Showing posts with label geometry. Show all posts
Showing posts with label geometry. Show all posts

Sunday, December 11, 2011

The Ionic Order


The Ionic order originated, unsurprisingly, in seafaring Ionia in the early 6th century BC. Culturally, the Ionians were thoroughly Greek and quite naturally they spoke Ionian, a Greek dialect.


Ionia itself was a small but economically and culturally powerful Greek province, actually a ridiculously small coastal enclave (no more than 90 x 55 miles in extent, located near Smyrna in present-day Anatolia, Turkey) that also encompassed the islands of Samos and Chios. Its major city, Miletus, was an important commercial center and Phocaea was a great port. Both cities spawned colonies, spreading Ionian influence throughout the eastern Mediterranean, and with Samos were the backbone of Ionian power and influence.

This loose confederation for a time banded together to form the Ionian League, which was an early and great center of Greek civilization. Its legacies are staggeringly outsized: the foundations of Greek philosophy, geometry and mathematics with the Ionian school of the eminent Thales and his followers Anaximander, Anaximenes, Heraclitus, Anaxagoras, Archelaus, and Diogenes of Apollonia; the mystery school founded by Pythagoras of Samos, the great geometer and philosopher; and generations of brilliant artists and architects who deeply influenced the development of Hellenic art. In fact, if Western civilization was born in ancient Greece, then Greek civilization can be said to have been born in Ionia.



This Ionian cultural and intellectual explosion ignited in the 6th century BC, at the birth of the Ionic order, and Ionic temples began to appear on the Greek mainland in the following century. According to Vitruvius, the architects Rhoikos and Theodorus of Samos built the first of the great Ionic temples at Samos, dedicated to Hera, circa 580-560 BCE (above, its floor plan and a surviving capital). Though it stood but a decade before being leveled by an earthquake, the temple of Hera at Samos was a remarkably ambitious undertaking, famous throughout the Greek world. Quite simply, it was the first great Greek temple, its footprint large as a soccer field, rivaling in scale and architectural ambition the temples of Egypt. (Below, the temple of Artemis at Ephesius, comparable in period and scale.)


Here, at the very birth of the Ionic order, we are confronted with a truly monumental construction that forces us to rethink our notions of the scale of the Greek temple. To give some sense of this temple's massive size, the Parthenon's stylobate or plinth measures approximately 70 x 31 meters, nearly a third smaller. As Nancy Mitford would say, the temple of Hera at Samos put Greece—or more truthfully, tiny Ionia—on the map.

The Ionic Column

Ionic column shafts, more slender than the Doric, usually stand eight to nine column diameters tall and may be fluted or smooth. When fluted, they traditionally carry 24 flutes, as opposed to 20 for the Doric. The flutes are slightly separated, leaving a thin strip of unfluted column between them known as a fillet, as opposed to the Doric, where the flutes abut at an acute angle. Finally, Ionic columns have a ringed base and square pad that raises them off the stylobate, or temple plinth, an element the Doric lacks completely.


Its capital is far and away the Ionic order's most remarkable feature, and the capital's most important characteristic is its bi-fold symmetry, or directional orientation, in contrast to the Doric's radial unity. That the Ionic capital has clearly defined faces and sides is a crucial observation to keep in mind as we go digging through the byways of ancient civilizations looking for its predecessors. If it is a truism that "the lie is different at every level," then it also holds that the truth is also different at every level, and builds accretively, and the Ionic order presents us with a multitude of precedents and influences.

The Ionic capital's volutes, also popularly called "eyes," have led to the painfully simplistic speculation that the Ionic column can be interpreted anthropomorphically, with the fluted shaft depicting a woman's toga-clad body and the capital her head (and, one supposes, the volutes must then depict Princess Leia's hair).


