古典建築是如何始於古希臘的？

Hello and welcome to the first of four presentations on the history and development of classical architecture. No study of architecture is complete without knowledge of the buildings of Greek and Roman antiquity. In this first segment, we will familiarize ourselves with the expressions and approaches of these two cultures, which together laid the foundations of Western classical architecture. In subsequent sessions, we will continue with the discussion of Italian Renaissance and Baroque architecture, followed by the 18th and 19th centuries, and conclude with a look at contemporary classicism. Before we begin, it is worth noting that the term classical architecture is an essentially modern one, thought to have first been used in the late 16th century. The Greeks referred to ordinary construction, employed in the making of houses, sheds, and other private or perfunctory structures, simply as tektoniki, or building. Construction of sacred and public edifices, such as temples, council houses, market colonies, theaters, etc., was called architektoniki, meaning principled or refined building. Today, the word architecture is applied to all buildings, and for that reason, the term classical is necessary to distinguish the more formal edifices. This map shows the extent of the ancient Greek world, which stretched westward along the shores of the Mediterranean, and eastward to include parts of North Africa and the Black Sea. After the death of Alexander the Great, Greek territories extended as far east as the Indus River and Afghanistan. The original Greek homeland on either side of the Aegean Sea is indicated in the boxed area. This more detailed map of the Greek homeland shows the approximate concentration of the principal Greek tribes. Green indicates Dorian areas, yellow indicates the Ionian territories, and cream shows the Aeolian region. The ancient Greeks were prolific writers. We know from the Roman author Vitruvius that a number of treatises on architectural subjects had been written and were kept in libraries, including the famous library of Alexandria in Egypt. Unfortunately, no original texts have survived. Many works are thought to have perished in a conflagration or a series of conflagrations that destroyed the library. Much can be deduced, however, about the development of Greek architecture from the buildings that survive. The origin of the Western classical tradition can be traced back to the Greek peripteral temple, which comprised an inner walled sanctum and surrounding colonnades. The Greek term for the inner temple was naos, and a flanking row of columns was called the pteron. Both of these words have nautical associations, as naos also means ship, and pteron referred to a bank of oars. Strictly speaking, pteron means wing, as in a bird's wing, and it may have been adopted on account of the rhythmical motions of the oars of a trireme, or galley, which resembled a bird in flight. The reason for these nautical analogies in Greek architecture may be that the ancients understood the temple as a vessel or ship for the transport of a deity to the earthly realm. Both words survive in terms used for the architecture of Christian churches, the principal space being the nave, from the Latin word navis, which itself derives from the Greek naos, and the flanking colonnades being the aisles from the Latin ali, meaning wings. From an archaeological point of view, the Greek temple's forms can be traced back to the Megaron Hall of the great Bronze Age, or Mycenaean palaces, an example of which is shown in the plan on the left. In these large directional spaces with their round hearths and pillared porches, the kings who led the Greek expedition against Troy held court and performed sacred rituals. After the collapse of this early Greek culture in the 12th century B.C., a Dark Age ensued, which according to tradition was brought on by an invasion of people from the northern reaches of the Greek world known as the Dorians. During this period of collapse and hardship, the first glimmerings of the Greek democratic state, or polis,began to appear, and small settlements grew around temples that closely resembled the layouts of the old Mycenaean palatial halls. Eventually these linear buildings were surrounded by colonnaded wings and assumed their familiar peripteral configuration. This map of the Mycenaean world, that is to say the Greece of the 13th century B.C., which is recounted in the Homeric epics written some five centuries later, shows the principal palatial centers of Mycenae, Pelos, Sparta, and Troy, which is indicated at top right on the eastern shores of the Aegean Sea. All Mycenaean urban centers were clustered around palaces such as the one at Pelos, which is here shown in plan, and featured a large megaron hall surrounded by royal living quarters. The typical Mycenaean megaron resembled the later Greek temple, but it was distinguished by a throne room with a large round hearth and four columns