Six Direction Body Architecture

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In this drawing the goal (the entrance) is clear, but the approach is diverted from the line of sight.

Approaching the entrances to the Carpenter Center, lines of

Approaching the entrances to the Carpenter Center, lines of

Centre Line Ramp
entrances are curved. At the start of either ramp the line of passage to the entrance does not follow the line of sight.

Sometimes a line of passage does not have an obvious

Sometimes a line of passage does not have an obvious

goal which can be seen. Interplay between lines of sight and lines of passage can create a sense of mystery in the experience of a work of architecture.

Sometimes a work of architecture presents a choice of lines of passage, each of which has to be assessed by sight.

Measuring

The word geometry derives from two Greek words, for earth (ge) and measure (metron). Measuring the world is essential to life; people measure their environment all the time, and in lots of different ways. Measuring with a ruler or tape measure is only one of those ways, and an artificial one. The more immediate ways in which people measure the world is with their own bodies.

People measure distance by walking. They may do it consciously by counting their paces; but they also do it subconsciously, merely by walking from one place to another. In connection with walking, people estimate distance or the height of a step with their eyes, and assess the amount of effort needed to cover the distance or climb the step.

People estimate the width of doorways and passageways, estimating whether there is space to pass others.

People estimate the height of openings to assess whether or not they must stoop to pass through.

People are conscious of the size of a room, and can estimate what it will accommodate. They do this primarily by

means of sight, but the acoustic of a space can also indicate its size. People also subconsciously calculate how the size of a room, and the distances between pieces of furniture in it, can influence social interrelationships within it.

People might estimate the height of a wall to assess whether it may serve as a seat; or of a table to assess its use as a work bench.

People literally measure out the lengths of their own bodies on the beds in which they sleep.

A person stands by a window conscious of the heights of the cill and of the head, and of whether the horizon can be seen.

People set the scale of a work of architecture in comparison with their own stature as human beings, and with the ways in which their bodies may move.

These are all transactions between people and works of architecture. People set the measure of the buildings they use; but buildings also set the measure of the lives they accommodate. People take measure from the works architecture they inhabit, and use their measurements to make different types of assessment.

In the late fifteenth century Leonardo da Vinci constructed this drawing illustrating the relative proportions of an ideal

People measure the world with their movement, their bodies, and their senses. A stair measures a difference between levels in equal steps.

human frame as set down by the Roman writer on architecture, Vitruvius. It suggests that in its ideal form the human frame conforms to geometric proportions; it also suggests that the measurements of the human frame are tied in with those of nature, and the universe.

the different postures that the human frame adopts: sitting, leaning, working at a table..

Reference for The Modular:

Le Corbusier (translated by de Francia and Bostock)—The Modulor, 1961.

In the middle of the twentieth century Le Corbusier contrived a more complex system of proportions relating the human frame to those of other natural creations. He used a special proportion called the Golden Section. His system, called The Modular, allowed for

Earlier in the twentieth century, however, the German artist and dramatist, Oskar Schlemmer, had recognised that the human frame also measures the world in its movement and projects its measure into the space around it.

A large doorway exaggerates the status of the occupant, and diminishes the status of the visitor.

A small doorway diminishes the status of the occupant, and enhances the status of the visitor.

A human-scale doorway puts the occupant and visitor at equal status.

Six-directions-plus-centre

A human being has a front, a back, and two sides; generally speaking, the ground is below, and above is the sky. Each stands (or sits, or lies) at the centre of its own set of these six directions.

These observations seem almost too obvious to bother stating, but they are simple truths that have fundamental ramifications for architecture. Six directions condition our relationship with the world, in which each of us is our own mobile centre. They condition our perception of architecture— how we find and occupy places, how we relate ourselves to other places—and play into the conception of architecture, presenting a matrix for design.

One way in which architecture can relate to the six-directions-plus-centre is by the evocation of resonance between an enclosure and its occupant, by making it a place which responds to (or deals with in some way) each of the six directions. An ordinary cell, with its four walls, ceiling and floor, conforms to this. In such places each of us can compare the orientation of our own six directions, and the position of our own centre, i i i

with those of the room, finding places where our six directions are in either formal accord or relaxed interplay with those of the room. By its six sides a place (a room, a building, a garden) can set out a two- or three-dimensional orthogonal framework, the power of which lies in its provocation in us of a sense of relationship.