Others have proposed that the spiraling volutes depict rams' horns or nautilus shells, and here we are moving much closer to the truth. The principle underlying both those physical forms is the Fibonacci sequence, a simple arithmetic progression that regulates and balances natural growth, including those rams' horns and nautilus shells and all matter of things from artichokes to tree branches, pine cones, fern heads and so on. You'll also find it topping the neck of classical string instruments such as violins, violas and cellos—the element called, appropriately enough, the scroll.

Fibonacci, otherwise known as Leonardo of Pisa, was the first to publish this arithmetical progression (0; 0+1=1; 1+1=2; 1+2=3; 2+3=5; 3+5=8; 5+8=13...) in 1202, gaining him lasting fame. But alas, like just about every other bit of knowledge of this nature, he was simply publicly disseminating elements of the ancient occult knowledge of the Egyptian mystery schools for the first time. In truth, this formula was part of the Sacred Geometry of ancient Egypt and was also known to the ancient Vedic civilization as well.

Out of Egypt (again)

And how did the occult knowledge of the ancient mysteries, the precious high knowledge of the Egyptian priestly class, escape Egypt to become known to Greece and then to the West? Through those wily, intrepid seafarers, the ancient Greeks, of course. A number of very old, very famous Greeks—among them Thales, Plato and Pythagoras—made quite some names for themselves after traveling to Egypt to become initiates of the mysteries.

These renowned sages were hardly solitary pilgrims. As we mentioned in our earlier post on the Doric, Greeks and Egyptians were carrying on a robust economic and cultural trade in the Archaic period and the Ionians were at its forefront; when the first Ionic temples were being built, Ionia was in the midst of an Egyptian trade boom.


According to Pliny, the very form of the great temple of Hera at Samos, a grid of 8 x 21 columns covering roughly 50 x 100 meters, evokes the Egyptian Labyrinth at Hawara, a vast mortuary complex of twelve courtyards (and according to Heroditus, who had visited) over 3000 rooms built for Pharaoh Amenemhat III, the last great king of the 12th dynasty. The Labyrinth (above, a recent computer-generated reconstruction) was one of the wonders of the Ancient world and far more famous in antiquity than the Great Pyramid. Tragically, the Romans used Hawara as a quarry and with customary thoroughness so completely effaced the complex that, even after major excavations, reconstructions of the Labyrinth are still based almost entirely on ancient descriptions. Nonetheless, Pliny specifically mentions the temple of Hera at Samos, with its dense grid of columns, as one of the world's great labyrinths, comparable to the Egyptian Labyrinth, famed for being so bewildering that one had to visit with a ball of string or a native guide. (Below, a view of the pronaos of the temple of Artemis at Ephesius.)


When we seek out the earliest recognizable Archaic precursors to Ionic columns, we first find ourselves on the isles of Lesbos and Troas, birthplace of the Aeolic capital, composed of two robust volutes bracketing a palmette. The Lesvs were great poets and pre-eminent seafarers and colonized the coast of Asia Minor (Anatolia, or contemporary Turkey), and the Aeolian city of Smyrna was admitted to the Ionian league circa 650 BC, unsurprisingly bringing us full circle back to Ionia's doorstep. (Below, an Aeolian capital from Neandria in Troas, an ancient city on the Turkish coast not far from Ionia).


Obviously, this abstracted floral motif is an adaptation of Egyptian lotus and papyrus capitals, and indeed one of the earliest recognizably Ionic capitals, which rotates the Aeolian volutes to link them horizontally, creating a pad, has been found in the Greek enclave of Naukratis in Egypt (below).



Assyrian Roots

Formally, the Ionic capital's directionality indicates that its earliest precursors were cap blocks meant to span and support beams and lintels, a construction technique most elaborated in Assyrian architecture. Egypt had fallen under control of the Assyrian empire in Archaic times, and a simple glance at early Assyrian capitals and particularly those used at Persepolis (both below), indicates that the ultimate inspiration for the Ionic springs from Assyria. (Though Persepolis was begun a century after the appearance of the Ionic, its architecture exhibits an extremely high level of refinement, indicating a long prior tradition.)