in an atrium-like configuration. The architecture and details, however, had little in common with later Greek forms. From these palatial centers of the 13th century B.C., the Greek expeditionary forces set out to subdue Troy, a story made famous by the poet Homer, who lived in the 8th century B.C., half a millennium after the event. Our image of the Trojan War is influenced by much later classical Greek representations so that we tend to picture the Mycenaeans as classical Greeks. In truth, the Mycenaeans were very different. They spoke an early form of Greek, dressed differently, and used different forms of armor in combat. For instance, full-body bronze suits, helmets made of boar's tusks, and cowhide shields. After the destruction of Troy around 1250 B.C., this early Greek culture collapsed and a dark age ensued, which lasted from the 12th to the 8th century B.C. The Greek mainland was overrun by a new wave of Greek-speaking people, the Dorians. New settlements were established in the ruins of the old Mycenaean palatial centers. The new towns appeared to have been ruled by chieftains, whose huts were more primitive than the old megaron halls, but they retained a similar linear, open-fronted configuration. It is thought that the chieftains performed priestly duties, and gradually these megaroid structures were transformed into proper temples. Out of these early primitive megaroid structures of the Greek Iron Age, the peripteral temple emerged in the 6th century B.C., and it came to be surrounded by colonnades, or ptera. The principal type took root across the Greek world in all regions, regardless of tribal affiliation. We will here look closely at the nature and development of Greek Doric forms, from their emergence and connection with the first mud-brick temples with wooden superstructures, to their translation into refined stone edifices. While we are familiar with the forms of the Greek temple through the surviving stone-built examples, it is clear that the forms making up the superstructure of the buildings are references to original constructions that made use of timber and other perishable materials. The carpentry that is conjured in Doric buildings is clear in the rows of triglyphs, which represent beam ends, and the metopes, or spaces between them. The art of the carpenter, or tecton, is also apparent in the small pegs, known as gutae, set beneath the triglyphs, and in the rows of mutuals, or rafters, that support the projecting cornice. A recollection of the art of the ancient Greek carpenter is preserved in the present-day term tectonics. This was an important aspect of Greek architecture, for although by the 6th century B.C. temples were made mostly of stone, it was important to represent the original wooden forms and connections as reconstructed in the 18th-century drawing on the left of the slide. As wood is tactile, has familiar tensile strength, and can be easily cut, hewn,and carved, unlike stone, which is brittle, the practical forms produced by a carpenter are more capable of representing the essence of construction. For this reason, Greek masons chose to preserve in their stone temples references to carpentry. Today, when we say that a building is tectonic, we mean more generally that its forms evoke their origin in straightforward constructional methods and materials. An aspect of Greek architecture that is less familiar today is the polychromy of its roof structure, which is to say the entablature and crowning pediment. In Doric temples, the triglyphs, regulae, and mutuals were always painted a bright blue, and the taenia, corona, and other mostly horizontal members were painted a vibrant red. Traces of color are sometimes preserved on ancient buildings. The images on the left and below are from recently excavated tombs in the northern Greek region of Macedonia. The photograph on the right shows a full-scale replica of part of an early Doric building produced by a team of University of Notre Dame students working under my direction. The blue-painted details employ a copper pigment that is often used in agriculture as a fungicide and is thought to have been passed down since antiquity. On the left are examples of blue-copper sulfate being used protectively on tree limbs, and blue-copper silicate paint being applied to beemends with exposed end grain. It is my belief that the formulaic rendering of triglyphs, regulae, and mutuals as bright blue components by the ancient Greeks can be explained as an evocation of the protective fungicidal treatment of vulnerable wooden members on temples before they were translated into stone. Like the blue-painted members, some components of the Doric roof structure were invariably painted red. In some cases they were also given a characteristic molded profile of the sort often associated with copper flashing. These components tend to be horizontal and are situated along edges where one would normally expect flashing to occur. In its original state, copper has