In relating to a place that has a front (an in front), a back (a behind), two sides (a left and a

The tank in Damien Hirst's Away from the Flock forms a three-dimensional orthogonal frame around the sheep. Each face of the tank implies an elevational view of the animal.

right), a top (the above), and sits on the ground (the below) we feel that in some way we are relating to something which is like ourselves, and which, to this extent, is created in our own image, and to which we can respond through comparison with our own six-directions-plus-centre.

The suggestion of accord between sets of six-directions-plus-centre can be a powerful identifier of place, especially when architecture sets up a centre which a person, or the representation of a god in human form, or a significant object, can occupy.

direction to dominate the space. Such a manifestation of direction might be reinforced in other ways, maybe by positioning the throne opposite the entrance, or by setting out a path—a red carpet perhaps— which identifies the monarch's route to and from the throne as well as emphasizing the forward direction from the throne.

The six directions are evident in human bodies, and these can be responded to in the architecture of spaces and rooms. The six directions are also manifest in

the conditions within which creatures live on the surface of the earth. The sky is above and the earth below; but each of the four

Often in such cases one of the six directions is dominant, usually the forward: as in the case of a soldier's sentry box which allows vision to the front while protecting his back and sides from attack, his top from rain or sun, and his feet from mud or the cold of the ground; or as in the case of a throne room, where the position of the throne against one of the four walls, rather than at the geometric centre of the room, allows the monarch's forward

horizontal directions has its own character. Each of the four cardinal points of the compass relates to the movement of the sun. In the northern hemisphere the sun rises in the east and sets in the west, it is at its highest in the south, and never enters the northern quarter. Works of architecture can be oriented to these terrestrial directions as well as to those of anthropomorphic form. In this way buildings mediate geometrically between human beings and their conditions on earth. Any four-sided building on the surface of the earth relates in some way, roughly or exactly, to these four cardinal points of the compass. Any four-sided building is likely to have a side which receives morning sun, a side which receives midday sun, and a side to the setting sun; it will also have a side to the north which receives little or no sun. These four horizontal directions have consequences in the environmental design of buildings, but they also tie architecture into the matrix of directions which cover the surface of the earth (and which are formally recognised in the grids of longitude and latitude by which any position on the surface of the earth is defined).

The four-sided building is directly related to the directions on the surface of the earth as it spins through time; and each side has a different character at different times of day. But such a building can be significant in another way too; for if its six directions are considered to be in congruence with those of the earth—its four sides face each of the four terrestrial directions implied by the movement of the sun, and its verticality accords with the axis of gravity which runs to the centre of the earth— then the building itself can be considered to identify a centre— a significant place that seems to gather the six directions of the earth into its own, and provide a centre which the surface of the earth does not.

In these ways the geometry of the six-directions-plus-centre can be seen to be inherent at three levels of being: in ourselves as human beings; in the original nature of the world on which we live; and in the places that we make

Reference for the Vitra Fire Station:

Vitra Fire Station', in Lotus 85, 1995, p.94.

trough architecture, which mediate between us and the world.

The six-directions-plus-centre are a condition of architecture, and as such are susceptible to the attitudes of acceptance and control mentioned in the chapter on Temples and Cottages: one can accept their pertinence and influence; or attempt to transcend them by exploring abstract and more complex geometries, or by tackling difficult concepts such as non-Euclidean, or more-than-three-dimensional space. Some might also argue that the submission of the world's surface to the rule of four directions, or three dimensions, is simplistic; that the movement of the sun through the sky is more complex than the cardinal directions suggest; and therefore that architecture either should not necessarily pay heed exactly to the matrix that the six-directions imply, or should look for more subtle indicators for the positioning and orientation of buildings.

Nevertheless, the notion of six-directions-plus-centre is useful in analysing examples of architecture of many kinds and characters. Its power is found in examples that range from the ways in which directions, axes and grids can be introduced into landscapes to make it easier to know where one is, and how one might get from one place to another...

Even a fairly rough stone can, like a person, introduce the six-directions-plus-centre into the landscape.

Even a fairly rough stone can, like a person, introduce the six-directions-plus-centre into the landscape.

...through the vast stock of orthogonal works of architecture, to attempts to escape or test the boundaries of rectilinear architecture, as in the works of Hans Scharoun, or of Zaha Hadid. Even though distorted, as if by the force of some warp in the gravitational field, the four horizontal directions retain their power in the plan of Hadid's Vitra Fire Station.