In fact direct proof of influence and exchange can be found in the remnants of the temple of Hera at Samos, where one finds sculpted stones bearing much the same doubled volutes as at Persepolis (below).


Clearly, the Assyrian capital holds a welter of meanings—those at Persepolis have three tiers of symbols: bulls, volutes and lotuses, like a triple-scoop ice cream cone. Other Assyrian precursors depict flowers, humans and rolled papyrus or parchment scrolls (below).


The Ionic abstracts and conflates all these symbols, and this was its genius. The horned bull of Taurus of the Assyrians; the Egyptian lotus; the papyrus scroll, symbol of human intellect; the Fibonacci sequence, sacred geometry encoding nature's growth—all these meanings come together in the volutes of the Ionic capital—a great fusion of ancient knowledge and a symbol above all of the glory of ancient Ionia.

Next: the Corinthian order

Wednesday, November 30, 2011

The Doric Order


The Doric order is the earliest of the classical orders developed by Archaic Greek civilization and was by far the most popular of the three. The eponymous Dorians were the dominant tribe of the four main peoples forming ancient Greece and the Dorian dialect was spoken in a great southward arc stretching across the Aegean from Corfu to the lower Peloponnesian Peninsula and on to the islands of Crete and Rhodes. (Above, the Hephaisteion or Theseion, a remarkably preserved Doric temple located on the north-west side of the Agora of Athens.)


Already well-established in the 7th century BC, the Doric order reached its apotheosis with the stunning achievement of the Parthenon in 438 BC but eventually fell from favor by the end of the 2nd century BC. It would spawn both the Roman Doric, an embellished version with lighter proportions and the addition of the Ionic column base, and much later the highly simplified Tuscan order, developed in 16th century Italy by Serlio and Palladio and employed principally for rural architecture, as embodied by Palladio's villas.

In Di Architectura, Vitruvius, a Roman architect who practiced during the reign of Augustus Caesar, remarked that the Doric was masculine in character and wrote that its fundamental proportion, a column shaft six times its diameter, deliberately mirrored "the proportions, strength and beauty of a man's body." (The length of actual shafts varied between 4½ and 7 column diameters, with the shaft almost uniformly bearing 20 flutes.) He also noted that the Doric was suitable for temples dedicated to such masculine gods as Hercules and Mars, while the Ionic and Corinthian were more feminine.


Though in truth the Romans used the Corinthian order for just about everything, Vitruvius's comments reflect the Doric order's thicker, squatter proportions, its traditional lack of naturalistic or floral ornament and its underlying static, rectilinear æsthetic logic. There is no point in cataloguing Doric elements here for the umpteenth time, rather we will examine the Doric column and its all-important capital and attempt to discern greater meaning than the ancient tidbit Vitruvius has tossed us—and also something beyond the obvious modern observation that many of its rectilinear elements (triglyphs, abacus, mutules, and so on) almost certainly are inspired from earlier timber-frame construction techniques, translated into stone decoration.


The first things to remark about the Doric capital (above and below, examples from the Parthenon) are its remarkable simplicity and unity. In contrast to the elegant, complex bifold geometry of the Ionic and the florid outburst that is the Corinthian, the Doric capital is composed of two visually balanced elemental elements: a thick, squared slab called the abacus and a flaring, circular pillow beneath named the echinus. Usually, but not always, three concentric fillets transition the echinus to the shaft, known as annulets.


Even more so than the Ionic order, the origin of the Doric is too diffuse to pinpoint, but the Archaic Greek impetus to erect monumental dressed stone temples to their gods obviously sprang from the example of the sacred architecture of ancient Egypt, a civilization then already in terminal decline. In the late Archaic period the Greeks and Egypt were carrying out extensive trade and by the 7th century BC Greek neighborhoods and trading centers had become established in Egypt's most important cities.


