803..., the golden ratio which is

beloved of the circle-squarer, and of all those who seek to find, and then to penetrate, the secrets of the Great Pyramid. It is deep-set in Pythagorean as well as in Euclidean geometry.

— Thompson, 1917, page 649

15. On the Shapes of Eggs, and of certain other Hollow Structures[edit]

(1st p652 - 2nd p934 - Bonner removed)

Eggs are what Thompson calls simple solids of revolution, varying from the nearly spherical eggs of owls through more typical ovoid eggs like chickens, to the markedly pointed eggs of cliff-nesting birds like the guillemot. He shows that the shape of the egg favours its movement along the oviduct, a gentle pressure on the trailing end sufficing to push it forwards. Similarly, sea urchin shells have teardrop shapes, such as would be taken up by a flexible bag of liquid.

16. On Form and Mechanical Efficiency[edit]

Thompson compared a dinosaur's spine to the Forth Railway Bridge (right).

(1st p670 - 2nd p958 - Bonner p221)

Thompson criticizes talk of adaptation by coloration in animals for presumed purposes of crypsis, warning and mimicry (referring readers to E. B. Poulton's The Colours of Animals, and more sceptically to Abbott Thayer's Concealing-coloration in the Animal Kingdom). He considers the mechanical engineering of bone to be a far more definite case. He compares the strength of bone and wood to materials such as steel and cast iron; illustrates the "cancellous" structure of the bone of the human femur with thin trabeculae which formed "nothing more nor less than a diagram of the lines of stress ... in the loaded structure", and compares the femur to the head of a building crane. He similarly compares the cantilevered backbone of a quadruped or dinosaur to the girder structure of the Forth Railway Bridge.

17. On the Theory of Transformations, or the Comparison of Related Forms[edit]

Albrecht Dürer's face transforms were among Thompson's inspirations

(1st p719 - 2nd p1026 - Bonner p268)

Inspired by the work of Albrecht Dürer, Thompson explores how the forms of organisms and their parts, whether leaves, the bones of the foot, human faces or the body shapes of copepods, crabs or fish, can be explained by geometrical transformations. For example:

Thompson illustrated the transformation of Argyropelecus olfersi into Sternoptyx diaphana by applying a shear mapping.

Among the fishes we discover a great variety of deformations, some of them of a very simple kind, while others are more striking and more unexpected. A comparatively simple case, involving a simple shear, is illustrated by Figs. 373 and 374. Fig. 373 represents, within Cartesian co-ordinates, a certain little oceanic fish known as Argyropelecus olfersi. Fig. 374 represents precisely the same outline, transferred to a system of oblique co-ordinates whose axes are inclined at an angle of 70°; but this is now (as far as can be seen on the scale of the drawing) a very good figure of an allied fish, assigned to a different genus, under the name of Sternoptyx diaphana. Thompson 1917, pages 748–749

In similar style he transforms the shape of the carapace of the crab Geryon variously to that of Corystes by a simple shear mapping, and to Scyramathia, Paralomis, Lupa, and Chorinus (Pisinae) by stretching the top or bottom of the grid sideways. The same process changes Crocodilus porosus to Crocodilus americanus and Notosuchus terrestris; relates the hip-bones of fossil reptiles and birds such as Archaeopteryx and Apatornis; the skulls of various fossil horses, and even the skulls of a horse and a rabbit. A human skull is stretched into those of the chimpanzee and baboon, and with "the mode of deformation .. on different lines" (page 773), of a dog.

Epilogue[edit]

(1st p778 - 2nd p1093 - Bonner p326)

In the brief epilogue, Thompson writes that he will have succeeded "if I have been able to shew [the morphologist] that a certain mathematical aspect of morphology ... is ... complementary to his descriptive task, and helpful, nay essential, to his proper study and comprehension of Form." More lyrically, he writes that "For the harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty" and quotes Isaiah 40:12 on measuring out the waters and heavens and the dust of the earth. He ends with a paragraph praising the French entomologist Jean-Henri Fabre^[12] who "being of the same blood and marrow with Plato and Pythagoras, saw in Number 'la clef de voute' [the key to the vault (of the universe)] and found in it 'le comment et le pourquoi des choses' [the how and the why of things]".

*************************************

Divine Pattern in Nature

******************************************