Many works of architecture relate to the four horizontal directions, to the above and the below, and to the concept of centre, in simple and direct ways. The Greek temple is a particularly clear example. The six-directions-plus-centre

Akropolis Zeichnung

operate at various conceptual levels, even in a building whose form is as apparently simple as this.

First, as an object in the landscape, the building has six faces: one to the ground; one (the roof) to the sky; and four sides, each facing one of the four horizontal directions. In this regard the temple establishes itself as a centre.

Second, as an internal place, the cella of the temple has a floor and a ceiling, and four walls that relate directly to the four horizontal directions implied by the image of the god or goddess who was its essential reason for being.

Third, in the relationship between the inside space and the outside world, the doorway (the prime link between the two) allows one of the four horizontal directions (that of the face of the deity, which is reinforced by the longitudinal axis of the temple) to strike out from the inside and relate to an external altar, and maybe also (as a line of sight) to some remote object of significance— the rising sun, or the sacred peak of a distant mountain.

These three ways in which the six-directions-plus-centre are inherent to the architecture of the temple collaborate to reinforce the role of the temple as an identifier of place. The temple itself is a cell and a marker, but its orthogonal form channels the ways in which it identifies the place of the sacred image, making it also a centre.

But there is also a fourth way in which this essentially simple building type relates to the six-directions-plus-centre, one that is of special importance in thinking of architecture as identification of place. This is to do with the way that the directions of the building relate to those of a visitor or worshipper.

The geometry of an ancient Greek temple responds to the six-directions-plus-centre...

Regarding its external form as a body, we are aware (if we know the building, and are in its presence) when we are at the back, at the front, or at either of its sides. Relative to the building, we know where we are. But in addition to that relationship, we are also aware that there are significant places created by the power of the orthogonal geometry of the building; places that maybe draw us to them. The most important of these is that prominent direction which emerges from the god's statue through the door and strikes out into the landscape; we know when we are standing on this axis and perceive it as special; it excites in us a thrill of connection between our own directions and those of the god.

This powerful axis is established by the architecture of the temple. We are not left as detached spectators, but brought into involvement with the archi-...as does the geometry of a tecture of the building, made traditional church. part of it. It is exactly the same

power, that of the dominant axis, which prompts the practice of nodding reverently as one crosses the axis of the altar in a Christian church or a Buddhist shrine. It is the same power that draws us to stand at the exact centre of a circular space (the Pantheon in Rome, or under the dome in St Paul's Cathedral in London, or the amphitheatre at Epidavros in Greece).

These simple uses of the six-directions-plus-centre are basic, rudimentary, and seemingly universally recognised as constituting a power of architecture.

Social geometry

The geometry of social interaction between people is perhaps a function of the six-directions-plus-centre that each possesses.

When people congregate they identify their own places, in particular ways. In doing so they overlay a social geometry where they come together. As a process of identification of place, this is architecture in its own right, but while it consists only of people its existence is transient. Works of architecture can respond to social geometri-es, order them, and make their physical realisation more permanent.

When schoolboys spectate at a playground brawl between two of their number, they form a circle. When there is a formalised bout between two boxers, the area of their battle is defined by a rectangular platform with rope barriers around the edge. Though square it is called a ring, and the boxers' confrontation is represented by their possession of opposite corners.

People may sit in a rough circle around a fire in the landscape. In the ingle-nook of an Arts and Crafts house that social geometry is transformed into a rectangle, accommodated within the structure of the fabric of the house.

It may not be an example of social geometry, but the grid layout of graves in a cemetery is a function of the geometry of the human frame and the way in which the rectangular shape of the space it needs can be tessellated across the land.

A stone circle makes a people pattern permanent.

An ingle-nook formalises the geometry of social interaction around a fire. This imaginary example was drawn by Barry Parker, and is illustrated in the book he produced with his partner in architecture, Raymond Unwin—The Art of Building a Home, 1901.

Coloriage Ado Fille Imprimer

The radial arrangement of spectators on the slopes of a valley, watching sports or dramatic performances,was architecturally translated by the ancient Greeks into the amphitheatre, with its (more than semi-) circular plan, consisting of many tiers of concentric sitting steps.