a reddish hue. When exposed to the air, it oxidizes and turns green. This transformation is illustrated in the photographs of present-day roof flashing examples on the left. The green patina tends to streak down the sides of buildings staining the walls. A traditional remedy for this problem is to paint the flashing with an anti-oxidizing lead paint, which has a natural bright red color. On the lower left is a photograph of a modern chimney breast with flashing that has been treated with red lead paint. In ancient Greece, ship's hulls are known to have been coated with red lead paint as an anti-fouling measure. In my essay The Origins of the Doric Order published in the Classicist issue number 12, I advance the idea of Greek polychromy as originating in protective treatments of building materials. But let us return to our study of Greek architectural forms. This slide shows how the essential components of the peripteral temple are broken down in elevational terms to include a plinth or crepus, sometimes also called a crepedoma, the column or stilos, the entablature and the pediment. The crepus almost always consists of three steps. The topmost step is called a stylobate, meaning column tread, and the lower two steps together are the stereobate, meaning solid tread or foundation. The Greek Doric column has no base. It always includes flutes, which are the vertical concave striations in the round shaft, and it concludes with a capital, which will soon be described in detail. The entablature, a term that refers to the table-like system of horizontal roofing members, comprises an architrave or principal beam, the frieze of triglyphs and metopes, and the corona. Above the corona rises the gabled roof, which on the two short ends of a temple translates into pediments. The forms of the pediment include the triangular tympanum, which normally contains sculptures, the raking cornices and the raking sima. Note that the sima, or gutter, never extends across the bottom of the pediment. At the corner and apex of the pediment are acroteria, which literally translated mean things at the corners or acra. Usually these are floral motifs or zoomorphic features, often apotropaic, meaning representations of threatening mythological beasts like griffins, sphinxes, gorgons, etc., whose purpose was to ward off evil. The Greek Doric capital consists of a number of distinctive components. At the top there is a square block called an abacus. Beneath it is a round cushion-shaped component called an echinus, or urchin-like thing on account of its parabolic profile, which resembles that of a sea urchin. At the point of transition of the echinus to the neck, or traheleon, there are a series of shallow bands called annulates. A short distance beneath these and cut into the top of the shaft are a number of small furrows called the hypotraheleon grooves. These are sometimes shown in vase paintings as securing the ends of garlands strung between the columns on festival days. In this slide, the various components and terms of the Greek Doric entablature are indicated with original Greek terminology in italics. The column supports a massive principal beam, or architrahe. This carries the frieze with its rows of triglyphs and metopes, which rest on a sill called the tenia. Beneath the triglyphs are small blocks called regulae, which feature six small pegs, or nail heads, known as gutae. Above the triglyphs are the slightly sloped mutuals, or rafters, which have three rows of six gutae. On the right-hand side of the slide is an image representing the parts of the pediment, or gable. This feature is an important aspect of Greek architecture and Western classicism in general. There is the cornice, or corona, which means crowning element. The horizontal corona wraps the entire temple, but at its two short ends it is surmounted by raking coronas, which carry a raking sima. This last feature represents the original gutter. The raking corona rises from a small fillet that crowns the horizontal corona in such a way that its own fillet merges into it. This distinctive juncture is known as a split fillet and is important to draw properly when setting out a pediment. Note that the metopes often contain relief sculptures. At the Parthenon in Athens, the metopes contain representations of the battle between lapiths, a human tribe, and centaurs, mythological creatures that were part human and part horse. Where the pediment turns the corner, the sima usually concludes, giving way to rows of antifixae, which are set at the ends of the roof cover tiles. Note the split fillet where the corona is met by the raking corona. Also note the incorporation of relief sculpture in the square metopes. Despite the strict tectonic logic of its parts, Doric architecture is not without formal problems and contradictions. One such issue which arises from the translation of the original slender wooden components into robust stone simulacra is the so-called problem of the corner triglyph. It