summary in easy language

Part 1. Elements of Architectural T h e o r y

Chapter One Basic Terms: Point, Line, Plane, Voltlrne

Geometry is fhe means, creufed by ourspbea, wkwby we p m i v e the e x t d world mui express the world within us. -LE C Q R B U S ~

F~ur cenmies separate Gebastiano Serlio's (1475-1554) Five Books on Architecture, 1545 from Le Carbusids (1887-1965) The City of Tomorrow, 1924. Yet both books begin in ex- actly the same way, wfth illustrations of .the basic geometric elements. f i e two architects worksd in remote time periods, designed build- ings of widely diverse character, made use of

1 different structural and constructional technolo- gies, and related to very different political, social, and cultural pressures. Yet for both, simple geometry and the basic Pythagorean

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progressiQn of point, to line, to plane, to vol- ume constituted the fundamental ingredients of all architectural fonn.

Why were these terms useful to Serlb and Le Corbusier? Geometry did not provide spe- cific solutions to quesrions of style, building program, or construction, since the two archi- tects were as far apart as possible on these hues. Instead, both architects wed geomem to establish an archite&d order. A geometii- cally determined definition of form and arrangement of pat% made it possible for both

2.1 L@: Sebastiano Serlio, Five Book; oq Archihture. Book I, Plnte I , 1545.

1.2 Righf: Le Curbusier, The Cify of Tomorrow, Frontiyiece, 2924.

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architects to express architectural ideas clearly, and further, to allow for the development of sub-themes and variations which could be un- derstood against the strong underlying formal order, or datum.

In discussing the role of geometry in archi- tecture we shall begin, like Serlio and Le Corbusier, with a point. Equal in their relation- ship to all surrounding directions, points establish centers for groupings of form. While geometers define a point as dimensionless, in our discussion of architectural form we under- stand a point as any simple, singular thing: a pin pomnt, a dot on a page, a sugar cube, a cylin- drical oil drum, etc.

The line, mathemahcally defined, is a one- dimensional entity which can be infinitely extended in two directions. Rather than con- sider a l i e to be of a wholly different nature than the point, one can imagine a transforma- tional relationship which links the two. In other words, a line can be understood as a point which has been dragged or 'translated' in space.

The next term in the Pythagorean progres- sion is the plane, a two dimensional, unbounded surface which extends infinitely in all directions. How does a plane relate to the previous term, the line? Just as the idea of transformation helps us to understand the rela- tionship between the point and the line, the same strategy can be used to clarify the relation- ship between the line and the plane. A plane can be understood as a line that has been dragged in space.

In discussing architecture, ltnes and planes do not have infinite extension: nothing in the physical world has that property. Still, the im- plication of extension gives power to the form,

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and establishes a potent demarcahon between front and back, above and below, inside and outside, etc. Such distinctions permit further lwels of spatial stratification to take place. Lim- ear and planar structures can act in many different ways. As solid components of the built environment, walls separate one precinct from another; slabs establish boundaries between vertically stacked spaces; filters effect a trans- formation of spatial experiences from one side to the other; the neutral plane, or datum, acts as a background against which a free play of fonn can be deciphered.

As voids, linear elements often have even greater power to organize form. Axes can forge dear relationships across long &stances without the requirement of adjacency; views or vistas make possible an understanding of connection between even more loosely grouped elements; paths orchestrate a sequence of experiences along the procession and enlist memory in the task of reconstructing and assembling the expe- rience as a n entirety.

The last term in the Pythagorean progression is the volume. Again, we can use our notion of transformation to understand the relationship between the three-dimensional volume and the precedmg two-dimensional term. If a plane is dragged through space the resulting form is a cube or some other rectangular prism. Hence point, line, plane, and volume, the primary terms needed to establish an architectural order, are all related by a one simple act of transforma- tion, as Paul Klee's (1879-1940) drawing, "Point, Line, Plane, Volume" illustrates.