People arguing stand opposite each other; when they are friends, they sit next to each other. Both can have architectural manifestations.

In British politics, the confrontation of the Government and the Opposition is physically represented in the benches of

There is a social geometry to the space of togetherness...

the House of Commons, which face each other across the chamber, with the Speaker (or chairman of the debate) sitting on the axis between them.

The social geometry of the British House of Commons is a manifestation of the procedural relationship between the Government and the Opposition.

diametrical, opposition across the chamber.

It is a moot point whether such architectural arrangements affect the behaviour of members of parliament or of chapters. Some countries, nevertheless, have chosen to accommodate their parliamentary debates in circular rather than confrontational debating chambers, if only for symbolic reasons. This, as one example, is the debating chamber of the Finnish parliament in Helsinki, which was designed by J.S. Siren and built in 1931.

Some chambers for discussion are designed not for argument and opposition but for collective debate. This is sometimes manifested in their architecture. Chapter houses are meeting rooms attached to cathedrals and monasteries. Often they have a circular, or perhaps polygonal, plan which, architecturally at least, is non-confrontational and non-hierarchical. Even the central column,

...and to the space of confrontation.

which supports the vaulted ceiling, seems to block direct,

The circle is one of the most powerful symbols of human community; architecturally it seems to speak of people being equal and together in a shared experience of the world. It is the pattern made, loosely, by the people around their campfire; it is the pattern made by people sitting around a picnic; it is a pattern associated with conversation; and it is a pattern associated with spectating at some dramatic or ceremonial event.

Though he avoided many other types of geometry in his designs, even the German architect Hans Scharoun accepted the aptness of the circle as a frame for the social event of a meal. In the Mohrmann House, built in 1939, the dining area is the only place in the plan which

it is turned on a woodturner's lathe; a table is rectangular because it is made of regular-shaped pieces of timber.

has a regular geometric shape: a circular table is accommodated centrally in a semi-circular bay window between the kitchen and living room.

Geometry of making

Many everyday objects have a geometry that is derived from

Many everyday objects have a geometry that is derived from

Floreros Para Colorear

vase is circular because it is thrown on a potter's wheel; a wooden bowl is circular because

it is turned on a woodturner's lathe; a table is rectangular because it is made of regular-shaped pieces of timber.

There is geometry to laying slates on a roof...

The same is true of building. Often the materials and the way in which they are put together impose or suggest geometry.

When put together into walls, bricks, as rectangular objects themselves, tend to produce rectangular walls, and rectangular openings and enclosures. When using such materials it requires a definite decision to deviate from the rectangular.

There is geometry to laying slates on a roof...

The geometry of bricks conditions the geometry of things that are made from them.

vase is circular because it is thrown on a potter's wheel; a wooden bowl is circular because

...and to the ways in which pieces of timber can be joined together.

This drawing is based on one in: Drange, Aanensen & Brsenne—Gatnle Trebus, (Oslo) 1980.

The geometry of making is essential to the construction of buildings. In this traditional Norwegian timber house, as in many traditional houses from around the world, there is an interplay of social geometry and the geometry of making. Social geometry conditions the sizes and the layout of the spaces. But the shapes of those spaces are also conditioned by the materials available and their intrinsic qualities, and by current building practice.

The building is infused with the geometry of making, even though that geometry is not always exact and regular. The fabric of the walls and the structure of the roof is influenced by the sizes of timbers available, and their innate strength. The sizes of roofing tiles influence the design of the roof. The small panes of the window are conditioned by the sizes of pieces of glass. Even the small portions of masonry are conditioned by the shape of the bricks and the subtle and complex geometries of the stones available. And the bracket which holds the cooking pot has its own structural geometry, and describes a locus which is part of a circle as it is swung across the fire.

The geometry of making is not so much a power of architecture as a force which conditions building. The force is not active, but lies latent in materials that are available for building, and in plausible strategies for bringing materials together into building under the influence of gravity. As such the geometry of making is subject, in architecture, to the range of

Coal Mining Families Poor

attitudes mentioned in the chapter on Temples and Cottages. In producing an archetypal 'cottage', it may be said, the geometry of making is accepted, whereas in an archetypal 'temple' it is transcended. Within this dimension architects can adopt any of a range of attitudes to the geometry of making.