is a curious fact in Doric peristyle temples that the spacing of the columns at the corners is narrower than the spacing elsewhere. This is sometimes erroneously explained as a visual adjustment that compensates for the greater apparent width of the corner spaces when seen against the bright Mediterranean sky. Another argument is that it creates a useful impression of strength at the corners of temples, which are the most structurally vulnerable. The likeliest explanation, however, has to do with the coordination of the components of the frieze with the massive columns below. It is a rule of construction that a beam should rest directly on top of a post or column. This is exactly what happens in peristyle colonnades. Another rule is that the metopes, which are the spaces between triglyphs, should be perfectly square in elevation. In the drawing at the top right, columns and triglyphs align neatly, and the outermost column sits comfortably beneath the corner triglyph. This is made possible by the fact that the neck diameter of the column is the same as the width of the triglyphs, as well as the architrave on which the triglyph rests. In the middle image, the columns have been greatly thickened after being translated from wood into stone. Here, the spaces between the columns are shown as constant. This means, however, that the last metope before the corner triglyph would need to be widened, disrupting the regularity of the frieze. In the lower images, the problem has been corrected by keeping a regular frieze in which all metopes retain a square proportion, and by sliding the corner column inward. So much, then, for the Greek Doric type. Greek Ionic is an equally important and familiar formal system which diverges from the Doric in a number of essential ways. Although superficially similar to Doric architecture, Ionic is characterized by columns of more slender girth which have complex mouldings at the base and resolve into spiraling volutes in the capitals. The Ionic also lacks a frieze of triglyphs and metopes. Instead, it has rows of dentals which are small blocks representing the ends of smaller wooden beams. These originally sat directly above the architrave. Note also that the architrave has three fasciae or horizontal facets, the uppermost of which sometimes include roundels or rosettes. The Ionians were Greeks who lived primarily in the islands of the Aegean Sea and on the shores of Asia Minor, also known as Ionia, which border the Aegean Sea on the east. For reference, Ionia is located on the west coast of present-day Turkey. It is said that the Ionians were a remnant population of the Mycenaean Greeks who fled eastward during the Dorian invasion in the 12th century B.C. Perhaps because the Ionians wished to include a zone of sculptural reliefs on their temples to match the sculpted metopes used by their Dorian cousins, a frieze was introduced at a certain point in the system's development. This feature was popularized by the Athenians or Ionian in origin. An example of an Athenian Ionic building with a frieze is the Erechtheum shown in the lower images in this slide. Like their Doric counterparts, Ionic temples comprise the crepus or crepidoma of three steps, the lower two forming the stereobate, the uppermost step being the stylobate. The columns meet the stylobate with a base made of stacked concave and convex moldings, and their round shafts, which are more slender and deeply fluted than Doric examples, rise to a decorative necking or traheleon, on top of which the voluted capitals are placed. The entablature consists of the architrave, the decorated frieze, and the cornice, which consists of a corona and sima. This last feature is sometimes called a simation for the shape of its profile. The Ionic capital is without doubt the most distinguishing feature of this columnar type, with its spiral volutes, cushion-shaped pulvinus, and echinus with egg-and-dart moldings. The egg-and-dart is a typically Ionic motif, as is the lesbian leaf, also known as leaf-and-dart, which in the example on the lower right decorates the abacus above the volutes. The themes were evidently used because their profile closely matches the shape of moldings. Note the anthemia or honeysuckles at the junctures between volutes and echinus. Although shrouded in mystery, the origin of the volute is likely to be found in the so-called Aeolic capital, which prevailed in the Near East in the 7th and 6th centuries B.C. An example of this type is shown above left. Note that the Aeolic volutes converge at the neck, rather than connecting across the shaft as in mature Ionic capitals. A possible transitional early Greek Ionic capital is shown in the middle image. Another possible origin of the volutes is in a papyrus or vellum scroll which Greek texts were written on. This type of book used by the Greeks and Romans and which survives today in the Hebrew Torah may have been placed on top of the columns before a temple's dedication, inspiring