Types of Phyllotaxis

In botany, phyllotaxis is the arrangement of the leaves on the stem of a plant (from Greek phyllo-: leaf; -taxis, motion/orientation).

The four main types of phyllotaxis usually recognized are shown below. They can be detected and further classified by the number of visible spirals (parastichies) they display. All these patterns can be modeled by simple lattice-like mathematical structures. Spiral phyllotaxis is the most frequent. Other patterns may exist that are not quite as regular, and seldom mentioned by botanists.

Four Main Types of Phyllotaxis

Distichous Phyllotaxis

In distichous phyllotaxis, leaves or other botanical elements grow one by one, each at 180 degrees from the previous one.

The Phalaenopsis orchid exhibits distichous phyllotaxis.

 

 

 

 

 

 

 

 

 

Disney’s Mathmagic World

Disney actually made an animation that teaches the Golden Section, and in surprising detail too. I find it so impressive for something that was made in ’59.

The Secret Power of Beauty

Armstrong examines a variety of attempts at divining the secret of beauty, beginning with the more formalist approaches, ensuing with a more psychological and philosophical aspect of beauty.

Approaches:

1. Form: The S-shaped Serpentine Line

• conceived by English painter and engraver William Hogarth, in his book, “The Analysis of Beauty”, published in 1753
• named the Line of Beauty, for its balance of simplicity and intricacy, unity and variety
• theory was easily disputed, for beauty can exist in the absence of curves (e.g. the Petit Trianon in the Versailles gardens) and the same s-shaped curve may not appear to be as pleasing when it is applied in another context (e.g. David Garrick’s bloated stomach)

2. Function: The Perfect Fit between Form and Purpose

• the function of the object allows us to see the multitude of its parts as a coherent and unified whole, as the eye fits our visual and intellectual understanding of the object together
• hence, the judging criteria of the appearance of the object would be determined by the preconceived notion of the function of the object (e.g. a beautiful sofa would look comfortable and inviting to sit in)
• this theory was also disproved as our prior knowledge of the object’s function would limit our view and understanding of its beauty. Function is merely one of the citeria for judging beauty, not its only criteria.
• often, when we wish to provide a precise and neat account of beauty, we risk making a Procrustean mistake, a fallacy whereby we try too hard to fit a single line of reasoning into the idea of beauty
• there may be more than one type of function for a particular classification of objects (e.g. a glass, though its main function is a vessel to allow liquids to be drunk out of, may come in many forms – a tankard, a mug, a Venetian glass etc.)

3. Proportion: The Pythagorean Theory of Beauty

• Pythagoras believed that there was a certain order governing both the cosmos and the human world – in other words, the structure of our soul is the same as the structure of the universe – that we are meant to coest in harmony with the universe
• this idealistic notion of beauty is very appealing and has its grounds in the realm of architecture, as many of the world’s man-made structures were conceived with these very proportions derived from nature
• however, it is very easy to spot a flaw in the Pythagorean argument, as these proportions simply cannot be applied to everything (e.g. a car cannot be nine times longer than it is wide)
• nonetheless, the connection of beauty with morality and truth still exists today and it is perhaps this longing for a utopia ruled by balance and harmony that makes us believe that

4. The Law of the Whole: The Gestalt Theory

• the secret to beauty is actually very complex, and it requires the considerations of numerous aspects, which are impossible to disentangle (e.g. Fantin-Latour’s “White Cup and Saucer”, 1864)
• by viewing things in a holistic light, we glean a better understanding of our appeal to its individual parts and this may lead to a productive result
• in turn, the acceptance of this theory raises other questions: what is it about the way that different elements go together that makes something beautiful? Why does it appeal to us?]

5. The Complementation of Parts

• it is the coming together of two seemingly disparate elements that bring out the best in each other and this enriches our experience of apprecating beauty
• each part retains its identity while flourishing in the setting that the other provides

Bibiliography

1. Armstrong, John. The Secret Power of Beauty. England: Clays Ltd, St Ives plc, 2005.

2. de Botton, Alain. The Architecture of Happiness. New York: Pantheon Books, 2006.

3. Pye, David. The Nature & Aesthetics of Design. Great Britain: The Herbert Press Ltd, 1988.

4. Elam, Kimberly. Geometry of Design: Studies in Proportion and Composition. New York: Princeton Architectural Press, 2001.

5. Smith, Thomas Gordon. Vitruvius on Architecture. New York: The Monacelli Press, Inc, 2003.

6. Doczi, Gyorgy. The Power of Limits: Proportional Harmonies in Nature, Art & Architecture. Boston: Shambhala Publications, Inc, 1981.