The general influence of Egypt is clear, as is the direct precedent of the colonnade at Saqqara (above). We find fluting, inverted from the bowed ribs at Saqqara , and the same æsthetic/geometric/volumetric rigor, elegance and abstraction. What is so fascinating with the Doric is exactly this deliberate abstraction, this remarkable renunciation of naturalistic ornament—exactly as we find at Saqqara. In fact, the Doric appears deliberately conceived to embody austere geometry and clear, rectilinear volumetrics.


A circular echinus supporting a square abacus. (Above, a capital at the Temple of Zeus at Olympia.) The circle and the square: Heaven and Earth. The ancient Egyptian principle of "as above, so below" has been purified and abstracted and, I believe, a unity of opposites is being expressed. The Egyptian duality is transformed into a single, fusional idea, most clearly palpable in the overarching æsthetic sense—this volumetric, geometric, abstract rigor I keep referring to—that is the glue that bonds these constituent ideograms together: the concept of consciousness itself. Man, the abstract thinker.


It is no coincidence that the Doric temple makes its appearance in the midst of the intellectual ferment that also sees philosophy's tandem birth. In a nutshell, the Doric order expresses, quite self-consciously and deliberately, the celebration of man's conscious rationality, the blossoming of Greek thought. In fact a parallel to the first recorded Western cosmology, that of Anaximander of Miletus (an Ionian, about which we'll have more to say in our next post on the Ionic), can and indeed has been drawn, but I'm not in agreement with Robert Hahn that Anaximander's vision of the earth as a thick, cylindrical wafer suspended in space finds a literal equivalent in the actual cylindrical stone blocks, or "drums" that make up a Greek column—first of all, because they are construction components and not the column itself. This is like some future archeo-anthropologist concluding that skeletons from our era exhumed with polyester clothing were doubtless acolytes of string theory, because their garments are composed of a complex interweaving of imperishable threads. Columns were conceived to be—and were preferably executed as—monoliths; assembling them from stacked drums was an expedient, a quite-literal "short-cut" never meant to bring attention to itself, let alone be celebrated as a metaphor for the divine order of creation!

But Hahn's thesis isn't totally wrong, though misplaced (and mostly irrelevant as Anaximander was born too late to have any decisive influence on the development of either the Doric or Ionic orders): Anaximander's cosmology is congruent with the column's symbolism, as both share the idea of a centered infinity and, just as importantly, an axiality that can be linked to the cosmic axis of earth and the zodiac. Vitruvius is much closer to the nub of things in symbolically equating the Doric column with man, and thus the capital with man's head, the locus of consciousness. The columns (humanity) support the roof (heaven) which shelters the sanctuary (the abode of the gods). This is the fundamental cosmology being expressed in any Greco-Roman temple. Not by accident is the word pediment, denoting the triangularly shaped wall found between the cornice and the sloping roof ends of a Greek temple, a workman's corruption of the word pyramid.


Vitruvius was certainly correct, the Doric order is the measure of man, but not in the literal sense: What is being measured is not man's body but his mind.

Coming soon: the Ionic order.

Tuesday, September 20, 2011

The Measures of Man


Having lived well over a decade in the country that founded the metric system, or the SI (Système internationale d'unités), I do now think in terms of kilograms and litres and metres, but I can't say that I find the system more useful. Logical, Cartesian, yes obviously, but practical and intuitive, even elegant? Certainly not.

US Customary and English Imperial measures—the pound, the gallon, the yard—grew organically from usage and were refined with practicality foremost in mind, and their hallmark is units that are intuitively proportional: 16 ounces to the pound, four quarts to the gallon, twelve inches to the foot and three feet to the yard. The decimal-based metric system replaces all this with measures of technocratic precision, mathematical logic and an appeal to universality.

The English-speaking world has in part resisted this appeal to universality and often countries have retained their traditional weights and measures in parallel with their metric equivalents. Though Britain itself began conversion in 1995, certain defining measures remain unchanged: by law, beer can only be sold in pints and road signs can only record distance in miles. The metric system remains largely foreign in everyday use in the US, and as one who is constantly measuring and designing, I believe this to be a good thing, for the traditional foot is an extraordinarily powerful design tool, one whose innate proportions are unconsciously absorbed by native-born designers, enriching and supporting the process of creation.