Further affinities and distinctions can be found among these four basic elements. A line has only one more dimension than a point, yet

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they fundion very differently in the making of space. The primary task of points is the defini- tion of center, wide the primary task of lines is the definition of edge or perimeter. Points are finite and bounded; lines are by definition ex- tensible. Volumes and points, on the other hand, play similar roles in spatial organizations. Even though volumes are three dimensional and points are conceptually dimensionless they can be thought of as equivalent, but for the mat- ter of scale. The farther we are away from something, the more it tends to have qualities of a simple, dimensionless point. When seen from

1 the earth, stars appear to be minuscule points, although we know they are of vast size. When examined under a microscope, grains of sand appear to be volumetric, although under ordi- nary circumstances they appear as mere specks.

, Both points and volumes are bounded, finite and neutral towards the various directions which surround them. Hence, both can act as centers to organizations. On the contrary, lines

1.3 Left: Johannes Kcpler, Harmony of the spheres, Mvsterium

1.4 Right: J. de Barbari, 'Luca Paccioli,' (author of- Geometrial , detail, c.1500, Capodimonte Museum, Naples.

and planes are defined both by extension and by their ability to set u p distinctions between one side and the other. They do not act as cen- ters but as edges to organizations of form.

Let us examine two structures which are bounded and crisply defined in form, an Egyp- tian pyramid from more than four thousand years ago, and a cylindrical corn crib from a Midwestern farm in the 1990's. Both are under- standable as almost pure Platonic solids, or elemental three dimensional fonns. The term ''latonic solids' derives from the writing of Plato, the Greek philosopher who speculated that the four elements in nature, earth, air, fire and water, corresponded to geometrical solids whose faces are regular polyhedrons. For Plato, these were the tetrahedron, the cube, the icosahedron and the dodecahedron. The sphere was reserved as the shape of the heavens above. Believing in the cosmological significance of Platonic solids, Johannes Kepler constructed a model of the universe by nesting the Platonic

1.5 Cylindrical corn m'b, Midwestm farm, 1990s.

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fonna around a center. Luca Paccioli wrote a treatise on the innak proportional relationships among simple solids in Da Diving Proportlane. Both sought to reveal the underlying order in the "Harmony of the spheres" which govemed relationships of everything from astral bodies to musical harmony.

In every day discourse, 'Platonic forms' have come to mean volumes like spheres, cylin- ders, cones, pyramids, cubes, tetrahedrons, and the l&e which are generated when idealized two-dimensional forms (circles, squares, isosce- les trianzles, etc.) have been extended, rotated - or xeflected around their centers or axes. Both the cylinder and the pyramid, by their own mathematical definitions, make reference to their centers. Both are organized by a vertical axis, or conceptual center line, running from the ground through the top of the roof. Indeed, one symbolic purpose of the pyramid was to gather - - - - up the earth into a single point to meet the sun. Hence, at changing scales, the function of these

T ' Pyrurnik of Cheops (c. 2570 B.C.), Che)m (c. 2530 B.C.), and Mycerinus (c. 2500, B.CJ at Giza, Egypt.

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structures as points or volumes is interchange- able.

A more complex organization takes place in the next figure, the Pantheon, a arcular Roman temple. Like the corn crib in the previous ex- ample, the Pantheon is essentially a domed cylindrical volume, but here the form is hybrid. A cylindrical drum is united with a temple front, joined together by a thick masonry block. Unlike the previous example in which all sur- rounding directions were addressed with equanimity, the addition of the temple portico introduces an axis into the centralized organiza- tion. The axis stems from the elongation of the structure in one direction, and the path of movement into the buildings. Even so, the axi- ality of the path is resolved in the vast centralized interior space. In the interior an- other axis is revealed: the axis mundii or the vertical axis which links the earth to the sky, framed by the circular window or oculus at the top of the dome.

1.7 The Pl Rome, centut

ztheon, taly, 1st A.D.

1.8 The Pantheott, elevation and plan.

1.9 Stoa of Attalos 11, c. 150 B.C. the Agora, Athens, Greece.

1.10 Stoa of Attalos 11,

' 1.11 Haizs Schauoun.

I Hostel, Wroclaw, Poland, 1927.