The Scottish architect Charles Rennie Mackintosh designed many pieces of furniture; in some of them he exploited the geometry of making, refining it according to his aesthetic sensibility. This, for example, is a waitress's stool he

Cocoa Fermentation Box

designed in 1911; it follows the geometry of making, but this has been refined into a matrix of perfect cubes.

There is a constructional geometry too in the shingle and timber buildings designed by the American architect Herb Greene; but it is stretched almost to its limit, and distorted into animal-like forms. This drawing (right) shows part of his Prairie House, built in 1962, on which the shingles are like the feathers of a hen.

The geometry of making includes the geometry of structure, whether it is the timber structure of a medieval tithe barn, or the steel structure of a micro-electronics factory. The geometry of structure is said to be susceptible to mathematical calculation, though there seems to be an infinite variety of ways

Reference for Mackintosh furniture:

Charles Rennie Mackintosh and Glasgow School of Art: 2, Furniture in the School Collection, 1978.

Reference for the architecture of Herb Greene:

Herb Greene—Mind and Image, 1976.

of arranging a structure to span a particular space. Some are said to be efficient if they use material economically and without redundant members; some have an added quality called elegance. Whether there is a direct correlation between efficiency and elegance is a point of debate.

It is also the discipline which controls industrialised building systems. Systems consist of standard components that can be put together as a kit of parts. These parts include structural components, and various types of non-structural cladding panels which form the

The structure of a native American teepee has an innate conical geometry, which produces a circular plan.

The three-dimensional geometry of some medieval carpentry is quite complex. This is part of the scaffold of the spire of Salisbury Cathedral The drawing is based on one by Cecil Hewett in his book English Cathedral and Monastic Carpentry, 1985.

The structure of a native American teepee has an innate conical geometry, which produces a circular plan.

The geometry of making does not only apply to traditional materials such as brick, stone and timber; it applies just as much to buildings with steel or concrete structures, and to buildings with large areas of glass walls.

envelope of the building. The dimensional co-ordination that allows standard components to be manufactured in a factory, transported to a site, and then put together to make a building depends on careful and disciplined appreciation of the geometry of making.

Ideal geometry

The circle and the square may emerge out of social geometry or from the geometry of making, but they are also pure, abstract, figures. As such, they are sometimes thought to have an aesthetic or symbolic power (or both) in their own right. Some architects use them to instil their work with a discipline that is independent of (but perhaps also related to) the various geometries of being.

Ideal geometry does not only include the circle and the square and their three-dimensional forms—the cube and the sphere. It also includes special proportions, such as the simple ratios of 1:2, 1:3, 2:3 or more complex ratios such as and that known as the Golden Section which is about 1:1.618.

In his book, Architectural Principles in the Age of Humanism (1952), Rudolf Wittkower explored the ways in which Renaissance architects used ideal geometric figures and ratios in their designs. He also discussed why they believed that such figures and ratios were powerful.

One argument was that natural creations, such as the proportions of the human frame, or the relationships between the planets, or the intervals of musical harmony, seemed to follow geometric ratios, and that if the products of architecture were to possess the same conceptual integrity they too should be designed using perfect figures and harmonic mathematical proportions. Another argument was that through architecture a geometrical perfection could be achieved that was only hinted at in natural creations.

The application of geometry was seen as one way in which human beings could improve the imperfect world in which they found themselves. Geometric purity was thus seen as a touchstone of the human ability, or perhaps duty, to make the world better. It is in this sense that ideal geometry, as a way of imposing order on the world, is a characteristic of the 'temple'.

The result was that architects produced designs for buildings which were composed using perfect figures and geometric ratios.

This, for example, is a copy of Wittkower's diagrams of the geometric composition of the façade of the church of S.Maria Novella in Florence, designed by Leon Battista Alberti and built in the fifteenth century. They

show that the façade of the building may be analysed as a composition of squares. These have a role in the design which is independent of the building's geometry of making; the geometry is displayed on the front wall of the church, as on a screen.

Many architects have designed buildings in which the accommodation is enclosed within a square plan. This is different from composing the design of a façade as a two-dimensional pattern of squares, because it involves the third dimension, and perhaps also the fourth—time.

A square plan is not usually a result of accepting the geometry of making; a square space is not the easiest to frame with a structure; it requires purposeful intent, derived from something other than mere practicality, to make a plan square.