the Ionic capital. Or if not, Ionian builders may have deliberately stressed the resemblance. Unlike its Doric counterpart, the Ionic capital is uncompromisingly frontal so that its flanks are treated very differently. The candelabra-like shape of the side is called a baluster in English. It usually includes a center strap known as a balteus. In general, the Ionic capital is much more decorated than the Doric. At the Propylaea in Athens, shown at top left, gold floral motifs were set into the tops of the echinus. Like the profile of the Doric echinus, Ionic capitals employ complex curves using multiple centers. The downward swelling of the strip connecting the volutes, which is sometimes very pronounced, is called pulvination. Sometimes the top of the volutes also forms a continuous curve, as in the capital of the temple of Apollo at Bassae, shown in the lower left-hand image. The image on the right shows a corner condition in an Ionic porch. The capitals retain the frontal orientation and do not negotiate the 90-degree turn. This external corner problem is to the Ionic what the corner triglyph problem is to the Doric, a formal contradiction that requires resolution by effecting a tectonically irrational intervention. The most common solution is one that replicates the volute on the side of the capital, as seen in the top left image in a 19th-century building imitating the Erechtheum. In plan, this causes the balusters of the perpendicular volutes to intersect awkwardly. At the Erechtheum, this was no doubt deemed acceptable because the inner corners were hard to see from below. The earliest Ionic column bases varied a good deal from building to building, and many were treated as stacks of concave moldings known as scotiae and convex moldings known as toruses, which were set on square plinths. A common Ionic base type is made up of the sequence plinth, scotia, scotia, torus. Eventually, the sequence plinth, torus, scotia, torus prevailed, and it is known as the Attic base, as it was used predominantly in Athens and the surrounding region of Attica. An example of an Attic base is the one from the Erechtheum at the top right. This comparison of the Greek Doric and Ionic types shows how, despite their differences, the two systems have a similar overall structure and formal basis. Note, however, the drastically different proportions in the elevations shown on the left and right-hand side of the slide. Although the Greeks developed a large variety of columnar types, for instance the Aeolic, which we've already seen, or the Pergamene and Tower of the Winds variants, three prevailed and were retained in the Western classical canon, namely the Doric, the Ionic, and the Corinthian. We will now look at the Corinthian type. This particular architectural order appeared for the first time as a freestanding column inside the cella of the temple of Apollo Epicurus at Bassae. The building was designed by the architect Tictinus, author of the Parthenon in Athens. Although it is Doric on the exterior, the interior of the temple is surrounded by engaged Ionic columns. The single Corinthian column occupies the center of the cella at the far end and is therefore given pride of place. Note that the column has the same proportions and base as the Ionic engaged columns flanking it. For this reason, the Corinthian type is sometimes considered to be a variant or development of the Ionic. The capitol comprises two rows of acanthus leaves with scroll-like stalks called colliculi wrapped around a slightly flaring bell. The colliculi rise to the corners and spiral inward at the middle. The first attested use of the Corinthian type on the exterior of a building is the 4th century B.C. coragic monument of Lysicrates in Athens. This variant makes the acanthus leaves larger and employs Ionic details in the entablature. The drawing on the lower right illustrates the story told by the Roman architect Vitruvius about the invention of the Corinthian capitol according to which the Athenian architect Colimachus noticed the makeshift grave of a girl in the cemetery outside the gates of Corinth. The girl's maid had placed a basket on the grave filled with the child's favorite toys and put a roof tile on top of the basket to keep it from blowing over in the wind. An acanthus plant had grown at the base of the basket and coiled around it. Finding the effect pleasing to the eye, Colimachus sketched what he saw and soon translated it in architectural terms. In his discussion of the Greek columnar types, historian John O'Nions has suggested that the Corinthian type was originally associated with healing sanctuaries. He notes that the temple at Bassae, shown on the left, was dedicated to Apollo Epicurus in his guise as a healer, and the building was commissioned to celebrate the end of a plague. The monument of Lysicrates, too, was dedicated to commemorate the success of a boy's choir at the Dionysian festival in Athens, which took place in connection with