This first struck me while designing at Robert A.M. Stern Architects. Residential interiors were always detailed with custom millwork, and I spent a fair amount of time drawing various moldings and profiles, sketching them at full scale. When it came time to hard-line them, a metric ruler offered little utility when spec'ing dimensions—6 millimetres and a bit; something that looks, after careful examination, to be 17 millimetres and a tad more—one quickly finds oneself adrift in datapoints of imprecise precision. But when I measured with a foot ruler, I was always struck by how effortlessly each reveal and return corresponded to the proportional divisions of the inch—1/4 inch, 3/8 inch, 1/2 inch, 1/8 inch, 3/4 inch, etc.—and unerringly so. (Below: just a commonplace millwork plan found on the net illustrating the inch at its effortless best.)


The same intuitively occurs when designing just about anything of human scale for human use, and really this is quite obvious to anyone using the inch/foot—it is a supremely elegant proportional system as much as it is a unit of measure, and therein lies its beauty, its utility and its genius.

Our forebears (going all the way back to the ancients themselves, whose geometry was proportionally based) all knew this, and when Renaissance architects began to publish elevational drawings of ancient architectural elements, their measures were usually given in fractions/proportions, not in the units of any particular system of measurement. Indeed, the standard unit of measurement of the columnar orders is a column's diameter at its base; the "standard" shaft of a Doric column is 8 diameters high; the Ionic, nine, and so on. (At top, Palladio's elevation of the Doric order, from his Four Books.)


In the Age of Reason, French surveyors made a grandiose attempt to accurately survey a meridian arc so as to precisely calculate the new metre (defined technically as one-ten millionth of the Earth's meridian along a quadrant and poetically by Condorcet as "for all people for all time") which was to replace the venerable toise (defined by edict of Charlemagne in 790 as the distance between the fingertips of a man's outstretched arms, and in practice roughly equal to the English fathom, or six feet). The surveyors ran into some difficult terrain, causing them to fudge mightily, and so the new metre came up short by 1/5 of a millimetre per metre when all calculations were tallied and decimal points placed. The metre had already become well established when this came to light and consequently the error was never corrected, and so it is that today the earth's circumference measures 40,007,863 metres, not 40,000,000.

So much for technocratic precision and mathematical logic.


In practice, designing with the metric system is often tedious and unsatisfying. A chore, frankly, like anything arbitrary and imposed. Good design is underpinned by rhythms, convergences, and harmonies, and the simple decimal divisions of a metric ruler demand that those elements be calculated and plotted out quite laboriously, ex nihilo. Imagine scrapping musical notation for herz impulses per second and you have a rather good approximation of the practicality of the metre as a compositional or design tool.

Which reminds me of a certain well-known and quite imperious German architect who was obsessed by the square and the grid and who made van der Rohe look like a free spirit (well this is not so unusual; German designers are often seduced by the square and the grid, just as many French designers are enamored by the poetry of the arc in tension, as a quick flip though any German or French design magazine will illustrate). Anyway, in his last years this architect received a prominent commission for a museum of contemporary art and based his entire design upon the square. Square windows, cubic rooms, grids everywhere—an orgy of order!

His obsession went so far that he designed square-sectioned stairs—rise over run equal, making a perfectly diagonal flight. Perfect order, and perfectly disastrous. He ignored all protestations of impracticality and even danger—for as (almost) any architect knows, you don't go about reimagining the proportions of stairs every Monday morning on a whim; there are well-defined standards that are employed unquestioningly because they have been optimized for the body's movement and refined from vast experience. Long story short, the building opened and the stairs were swiftly condemned before someone was actually killed trying to use them. It is hard to conceive such folly occurring in a mind conditioned by fractions and proportions, rather than the arbitrary tyranny of the decimal point.