Architects may design a square plan for various reasons: maybe for the philosophical reasons outlined above; maybe because a square can seem to identify a still centre which relates to the six directions mentioned above; maybe as a kind of game—a challenge to fit accommodation within this rigid shape.

Architects are always looking for ideas which will give form to their work and direction to their design. Geometric ideas are some of the most seductive. To design within a square plan is an easy idea to grasp (and a way to break through the problem of getting started). But although it may seem a limitation, the square plan is also open to infinite variation.

There are many examples of square plans. They are rare in ancient and medieval architecture, but became more part

One very ancient example is of course the Egyptian pyramid. These tombs were generally built on land to the west of the Nile, between the river and the desert, and carefully oriented to what we know as the cardinal points of the compass. They are clear examples of architecture responding to the six-directions-plus-centre.

Below is the plan of the pyramid complex of Pepi II, at Saqqara in Egypt. The pharaoh's pyramid has been cut through to show the burial chamber at its centre. There are three smaller pyramids for his wives. The building to the right of the drawing is the valley temple, which was the ceremonial entrance to the complex and linked to the pyramid temple by a causeway which is too long to be included in the drawing in its full length.

Each side of the pyramid faces a direction with a different character. The temple build

ings and the ceremonial approach are to the east and link the pyramid to the river and the life of Egypt. The opposite side faces the desert. The south faces the sun when it is at its highest. The north side seems to have less symbolic significance, and was used for the physical access to the burial chamber, which was perhaps less important than the ceremonial entrance from the east. The pyramid is a centre where these directions meet, and the burial chamber lies at the centre of its geometric form. It is in this way that the ancient Egyptian pyramid was a powerful identifier of place.

Below are the plans of the principal floors of two square plan houses built in England in the 1720s. On the left is Mereworth Castle in Kent designed by Colen Campbell; on the right Chiswick Villa by Lord Burlington. Both architects were influenced in the choice of a square plan by the design on the right, which is of the Villa Rotonda designed by the Italian architect Andrea Palladio, and built some one-hundred-and-fifty or so years before the two English examples.

Palladio's plan is the most consistent of the three. As in the ancient pyramid, it gathers the four horizontal directions into a centre—the focus of the circular hall at the heart of the plan, from which the villa gets its name. (Unlike the pyramid, the trx trx

Michelangelo Anatomy Circle

sides of the Villa Rotonda do not face north, south, east, and west,

Reference for the Villa Rotonda:

Camillo Semenzato—The Rotonda of Andrea Palladio, 1968.

but northeast, southeast, southwest, and northwest.) The plan is not just one square, but a concentric series of five; the size of each successive one is determined by the radius of a circle circumscribed about the next smallest. The smallest circle is the rotonda itself; and each square (except for the second smallest) determines the position of some substantial part of the building. The largest square gives the extent of the steps which lead up to the porticoes on each side; their depth is determined by the second largest square; and the main walls of the villa are determined by the middle-sized square.

The cross-section through the Villa Rotonda is also a composition of circles and squares, though not such a simple one as in the plan.

Square plans have been used by architects designing in the twentieth century.

Charles Moore used the square as the basis of his plan for the Rudolf House II. As in the Renaissance examples Moore created a central place, which is here the living room, surrounded by subsidiary places: kitchen, dining room, bedroom, and so on. Perhaps for practical reasons, the plan is not so neatly arranged as that by Palladio.

The Swiss architect Mario Botta bases many of his designs on geometric figures. He has designed a number of private houses in Switzerland; these are often composed of squares and circles, cubes and cylinders.

Botta's design for a family house at Origlio, which was built in 1981, is a composition of rectangles and circles fitted into a notional square. On each

floor he uses the square in a different way. On this floor, the middle of three, the plan is nearly symmetrical, with the living room and fireplace at its heart.

The plan of this house at Riva San Vitale is also based on a square. The house is a tower

of five floors built on the sloping bank of Lake Lugano. It is entered across a bridge to the top floor (which is the one shown in the drawing).

In both these houses Botta also appears to have used another geometric figure—the Golden Rectangle—to help him in deciding the layout of the plans. The Golden Rectangle is one which has a particular proportional relationship between its two dimensions: the ratio of the short dimension to the long is equal to that between the long dimension and the sum of the two dimensions. This means that if one subtracts a square from a Golden Rectangle, one is left with another, smaller, Golden Rectangle. This ratio, known as the Golden Mean, is not a whole number, but approximately 1.618:1.