healing rites. The great tholos at the sanctuary of the healing god Asclepius at Epidaurus, designed by the sculptor Polyclitus and shown above right, also employs a Corinthian columnar type on the interior, the capital of which is shown below the drawing. According to O'Nions, the Corinthian type may have been favored for such buildings because of an etymological coincidence, wherein the first part of the word acanthus resembles the verb acin, meaning to heal. According to this theory, acin and anthos, or flower, gives us acanthus. The three Greek Corinthian capitals which we have reviewed, namely the Bassae, Lysicrates, and Epidaurus, or Polyclitus variants, are similar in their sparing use of foliage, prominent bell, and sharply angled abacus. The most obvious difference is the location of the floral motif, known as the floron, or anthos in Greek, either in the middle or at the bottom of the abacus. As noted earlier, other columnar types were also developed in Greece, among which was the so-called Tower of the Winds Corinthian variant. Named after the building in Athens where it was famously used by the architect Andronicus, the capital consists of a single row of acanthus leaves and a second, taller row of palm leaves. This type became a favorite of American colonial and federal period builders. Much has been written about Greek proportioning systems, and most of it remains speculative as almost nothing has come down to us on the subject in ancient literature. The Greeks are also known to have engaged sculptors as architects, and for this reason buildings tend to employ different, highly particularized proportions. This is evident, for example, in the plate from Mauchen Norman's Parallel of the Classical Orders that compares a number of Greek Doric buildings. Or these two plates from the same publication showing Greek Ionic and Corinthian examples. As you can see, the proportions of both columns and entablatures are quite different from building to building. In an effort to subjugate the Greek types into a Renaissance system, 19th- and 20th-century architects made the Greek Doric six base diameters tall and set the entablature at a third the height of the column. The Greek Ionic and Corinthian variants, being similar enough to their Renaissance counterparts, were fixed at nine and ten diameters tall. This kind of standardization, of course, is quite contrary to original Greek practice. However, a number of theories have been developed that seek to identify overriding, proportioning strategies used by the Greeks. One theory, advanced in 1861 by British engineer Edward Cressy and published in Joseph Guilt's Encyclopedia of Architecture, notes that in the elevations of many temples, the columns support a superstructure with an identical mass. Here, Cressy illustrates a four-columned or tetrastyle Doric temple with a pediment which is square in proportion. You can see that the columns fit roughly into 20 squares, if the base diameter is used to determine the size of a square. The sum total of the entablature and the pediment above the columns is also 20 squares. We see the same equivalence of columns and superstructure in this six-columned or hexastyle Doric temple facade with pediment, which assumes a 2 to 3 proportion. In this case, there are 36 squares respectively for the columns and superstructure. Here, the system appears to work for an octastyle pedimented elevation with 48 squares respectively for the columns and superstructure. In this case, the elevation assumes a 1 to 2 proportion. Cressy also applied the method to the more slender ionic type with similar results. Here, a tetrastyle ionic temple facade with pediment, all inscribed in a square, yields 38 over 38 squares. And here, a hexastyle ionic pedimented elevation gives 54 over 54 squares, and the whole is inscribed in a 2 to 3 rectangle. The problem with Cressy's proposed method is that it only works on pedimented elevations and cannot be universally applied to all Greek temples. Other proportioning systems are more likely to have been used by Greek architects. In the Greek mind, beauty was closely associated with number, at least since the days of Pythagoras in the late 6th and early 5th centuries B.C. The influence of Pythagoras on the development of Greek and more generally Western thought is immense. A native of the island of Samos in Ionia, he resettled in the Greek colony of Croton in southern Italy and is said to have studied with priests in Egypt and Phoenicia. He is best known for the trigonometric theorem that still bears his name. In Pythagoras's view of the world, number constitutes the very essence of being. Among his many observations, Pythagoras perceived that the first four numbers contain all the proportional intervals of music. According to legend, he made this discovery when he happened one day to be passing by a forge and noticed that the blacksmith's hammers rang out differently depending on their weight, making harmonious sounds