7. Hemenway, Priya. Divine Proportion: Phi in Art, Nature & Science. New York: Sterling Publishing Co., Inc, 2005.

8. Skinner, Stephen. Sacred Geometry: Deciphering the Code. New York: Sterling Publishing Co., Inc, 2006.

Interest in the Golden Section

Visual Aspect

I am attracted to the rationality that the Golden Section

provides: how it seems to explain a lot of phenomena

in Nature, in particular, dinergy in living things, which is

Nature’s basic pattern-forming process.

Philosophical Aspect

The theory of the Golden Section also seems to attest

to the reason behind why we find some things appealing

and attractive, and how we immediately feel at ease

upon looking at them. This contributes to the idea that

harmonious proportions form the basis of elegance and

beauty. I wish to investigate how much of our aesthetic

judgement is based on logic and reason as opposed to

instinct and intuition.

Visual Aspect

I am attracted to the rationality that the Golden Section provides: how it seems to explain a lot of phenomena in Nature, in particular, dinergy in living things, which is Nature’s basic pattern-forming process.

Philosophical Aspect

The theory of the Golden Section also seems to attest to the reasonbehind why we find some things appealing and attractive, and how we immediately feel at ease upon looking at them. This contributes to the idea that harmonious proportions form the basis of elegance and beauty. I wish to investigate how much of our aesthetic judgementis based on logic and reason as opposed to instinct and intuition.

Works designed with the Golden Ratio

Architecture

Ancient Civilisations

Carlos Chanfon Olmos, researcher of the UNAM, exposed in his Curso de Proporciòn, the presence of the golden ratio in a series of olmec heads, the Aztec calendar stone and a series of Aztec permission house plans.

In the fifties, Manuel Amabilis applied some of the analysis methods of Frederik Macody Lund and Jay Hambidge to several plans and sections of prehispanic buildings, such as El Tolocand La Iglesia of Las Monjas, a notable complex of Terminal Classic buildings constructed in the Puuc architectural style at Chichen Itza. According to his studies, their proportions derived from a series of successively inscribed pentagons, circles and pentagrams, just as the Gothic churches Lund studied do. Amabilis published his studies along with several self-explanatory images of various other precolumbine buildings with golden proportions in La Arquitectura Precolombina de Mexico, which was awarded the gold medal and the title of Academico by the “Real Academia de Bellas Artes de San Fernando” (Spain) in the “Fiesta de la Raza” contest of 1929.

According to John Pile, The Castle of Chichen Itza, built by the Maya civilization sometime between the 11th and 13th centuries AD to serve as a temple to the god Kukulcan, has golden proportions in its interior layout with walls placed so that the outer spaces relate to the center chamber as 0.618:1.

Islamic Architecture

A geometrical analysis of the Great Mosque of Kairouan (built by Uqba ibn Nafi c. 670 A.D.) reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz, who say it is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret.

Buddhist Architecture

The Stuppa of Borobudur in Java, Indonesia (built eighth to ninth century AD), the largest known Buddhist stupa, has the dimension of the square base related to the diameter of the largest circular terrace as 1.618:1, according to Pile.

Neogothic

According to the official tourism page of Buenos Aires, Argentina, the ground floor of the Palacio Barolo (1923), designed by Italian architect Mario Palanti, is built according to the golden section.

Gothic Era

Illustration of the Notre-Dame of Laon cathedral. According to Macody Lund, the superimposed regulator lines show that the cathedral has golden proportions.

The Golden Mean Philosophy

In philosophy, especially that of Aristotle, the golden mean is the desirable middle between two extremes, one of excess and the other of deficiency.

To the Greek mentality, it was an attribute of beauty. Both ancients and moderns realized that “there is a close association in mathematics between beauty and truth“. The poet John Keats, in his Ode on a Grecian Urn, put it this way:

Beauty is truth, truth is beauty, that is all
Ye know on earth, and all ye need to know.

The Greeks believed there to be three ‘ingredients’ to beauty: symmetry, proportion, and harmony. This triad of principles infused their life. They were very much attuned to beauty as an object of love and something that was to be imitated and reproduced in their lives, architecture, Paideia and politics. They judged life by this mentality.

In Chinese philosophy, a similar concept, Doctrine of the Mean, was propounded by Confucius.

History of the golden mean in philosophy

Crete

The earliest representation of this idea in culture is probably in the mythological Cretan tale of Daedalus and Icarus. Daedalus, a famous artist of his time, built feathered wings for himself and his son so that they might escape the clutches of King Minos. Daedalus warns his son to “fly the middle course”, between the sea spray and the sun’s heat. Icarus did not heed his father; he flew up and up until the sun melted the wax off his wings.