Tuesday, July 26, 2011

Architectural Symbolism 101: Geometry


Classical architecture has a rich and intricate symbolic repertoire reaching back to earliest recorded history that today can be compared to ancient Latin: a language once in common currency but today understood only by a few adepts. Well into the 19th century, the educated viewer could read a building as one reads a book, but today the language of classicism is largely mute to us, much of its meaning lost and eroded by time and the relentless evolution of human societies.

The first step in deciphering the meaning of the built world is to understand a structure's geometry—both its two-dimensional plan and in three dimensions. The origins of geometry—literally, "the measure of the earth"—are as obscure as the origins of civilization, and much that was "discovered" by the likes of Pythagoras was actually obtained from the priestly caste of ancient Egypt—their own knowledge so lost in the mists of time that it was attributed to Toth, god of language and knowledge—and was simply openly disseminated by the Greeks for the first time.


Pi, for example, often attributed to Archimedes, is clearly encoded in the measures of the Great Pyramid of Giza (and was also known in ancient China, the Indus Valley and in Sumer). Likewise, the symbolic meaning of geometry and number can be traced through the Greeks and the other ancient civilizations of the Mediterranean basin to Egypt and Sumer, and when we continue farther in time we encounter the evidence of monolithic civilizations destroyed by the last ice age, about which so much has been projected and too little known. The point here is to identify the origin of the symbolic meaning of geometric figures: Egypt, transmitted to us via the Greeks and their neighbors.

We will use a very simple example to illustrate geometry's symbolic power: the Bosquet of the Three Fountains in the gardens of Versailles (depicted in the watercolor reproduced at the top of the post). This elaborate garden-within-a-garden was built in the early 1700s by order of Louis XIV, and tradition holds that the king acted as his own architect and directed the bosquet's design.

The bird's-eye-view watercolor above was commissioned by the Société des Amis de Versailles to aid in fund-raising efforts to rebuild the bosquet. As you can see, the garden is laid out on three levels: each parterre with its central fountain is linked by grass steps, ramps and low cascades to the level below. Like the other baroque bosquets in the park of Versailles designed for Louis XIV, the Three Fountains is rigidly geometric and features elaborate water displays.

Though difficult to see in this small reproduction, the highest, farthest fountain has a circular basin; the middle basin is square and the lowest is octagonal. And here we have the crux of the bosquet's symbolic meaning: the circle (and its three-dimensional counterpart the sphere) represents the arcing vault of the heavens.


The square represents the earth, literally its four "corners," or cardinal directions (as well as the four known continents of the Renaissance age: Europe, Asia, Africa and America).


Finally, the octagon is the symbol of kingship, standing halfway between earth and Heaven, the square and the circle—a perfect geometric form that perfectly incarnates the French conception of the sovereign as the essential mediator standing midway between God and the people.


The traditional method of constructing an octagon begins with a square, upon which one inscribes the arc of a circle. Constructing an octagon also generates an infinitely regressing triangle, further adding to the figure's symbolic power (in fact, Louis XIV became linked to Descartes' idea of a centered infinity—with himself as the central point from which infinity was referenced, of course).

You will also notice that Louis XIV did not place the octagon between the circle and the square, as one would expect, but rather he employed it as the summation of a progression, or an equation: Heaven (circle) and earth (square) give rise to the king (octagon). And here we have a simple but profound insight into the mind of the Sun King: unsurprisingly, he considered himself and his position as the summation of the union of Heaven and earth, rather than as the mediator between them. No one ever said Louis XIV was afflicted by self-doubt.


Finally, what we have here is a perfect symbolic expression of absolutism—no surprise really, as the bosquet was conceived by the man who literally defined the age. France, in the Age of Louis XIV, superceded Italy to claim first place among the powers of Europe in all spheres, including for the first time, culturally. Though it used the art and architecture of Italy as its template, France constructed its cultural hegemony upon the foundations of absolutism, not humanism, and Leonardo's humanist vision of man as the center and measure of all things was replaced by the idea of a single man—a king.