when played one after the other in various combinations. The weights of the hammers were 6, 8, 9, and 12 pounds so that when they were struck, the musical tones they rang were a fourth, fifth, and octave, in other words, four, five, and eight notes above a bass tone, do being the bass, fa, sol, and do. The mathematical ratios of these tones are respectively 3 to 4, 2 to 3, and 1 to 2, which Pythagoras's followers later showed can be measured out accurately by comparing the ratios of lengths of musical strings. So, for example, if we strike a string and sound a bass note, do, then put our finger three-fourths of the way along the string and strike it again, it will sound fa, which is the fourth note away from the bass note. Similarly, if we put our finger two-thirds of the way along the string, we will sound sol, which is a fifth note away from the bass, and finally, if we strike the string with our finger half the way along, we will sound another bass note, or do, which is eight notes away from the original bass note, octo meaning eight in Greek. In other words, when struck in sequence, these chord lengths sound a perfect musical fourth, a perfect fifth, and an octave. The Pythagoreans quickly recognized that when translated into lengths, these same ratios formed rectangles that had long before been considered beautiful. In other words, windows, doors, or building plans and elevations that exhibited four to three, three to two, or two to one proportions were deemed to be the fairest, just as fourths, fifths, and octaves were on the musical scale. This startling revelation reinforced Pythagoras' belief that number and harmony were key to understanding the essence of the world. The consonances on which the Greek musical system was based, fourth, fifth, and octave, can thus be expressed by the arithmetic progression one, two, three, four. Just as startling to the Pythagoreans would have been the observation that these same ratios, one to two, two to three, and three to four, are also manifest in the principle of proportional means. The Greeks held that there are three different types of proportional means, arithmetic, geometric, and harmonic. A mean is a number, or a measure, when units of length are involved, situated between two extremes so that its relationship to the extremes creates a regular progression. For instance, if one takes the ratio six to twelve, in which the second number is twice the first one, we can deduce a mean that relates to the extremes so that each term exceeds the previous one by the same amount. Six is to nine, as nine is to twelve. Here the mean exceeds the first extreme by the same amount, three, that the last one exceeds the mean. This is called an arithmetic progression. By contrast, a geometric progression is one in which the mean exceeds the first number by the same ratio as the last one exceeds the mean. For example, if we take the numbers four and nine, the geometric mean will be six, giving us the progression four, six, nine. The third method of deriving a mean makes use of what is known as a harmonic progression, in which the mean exceeds one extreme and is exceeded by the other by the same fraction of the extreme. For example, if we begin with the numbers six and twelve, the harmonic mean will be eight, giving us the progression six, eight, twelve. Here eight exceeds six by a third of six, and twelve exceeds eight by a third of twelve, or four. The proportions encrypted in the Parthenon in Athens are an example of the use of proportional means. Analysis of the building's plan and short elevations reveal that the ratio of short and long dimensions is consistently four to nine. The geometric mean between nine and four is six, which gives the equal ratios nine to six and six to four. This translates as three to two, in other words, a perfect musical fifth. All the important relationships in the Parthenon thus relate back to the Pythagorean musical scale. Greek number theory is not confined to rational progressions, but also includes irrational numbers, which cannot be written as simple fractions. As we have seen in the Pythagorean system, the first four numbers yield the most important ratios. One ratio, however, that the Greeks were aware of and may have deliberately employed in buildings is the so-called golden ratio. The ratio is produced if a line segment is divided in two parts, so that the ratio of the larger part to the smaller equals the ratio of the whole segment to the larger part. A common way to construct this ratio is by taking the distance from the middle of a square's side to an opposite corner and striking an arc from that corner until it aligns with the square's side, then raising a new rectangle. The ratio of the short side of the new rectangle to the long one will be 1 to 1.618. Although ratios yielding irrational numbers are often found in nature, for instance, the successive arcs of a nautilus shell, the golden ratio occurs in the human body, most notably in the