Delphi

Another early elaboration is the Doric saying carved on the front of the temple at Delphi: “Nothing in Excess”.

Pythagoreans

The first work on the golden mean is often attributed to Theano, wife of Pythagorus.

Socrates

Socrates teaches that a man “must know how to choose the mean and avoid the extremes on either side, as far as possible”.

In education, Socrates asks us to consider the effect of either an exclusive devotion to gymnastics or an exclusive devotion to music. It either “produced a temper of hardness and ferocity, (or) the other of softness and effeminacy“. Having both qualities, he believed, produces harmony; i.e., beauty and goodness. He additionally stresses the importance of mathematics in education for the understanding of beauty and truth.

Plato

Something disproportionate was evil and therefore to be despised. Plato says, “If we disregard due proportion by giving anything what is too much for it; too much canvas to a boat, too much nutriment to a body, too much authority to a soul, the consequence is always shipwreck.”

In the Laws, Plato applies this principle to electing a government in the ideal state: “Conducted in this way, the election will strike a mean between monarchy and democracy …”

Aristotle

In the Eudemian Ethics, Aristotle writes on the virtues. His constant phrase is, “… is the Middle state between …”. His psychology of the soul and its virtues is based on the golden mean between the extremes. In the Politics, Aristotle criticizes the Spartan Polity by critiquing the disproportionate elements of the constitution; e.g., they trained the men and not the women, and they trained for war but not peace. This disharmony produced difficulties which he elaborates on in his work. See also the discussion in the Nicomachean Ethics of the golden mean, and Aristotelian ethics in general.

Quotations

“In many things the middle have the best / Be mine a middle station.”
— Phocylides

“When Coleridge tried to define beauty, he returned always to one deep thought; beauty, he said, is unity in variety! Science is nothing else than the search to discover unity in the wild variety of nature,—or, more exactly, in the variety of our experience. Poetry, painting, the arts are the same search, in Coleridge’s phrase, for unity in variety.”
— J. Bronowski

“…but for harmony beautiful to contemplate, science would not be worth following.”
— Henri Poincaré.

“If a man finds that his nature tends or is disposed to one of these extremes…, he should turn back and improve, so as to walk in the way of good people, which is the right way. The right way is the mean in each group of dispositions common to humanity; namely, that disposition which is equally distant from the two extremes in its class, not being nearer to the one than to the other.”
— Maimonides

Jacques Maritain, throughout his Introduction to Philosophy, uses the idea of the golden mean to place Aristotelian-Thomist philosophy between the deficiencies and extremes of other philosophers and systems.

Confucius in the Analects taught excess is similar to deficiency(過猶不及). A way of living in the mean is the way of Zhongyong(中庸之道).

Gautama Buddha taught middle Way.

Zhuangzi had a similar idea.

Hu Shi wrote an article ‘the tales of Mr. Chabuduo’(差不多先生傳) on people who did not mind accuracy on matters.

http://en.wikipedia.org/wiki/Golden_mean_(philosophy)

Ancient Greek Architecture

Greek life was dominated by religion and so it is not surprising that the temples of ancient Greece were the biggest and most beautiful.They also had a political purpose as they were often built to celebrate civic power and pride, or offer thanksgiving to the patron deity of a city for success in war.

The Greeks developed three architectural systems, called orders, each with their own distinctive proportions and detailing. The Greek orders are: Doric, Ionic, and Corinthian.

The Doric style is rather sturdy and its top (the capital), is plain. This style was used in mainland Greece and the colonies in southern Italy and Sicily.

The Ionic style is thinner and more elegant. Its capital is decorated with a scroll-like design (a volute). This style was found in eastern Greece and the islands.

The Corinthian style is seldom used in the Greek world, but often seen on Roman temples. Its capital is very elaborate and decorated with acanthus leaves.

DORIC ORDER

Parthenon – temple of Athena Parthenos (“Virgin”), Greek goddess of wisdom, on the Acropolis in Athens. The Parthenon was built in the 5th century BC, and despite the enormous damage it has sustained over the centuries, it still communicates the ideals of order and harmony for which Greek architecture is known.

IONIC ORDER

Erechtheum – temple from the middle classical period of Greek art and architecture, built on the Acropolis of Athens between 421 and 405BC. The Erechtheum contained sanctuaries to Athena Polias, Poseidon, and Erechtheus. The requirements of the several shrines and the location upon a sloping site produced an unusual plan. From the body of the building porticoes project on east, north, and south sides. The eastern portico, hexastyle Ionic, gave access to the shrine of Athena, which was separated by a partition from the western cella. The northern portico, tetrastyle Ionic, stands at a lower level and gives access to the western cella through a fine doorway. The southern portico, known as the Porch of the Caryatids (see caryatid) from the six sculptured draped female figures that support its entablature, is the temple’s most striking feature; it forms a gallery or tribune. The west end of the building, with windows and engaged Ionic columns, is a modification of the original, built by the Romans when they restored the building. One of the east columns and one of the caryatids were removed to London by Lord Elgin, replicas being installed in their places.

