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.











October 29, 2009 at 9:42 pm Leave a comment

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.

October 20, 2009 at 6:59 am Leave a comment

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.


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

September 9, 2009 at 3:11 am Leave a comment


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.

September 6, 2009 at 8:58 am Leave a comment

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.

September 6, 2009 at 8:34 am Leave a comment

Works designed with the Golden Ratio


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.


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

In his 1919 book Ad Quadratum, Frederik Macody Lund, a historian who studied the geometry of several gothic structures, claims that the Cathedral of Chartres (begun in the 12th century), the Notre-Dame of Laon (1157–1205), and the Notre Dame de Paris (1160) are designed according to the golden ratio. Other scholars argue that until Pacioli’s 1509 publication (see next section), the golden ratio was unknown to artists and architects.

A 2003 conference on medieval architecture resulted in the book Ad Quadratum: The Application of Geometry to Medieval Architecture. According to a summary by one reviewer:

Most of the contributors consider that the setting out was done ad quadratum, using the sides of a square and its diagonal. This gave an incommensurate ratio of [square root of (2)] by striking a circular arc (which could easily be done with a rope rotating around a peg). Most also argued that setting out was done geometrically rather than arithmetically (with a measuring rod). Some considered that setting out also involved the use of equilateral or Pythagorean triangles, pentagons, and octagons. Two authors believe the Golden Section (or at least its approximation) was used, but its use in medieval times is not supported by most architectural historians.

The Baroque and the Spanish empire

Jose Villagran Garcia has claimed that the golden ratio is an important element in the design of the Mexico City Metropolitan Cathedral (circa 1667–1813). Carlos Chaflon Olmosclaims the same for the design of the cities of Coatepec (1579), Chicoaloapa (1579) and Huejutla (1580), as well as the Merida Cathedral, the Acolman Temple, Cristo Crucificado by Diego Velazquez (1639) and La Madona de Media Luna of Bartolomé Esteban Murillo.

Modern Architecture

Mies Van der Rohe

Mies created for Dr. Edith Farnsworth from 1945-51 a one room weekend retreat that is widely recognized as an iconic masterpiece of modernist architecture.

Mies created for Dr. Edith Farnsworth from 1945-51 a one room weekend retreat that is widely recognized as an iconic masterpiece of modernist architecture.

The Farnsworth House has been described as “the proportions, within the glass walls, approach 1:2” and “with a width to length ratio of 1:1.75 (nearly the golden section)”and has been studied with his other works in relation to the golden ratio.


Le Corbusier

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier’s faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as “rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned.”

Le Corbusier explicitly used the golden ratio in his 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. 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.[40]

In The Modulor: A Harmonious Measure to the Human Scale, Universally Applicable to Architecture and Mechanics Le Corbusier reveals he used his system in the Marseilles Unite D’Habitation (in the general plan and section, the front elevation, plan and section of the apartment, in the woodwork, the wall, the roof and some prefabricated furniture), a small office in 35 rue de Sèvres, a factory in Saint-Die and the United Nations Headquarters building in New York City. Many authors claim that the shape of the facade of the second is the result of three golden rectangles; however, each of the three rectangles that can actually be appreciated have different heights.

Villa Savoye, designed by Le Corbusier in 1929

Villa Savoye, designed by Le Corbusier in 1929

The blueprint for Villa Savoye

The blueprint for Villa Savoye

Post-modern architecture

Mario Botta

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.

In this house Botta designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.

In this house Botta designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.

Art History


Leonardo Da Vinci's illustration of a human head from De Divina Proportione

Leonardo Da Vinci's illustration of a human head from De Divina Proportione

De divina proportione, written by Luca Pacioli in Milan in 1496–1498, published in Venice in 1509,[20] features 60 drawings by Leonardo da Vinci, some of which illustrate the appearance of the golden ratio in geometric figures. Starting with part of the work of Leonardo Da Vinci, this architectural treatise was a major influence on generations of artists and architects.

Vitruvian Man, created by Leonardo da Vinci around the year 1492,is based on the theories of the man after which the drawing takes its name,Vitruvius, who in De Architectura: The Planning of Temples (c. I BC) pointed that the planning of temples depends on symmetry, which must be based on the perfect proportions of the human body. Some authors feel there is no actual evidence that Da Vinci used the golden ratio in Vitruvian Man;however, Chanfon(1991) observes otherwise through geometrical analysis. He also proposes Leonardo da Vinci’s self portrait, Michelangelo’s David(1501–1504), Albrecht Dürer’s Melencolia and the classic violin design by the Masters of Cremona, as having similar regulator lines related to the golden ratio.

Da Vinci’s Mona Lisa (c. 1503–1506) “has been the subject of so many volumes of contradicting scholarly and popular speculations that it virtually impossible to reach any unambiguous conclusions” with respect to the golden ratio, according to Livio.