relation of the height of a person's navel from the floor to the height at the top of the head. It is thought that the sculptor Phidias, who oversaw the works on the Parthenon, employed the golden ratio to determine the relative heights of the entablature and pediment apex. For this reason, the golden ratio is today indicated with the Greek letter phi, which is the first letter in Phidias's name. In this geometric overlay, you can see how the last set of diminishing rectangles appear to precisely situate the entablature in the elevation. One objection that has been raised with regard to this theory is that the base of the larger inscribing rectangle extends below the crepidoma, or three steps of the temple's substructure to the bottom of the stereobate, and does not account for the crepidoma's own outward projection, or the subtle curvatures employed in the elevation. A characteristic of ancient Greek art and architecture is a tendency towards typological and geometric purity. This was driven by an interest in ideality, which permeated the arts, especially Athens, during the classical period when, under the leadership of the statesman Pericles, the Acropolis was reconstructed and embellished. The interest in typological purity is perhaps most evident in the Greek art of sculpture, where the human body, both male and female, is rendered in an unblemished, perfect state as divine will itself might have intended. It is also evident in the habit of reducing buildings to almost diagrammatic simplicity. Thus the temple is a reductive image of a dwelling. The tholos is a perfect circle, and the theater a perfect half-round shape. The Greek obsession with ideality can be observed even when new building types were created to serve the expanding polis, or city. In this slide are two reconstructed civic buildings of the 3rd and 2nd centuries B.C., which employ pristine reductive forms in a perfect fusion of form and function. As Greek cities grew and expanded and democratic systems of government were refined, new buildings were added to the urban repertoire. Late Greek cities usually had an ecclesiasterion, or assembly hall, and a boulefterion, or council hall. The building types were standard even when their architects imbued them with highly particularized proportions. Greek city planning was either informal, with buildings arrayed in a seemingly casual manner as a collection of freestanding objects, or it was based on a gridded street pattern. An example of the informal approach is the Acropolis in Athens, shown here. At first glance, the various buildings, including the Parthenon, Erechtheum, and Propylaia, appear to ignore each other's orientation and proportions. After careful study of the Acropolis and other Greek urban settings, in the 1930s, architect Constantin Doxiadis advanced the intriguing theory that despite their apparent informality, many sites appear to have deliberately set buildings so that they create visual contiguities when viewed from important vantage points, for instance, the eastern threshold of the Propylaia, or gatehouse of the Acropolis. In Doxiadis's rendering below right, you can see these contiguities, which even include features in the distant landscape, such as Mount Lycabettus. The second approach to the layout of cities was the gridded street plan. This method was perfected by the Ionian architect Hippodamus, who was said to have produced the plans of the cities of Miletus, Praene, Piraeus, and more. In this view of Miletus, you can see all the familiar Greek building types set into an orthogonal urban matrix. A similar urban matrix is seen at Praene, with temples, theater, stoas, council house, etc. worked into a gridded pattern of urban blocks. The relationship of these components can be appreciated even better in this plan of the city's civic center. The Greeks, of course, did not think in planimetric terms, but instead perspectively. This orthographic drawing renders the urban spaces and buildings as a pattern so that one does not appreciate the sculptural nature of many of the civic edifices. This drawing by Doxiaris shows that Hippodamus merged his new gridded approach with the older perspectival system of sizing and placing buildings so that they form visual contiguities. Often the pursuit of ideality in Greek architecture involved adaptation of typologically pure forms to complex sites. Examples of localized level changes are the Propylaia and Erechtheum on the Athenian Acropolis. More usually, however, uneven or difficult terrain was shaped into stepped terraces that allowed buildings to stand as perfect sculptural objects. This late Greek approach to architecture was adopted by the Romans in the period of the Republic. Sanctuaries like that of Fortuna at Perneste were configured on the model of Hellenistic terraced sites with geometrically pure buildings and spaces deployed on either side of the central approach. We will look more closely at Roman architecture in the next segment.