The Temple of Apollo at Didyma – The Greeks built the Temple of Apollo at Didyma, Turkey (about 300 BC). The design of the temple was known as dipteral, a term that refers to the two sets of columns surrounding the interior section. These columns surrounded a small chamber that housed the statue of Apollo. With Ionic columns reaching 19.5 m (64 ft) high, these ruins suggest the former grandeur of the ancient temple.

The Temple of Athena Nike – part of the Acropolis in the city of Athens. The Greeks built the Temple of Apollo at Didyma, Turkey (about 300 BC). The design of the temple was known as dipteral, a term that refers to the two sets of columns surrounding the interior section. These columns surrounded a small chamber that housed the statue of Apollo. With Ionic columns reaching 19.5 m (64 ft) high, these ruins suggest the former grandeur of the ancient temple.

CORINTHIAN ORDER

The Temple of Zeus at Athens – The oldest known example, however, is found in the temple of Apollo at Bassae (c.420 B.C.). The Greeks made little use of the order; the chief example is the circular structure at Athens known as the choragic monument of Lysicrates ( 335 B.C.). The temple of Zeus at Athens (started in the 2d cent. B.C. and completed by Emperor Hadrian in the 2d cent. A.D.) was perhaps the most notable of the Corinthian temples.

http://www.ancientgreece.com/s/Art/

Fashion in Ancient Greece

Around 1.200 B.C. waves of Dorian invaders swept into Greece from Illyria on the east of the Adriatic and brought about the downfall of the Mycenaean civilization. The following four centuries are known as the “Dark Age” of Greece. The period started with a civilization of people dressed in bell-shaped skirts and tightly fitted bodices, and ended with a race dressed in draped clothes, the costumes we now associate with the Greeks and the Romans.

From the seventh century B.C. onwards, we have vast quantities of reference material for the study of costume. Greeks were among the finest exponents of figurative sculpture. Never before had costume been portrayed with such meticulous care and precision. Statues, together with untold numbers of painted pots, give the historian a unique pictorial history of the development of a nation and its fashions. At the same time, we have the invaluable contribution of the written word. Such great Greek historians as Herodotus have given us very detailed descriptions of developments in fashion and the social significance of costume and their accessories.

THE DORIC AND THE IONIC COSTUME

During the periods under discussion, generally referred to as Archaic and Classical, there were two basic styles of costume for both men and women: Doric, in existence at the beginning of the Archaic period, and Ionic which was adopted later.

The most basic garment for women was the Doric peplos, worn universally up to the beginning of the sixth century B.C. Made from a rectangle of woven wool, it measured about six feet in width and about eighteen inches more than the height of the wearer from shoulder to ankle in length. The fabric was wrapped round the wearer with the excess material folded over the top. It was then pinned on both shoulders and the excess material allowed to fall free, giving the impression of a short cape. The pins used for fastening the shoulders of the peplos were originally open pins with decorated heads, but they were later replaced by fibulae or brooches.

Herodotus, explaining this development, tells us a rather macabre story:

“After a disastrous military campaign by the Athenian army, all the forces were put to death except one man who managed to escape, return to Athens and tell the women about their husbands’ fate. Devastated the women took the huge pins from their Doric peplos and butchered the man in anger and contempt. The men of Athens, Herodotus tells us, were so horrified that they declared that Ionic dress should be worn in the future. Whether the story is true is uncertain, but there was for sure a period when open pins went out of favor.

The drawing on the left is showing the Doric chiton, the basic garment worn by Greek women up to the beginning of the sixth century B.C. It was folded so that there was an overlap of material on the bodice; the cloth was secured in place (on the shoulders) by pins.

One way of draping the Doric peplos involved covering the pouching formed by the belt with another section of the woollen rectangle. This maiden from the Acropolis dates from the 6th century B.C. The statue was originally brightly painted.

The Ionic chiton which followed was similar to the Doric peplos in overall construction. The woolen fabric was replaced by thin linen or occasionally silk, reflecting the increased influence of Asia Minor on the country. Again the costume was made from a single rectangle of material but much wider than that used for the peplos, measuring up to ten feet wide. It had no surplus material and was measured exactly from shoulder to ankle. The enormous width required that eight to ten fibulae were needed to fasten the top edge, leaving an openwork seam either side of the neck which ran across the shoulders and down the arms to form elbow-length sleeves.