Reference for Botta houses:

Pierluigi Nicolin—Mario Botta: Buildings and Projects 1961-1982, 1984.

In the house at Origlio it appears that Botta used the Golden Mean to give the proportion between the central section and the side sections of the

house. In the Riva San Vitale house he seems to have used Golden Rectangles in a way similar to that in which Palladio used circles and squares in the Villa Rotonda, that is like Russian Dolls. The square near the middle of the plan accommodates the stair which connects the floors.

Le Corbusier also used the Golden Mean to give geometric integrity to his work. In his book

Le Corbusier ordered the elevation of this studio house with 'regulating lines'.

Vers Une Architecture (1923), translated as Towards a New Architecture (1927), he illustrated his geometric analyses of some well-known buildings and the geometric framework on which he had built some of his own designs. He did not only use the Golden Mean, and sometimes his 'regulating lines' (he called them 'traces regulateurs'), make a complex web of lines. This is a copy of his diagram of the geometric composition of one of the elevations of the studio house which he designed for his friend Amedee Ozenfant; it was built in a southern suburb of Paris in 1923. Rather like in Alberti's S.Maria Novella (shown above), the geometry is displayed on the elevation of the house, as on a screen.

Complex and overlaid geometries

Many twentieth-century architects have used ideal geometry to lend rationality or integrity to their plans, sections and elevations. Some, seemingly bored with simple relationships, have experimented with complex arrangements in which one geometry is overlaid on another.

In some of the house designs by the American architect Richard Meier, the places of dwelling are identified by the spaces which result from a complex interplay of orthogonal geometries.

This, for example, is Meier's design for the Hoffman House, built in East Hampton, New York State, in 1967. The idea for the plan seems to have been generated from the shape of the site, which is an almost perfect square. The diagonal across the square determines the angle of one of the elevations of one of the two main rectangles on which the plan of the house is based.

Each of these two rectangles is a double-square. One is set on the diagonal of the site; the other is parallel to the sides

of the site. They share one corner. Their geometric interrelationship determines the position of almost everything in the plan.

Places—living room, kitchen, dining area, and so on—are allocated zones which are defined by the interaction of the overlaid geometries. The positions of basic elements—walls, glass walls, defined areas, columns—are determined in accord with the complex armature of lines which the geometries of the rectangles create. To help in this game the squares are sometimes subdivided to make the geometry even more complex, and thus identify a greater range of different places within the armature.

One interpretation of the geometry which provides the armature of the ground floor of this house is shown in the drawing on the right. The actual plan is below.

In this version one of the squares is divided into thirds in both directions, giving nine smaller squares. The intersections of the third-lines give the positions of the columns set in the glass wall which lights the living room and dining area. The fireplace is positioned on the one corner which the two rectangles share. The en-trance—itself a square—seems to be generated by an interaction of the centre line of one of the double-squares with the side of the other, and sits in an axial relationship with the fireplace and the seating in the living room. An alcove in the living room is created by a projection of the middle third of the divided square to meet the corner of the other double-square. And so on.

This may seem complicated, and is certainly difficult to follow when explained verbally. If this is the way that Meier progressed his design for this house, which seems plausible, then he was using geometry as the framework for design decision, a hybrid of that used by

Joseph Rykwert (Introduction)—Richard Meier Architect 1964/1984, 1984, pp.34-37.

Vitra Fire Station Plan

This apartment building in a suburb of Tel Aviv is a complicated spiral composition of fragmented circles and rectangles. The places of dwelling are dispersed amongst the spaces which result from the overlaid geometries.

Reference for Tel Aviv apartments by Zvi Hecker:

L'Architecture d'Aujourd'hui, June 1991, p.12.

Alberti and Palladio. Geometry is used in this way to suggest formal and perhaps also aesthetic integrity. In the overlaying of geometries Meier adds a further dimension—intricacy in the quality of the spaces which are created.

Meier's geometric overlays may seem complex, but some other architects have used geometric frameworks more complex than that in the Hoffman House.

On the left and below, as one example, are the section and plan of an apartment building in the Tel Aviv suburb of Ramat Gan in Israel. The architect of this complicated building was Zvi Hecker, and it was built in 1991. It is formed of a spiral of fragmented circles and rectangles, with dwelling places disposed in the spaces which result from the geometric over lays.

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