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Centralized buildings may grow complex but still make reference to the central point around which they are organized. Linear build- ings may also have more elaborate internal configurations. The simplest form of a linear structure is a wall; a bar of matter which is con- sistent throughout its length, like a bar of chocolate. However, the width of the bar itself may yield a more intricate organization. The space between the two exterior surfaces of a wall, called pochh, can be carved to shape spa- . * tial figures, i k e the holes formed in a thick mass of Swiss cheese. A linear or 'bar' building can be permeable, like a comb, in which the surface is not continuous but marked by a regular rhythm of vertical members. A bar-building can be cellular, like a Tootsie Roll, in which the line is comprised of like units which are serially repeated. A bar-building can be layered, like a Snickers bar, with one surface treatment on the outside, then another, and another to comprise. the whole. A Greek stoa is an example such a

1.12 Josr!ph Pa.xton, "I--tsworfh,

land, the mhouse, 540.

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structure. The side facing the market square or agora is columnar in structure. The outer wall, on the contrary, is solid and frequently houses a cellular band of shops. The Greenhouse at Chatsworth by Joseph Paxton (1801-65), is an- other example. Attached to a heavy stone wal which girdles the estate is a delicate glass anc metal structure. The transparency of the green house enclosure allows both surfaces to be reac simultaneously and strongly differentiates thf inside from outside.

While centralized buildings insist on the equality of all space surrounding them, linear buildings darify difference from one side of the line to the other. When a line bends or deflects it no longer simply desmbes a boundary but it begins to define a space. Alvar Aalto's (1898- 1976) Baker House Dormitory in Cambridge, Massachusetts, gently curves in response to the natucal form of ihe Charles River, forming a garden on the river side of the complex. Differ- ences between inside and outside are further

1.14 Baker House,

-- plan.

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1.15 John Hejduk, Wall House I, plan, 1968-74.

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I 1.16 John Hejduk, Wall House I,

I preliminary I sketches.

refined by the use of hard, angular forms on the street side in contrast to the gentle undulations of the river side. Moreover, the river side col- lects private dormitory rooms while the city side is mostly comprised of circulation, (corri- dors, stairs, and elevators,) and collective spaces such as living rooms and bathrooms. At entry, a shaft of space cuts a void through the build- ing, pushing matter out from the bar to form a dining pavilion. Hence, all components of the dormitory derive from transformations of a simple bar.

In Hans Scharoun's (1893-1972) Hostel in Wroclaw, Poland, the bar of dormitory rooms frames exterior space not so much by bending as by 'twisting.' The open, balconied edge of the long wing threads through the irregularly shaped central hall and reemerges on the oppo- site side of the short wing, almost as if the particularity of the central space arose through the pressure of the deformation. As in the Baker H o w , not only form but also program, or the functional requirements of a building, are satisfied through this move. Rooms for couples are arranged in the wider bays of the short wing and rooms for bachelors are arranged in the narrower bays of the long wing. The specially shaped central piece houses common living and dining activities.

In the previous two examples we observed how a relationship between linear elements and centric objects can arise from a transformative process. In the Baker House, the trajectory of the enhy path pushed the rooms at the lower floor outside the bar to form a pavilion piece in the garden. In the Hostel, a twisting of the bar yielded a special condition, much as in the twisting of a Mobius strip. In Wall House I,

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John Hejduk @. 1929) reduces the wall to the simple condition of datum. Against the neutral surface of the wall a more particular collection of objects is arranged, not through transforma- tion but through collage. The wall emphasizes the difference from one side to the other. Public activities are arranged on one side, private spaces on the other; one side confronts an urban condition, the other side a landscape; one side makes use of a tectonic, or conshvctional sys- tem, of heavy, compressive materials, while the other side uses a tectonic system of light, tensile materials, suspended from the wall; one side is comprised of opaque, neutrally colored materi- als, while the other side is comprised of reflective, transparent surfaces. Hejduk de- scribes his project in the following way: "on one side of the wall (the past), the circulatory elements- rmnp, stair, elevator- were placed. They were volu- metric, opaque, monochromatic, i n perspective with the structure grounded. The color was white, grey, black; the materials reinforced concrete, steel a n d ce- ment. Once the single inhabitant passed fhrargh the Fs+F- ---- , wall he was in a space overlooking a landscape (trees? water? earth? sky?) which was basically pri- vate, contemplative and refectiwe. There were three suspendedfloors cantileveredfrom the cotlectiv~ ele- -A,' ments. The materials on this side of the wall were YT 7 glass mtd rejective metal; a fluidity was sought af- ter. Whereas the collective side was hard, tough, mnnete, the private side was inwardly rejective, a light shattering intofragments, mirror image8 mou-