The Tempietto chapel at the Monastery of Saint Peter in Montorio, Rome, built by Bramante, has relations to the golden ratio in its elevation and interior lines.


Matila Ghyka and others contend that Georges-Pierre Seurat used golden ratio proportions in paintings like La Parade, Le Pont de Courbevoie and Une Baignade, Asnières. However, there is no direct evidence to support these claims.


French mathematician, Henri Poincaré, taught the properties of the golden ratio to Juan Gris, who developed Cubism featuring them.

In 1924 Gris delivered a paper at the Sorbonne, Les Possibilités de la peinture (On the Possibilities of Painting), which was later translated and widely published. He died in Paris on May 11, 1927.

One of Gris’s most famous pronouncements was made in 1921:

“I consider that the architectural element is mathematics, the abstract side; I want to humanize it. Cézanne turns a bottle into a cylinder, but I begin with a cylinder and create an individual out of a special type:I make a bottle – a particular bottle – out of a cylinder.”

Section d’Or

Section d’Or was a Paris-based association of Cubist painters; the group was active from 1912 to about 1914.

The group’s name was suggested by the painter Jacques Villon, who had developed an interest in the significance of mathematical proportions such as the ancient concept of the golden section, the section d’or. The name thus reflects the Cubist artists’ concern with geometric forms, although Villon and Juan Gris were the only Cubists who directly applied such concepts to their work. The principal members of the group were Robert Delaunay, Marcel Duchamp, Raymond Duchamp-Villon, Albert Gleizes, Juan Gris, Roger de La Fresnaye, Fernand Léger, André Lhote, Louis Marcoussis, Jean Metzinger,Francis Picabia, and André Dunoyer de Segonzac.

In 1912 the group first exhibited together at the Galerie la Boétie in Paris, and it also published a short-lived magazine entitled Section d’Or.


The Sacrament of the Last Supper, 1955 by Salvador Dali

The Sacrament of the Last Supper, 1955 by Salvador Dali

The canvas of this surrealist masterpiece by Salvador Dali is a golden rectangle. A huge dodecahedron, with edges in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.

De Stijl

Composition in Red, Yellow and Blue, 1926 by Piet Mondrian

Composition in Red, Yellow and Blue, 1926 by Piet Mondrian

Some works in the Dutch artistic movement called De Stijl, or neoplasticism, exhibit golden ratio proportions. Piet Mondrian used the golden section extensively in his neoplasticist, geometrical paintings, created circa 1918–38. Mondrian sought proportion in his paintings by observation, knowledge and intuition, rather than geometrical or mathematical methods.


Neoclassicism and romanticism

Leonid Sabaneev hypothesizes that the separate time intervals of the musical pieces connected by the “culmination event”, as a rule, are in the ratio of the golden section. However the author attributes this incidence to the instinct of the mucisians: “All such events are timed by author’s instinct to such points of the whole length that they divide temporary durations into separate parts being in the ratio of the golden section.”

In Surrey’s Internet site, Ron Knott exposes how the golden ratio is unintentionally present in several pieces of classical music:

An article of American Scientist (Did Mozart use the Golden mean?, March/April 1996), reports that John Putz found that there was considerable deviation from ratio section division in many of Mozart’s sonatas and claimed that any proximity to this number can be explained by constraints of the sonata form itself.

Derek Haylock claims that the opening motif of Ludwig van Beethoven’s Symphony No. 5 in C minor, Op. 67 (c. 1804–08), occurs exactly at the golden mean point 0.618 in bar 372 of 601 and again at bar 228 which is the other golden section point (0.618034 from the end of the piece) but he has to use 601 bars to get these figures. This he does by ignoring the final 20 bars that occur after the final appearance of the motif and also ignoring bar 387.

Contemporary music

James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.

Ernő Lendvai analyzes Béla Bartók’s works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. In Bartok’s Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry.

The golden ratio is also apparent in the organisation of the sections in the music of Debussy’s Image, Reflections in Water, in which “the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position.”

The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the golden section. Trezise finds the intrinsic evidence “remarkable,” but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.

This Binary Universe, an experimental album by Brian Transeau (aka BT), includes a track entitled “1.618” in homage to the golden ratio. The track features musical versions of the ratio and the accompanying video displays various animated versions of the golden mean.

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for apatent on this innovation.

According to author Leon Harkleroad, “Some of the most misguided attempts to link music and mathematics have involved Fibonacci numbers and the related golden ratio.”

August 11, 2009 at 1:37 am 2 comments

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


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.


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


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


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.


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 …”


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.


“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 AristotelianThomist 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.


August 11, 2009 at 12:34 am Leave a comment

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