The garment on the left, the Ionic chiton, was an alternative to the Doric chiton. The main difference lay in its greater width. The top edges of the material were fastened on the shoulder with clasps, so giving the wearer greater opportumity for the display of jewellery.

The dish illustrates the doric chiton as worn by both a man and a woman

Forms of male dress throughout the Archaic and Classical periods were very similar to women’s. The two basic garments, the chiton and the himation, were adopted by both sexes. The full-length chiton was worn by all Greek men until the fifth century B.C. when, except for the elderly, it was abandoned in favor of a shorter version, which was knee-length. It was sometimes pinned on the left shoulder only, leaving the right shoulder and arm free. Soldiers worn a chiton on which both shoulders and arms of the garment were sewn, leaving a wide slit for the head. It was generally tied at the waist and a certain amount of material was gathered above the belt to draw the hem-line to mid-thigh level.

It is a popular misconception that Greek costumes were white.This idea most probably arose because most Greek statues are of marble, bronze, or some other monochromatic material, and even the ones which were originally polychromatic had lost their colors by the time they were discovered.

During the Archaic period, clothes were generally white or off-white, commoners were forbidden to wear red chitons and himations in theaters or public places, but by the fifth century costumes were decorated with a wide range of colors.

Homer tells us of extravagant costumes woven with threads of silver and gold.

Pottery, statues and the written word have given us some knowledge of their decorative themes. One of the most common designs for borders was the Greek key pattern which has been used as a decorative motif ever since. More complex borders depicted themes ranging from animals, birds, and fish to complex battle scenes. The colored threads for these embroideries appear to have been limitless. Herodotus mentions yellow, violet, indigo, red and purple in a single garment.

http://www.annaswebart.com/culture/costhistory/ancient/

Education in Athens

The School of Athens, by Raphael, 1509-10. It is regarded as the perfect embodiment of the classical spirit of the High Renaissance

The goal of education in Athens, a democratic city-state, was to produce citizens trained in the arts of both peace and war.

In ancient Athens, the purpose of education was to produce citizens trained in the arts, to prepare citizens for both peace and war. Other than requiring two years of military training that began at age 18, the state left parents to educate their sons as they saw fit. The schools were private, but the tuition was low enough so that even the poorest citizens could afford to send their children for at least a few years. Until age 6 or 7, boys generally were taught at home by their mother.

Most Athenian girls had a primarily domestic education. The most highly educated women were the hetaerae, or courtesans, who attended special schools where they learned to be interesting companions for the men who could afford to maintain them.

Boys attended elementary school from the time they were about age 6 or 7 until they were 13 or 14. Part of their training was gymnastics. Younger boys learned to move gracefully, do calisthenics, and play ball and other games. The older boys learned running, jumping, boxing, wrestling, and discus and javelin throwing. The boys also learned to play the lyre and sing, to count, and to read and write. But it was literature that was at the heart of their schooling.

The national epic poems of the Greeks – Homer’s Odyssey and Iliad – were a vital part of the life of the Athenian people. As soon as their pupils could write, the teachers dictated passages from Homer for them to take down, memorize, and later act out. Teachers and pupils also discussed the feats of the Greek heroes described by Homer.

The education of mind, body, and aesthetic sense was, according to Plato, so that the boys. From age 6 to 14, they went to a neighborhood primary school or to a private school. Books were very expensive and rare, so subjects were read out-loud, and the boys had to memorize everything. To help them learn, they used writing tablets and rulers.

At 13 or 14, the formal education of the poorer boys probably ended and was followed by apprenticeship at a trade. The wealthier boys continued their education under the tutelage of philosopher-teachers.

Until about 390 BC there were no permanent schools and no formal courses for such higher education. Socrates, for example, wandered around Athens, stopping here or there to hold discussions with the people about all sorts of things pertaining to the conduct of man’s life. But gradually, as groups of students attached themselves to one teacher or another, permanent schools were established. It was in such schools that Plato, Socrates, and Aristotle taught.

The boys who attended these schools fell into more or less two groups.

Those who wanted learning for its own sake studied with philosophers like Plato who taught such subjects as geometry, astronomy, harmonics (the mathematical theory of music), and arithmetic.

Those who wanted training for public life studied with philosophers like Socrates who taught primarily oratory and rhetoric. In democratic Athens such training was appropriate and necessary because power rested with the men who had the ability to persuade their fellow senators to act.

http://www.crystalinks.com/greekculture.html