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ing along polished sulfaces of metal." (Mask of m-

Medusa, by John Hejduk.) The plan of the Hellenistic town Pergamon

illustrates an entire context that can be deci- w , , phered as a transformation of very simple

objects and linear forms to yield a much more

1.17 John Helduk, Wall House LI, axonometric and plan, 1968-74.

1.18 Douglas GraJ

@ 3 Diagrams of =,, - 4 the Upper

City, Pergamon.

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complex order. In the broadest terms, the urban form of Upper Pergamon can be described as radial, taking advantage of the slope of the hill to carve out the fan-shaped space of the theater, around which secondary spaces splay out like plumes in a fan. A more precise description of interconnections among various spaces in the plan can be construed by imagining acts of for- mal transformation, just as we earlier read relationships between points, ling, planes and volumes by positing spatial translation from one dimension to the next.

Our reading of the plan begins with the Caracalla Temple at the lower left. The Temple, much like a sugar cube, is a discrete, bounded object, although its permeability along the axis of entry implies extension and organizes a shaft of space in front of it. Just beneath the Temple, a bar-like building, or stoa, extends rightward, as if the temple had been dragged to yield a new linear form. The next element, moving

counter-clockwise, is the Market. Here the stoa bends, enveloping in its perimeter the plane of the market area. A few isolated buildings flank the lower edge of the Market, as if the bar had been fragmented to yield discrete objects once again. The plane defined by the Market stoa slips to the left to act as a platform for the Pergamon Altar. Meanwhile, the stoa, which defined perimeter in the Market Complex, shrinks inwards to define the Altar itself. The voided center of the Altar is as bounded and figural as the first term in o w progression, the Temple. However, now the figure is defied by a n endosed void rather than a space-displacing solid. While the Temple had a voided perimeter defined by the space of the colonnade and a solid center, the cella, the Pergamon Altar has a solid perimeter and voided center. Hence, form presents itself in its two purest manifestations, as pure solid and pure void. In the Library Complex the bent bar of the Altar splits apart,

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hattered by the path from the Citadel Gate. In chaotic and without structure. Understanding he process, the center is freed from the the various spaces as the product of translations ngirdling wrapper. A monument marks the of objects in space, the foldings of edge-defining

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>li&val location of tenter, the Athena T, rl

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the center while the true bars, scale changes, and reversals of figure and emple, shifts downwards to ground makes it possible to see how affinities of

lign with the central void of the Pergamon Al- form ,and mean& might be constructed. It is u to which it formally relates as a figure1 worth mentioning that Pergamon did not his- round reversal. The last major space, the torically emerge according to an overarching r a j a n w reconstitutes the wrapper and ideal urban strategy of constructing variations on

centering of the Pergamon Altar at an expanded centerlperimeter relationships. Pergamon '$ale. However, here a solid object, the temple, grew up over time, responding not only to the marks center. To complete the circuit, one inherent pressures that buildings have upon merely needs to slip the temple of the one another; but also to complicated landform, Trajaneum downwards to yield the Caracalla or topography; to precise religious, political and Temple, our starting point. social ordering5 or hierarchy; and to the neces-

Reading the plan of Pergamon as a series of sity to make connections to other parts of the transformations performed on fairly straightfor- town. Still the formal structure yields a 'read- ward elements makes it possible to decipher ing' of the relationship among elements on the

.20 'ergamon,

rder in a plan that at first glance might seem site.