## Geometry In Architecture

There are many different ways in which geometry plays a part in architecture.

The previous chapter, Temples and Cottages, discussed some of the different attitudes that a designing mind can adopt towards the conditions within which architecture is done. In particular it identified the attitudes of control and acceptance as they can be exemplified in an archetypal 'temple' and an archetypal 'cottage'.

The architectural uses of geometry can be discussed in these terms too. There are ways to use geometry that emerge out of the conditions of being, and there are others that may be imposed or overlaid upon the world. The latter, termed 'ideal' geometries, are the subject of the last section of this chapter; the chapter begins with some of the geometries 'of being'.

The word geometry, as a subject in school for example, suggests circles, squares, triangles, pyramids, cones, spheres, diameters, radii, and so on. These play an important part in architecture; as abstract ideas they belong in the category of ideal geometries—their perfection can be imposed on the physical fabric of the world as a means for identifying place.

But geometries emerge from our dealings with the world too; geometry can derive from an attitude of acceptance as much as it can be associated with an attitude of control. Geometries of being are inherent to the identification of places.

Circles of presence

People and objects introduce geometry into the world just by being.

Every body has around it what might be called a 'circle of presence', which contributes to its own identification of place. When a body is in relationship with others, their circles of presence affect each other. When a body is put into an enclosure or cell its circle of presence is also contained and perhaps moulded.

An object standing on a flat landscape occupies its own space, but it also exerts concentric circles of presence, to which we can relate.

If one discounts electronic and radio presence, the broadest of these circles of presence is the visual, described by the distance at which the object is visible. This circle may stretch as far as the horizon, or it might be contained by a forest, or a wall.

In terms of sound this large circle of presence would be the distance at which a sound emanating from a body is audible; smell, smellable; radio waves, receivable.

The smallest circle of presence, physically, is described by the distance within which one is able to touch, and perhaps embrace, the body.

The most difficult circle of presence to determine rationally is the intermediate, the one within which one feels that one is 'in the presence' of the body. It might be said that it is this circle of presence that delimits the place of the body.

Architecture uses all three: the extensive circle of visibility; the intimate circle of touchabi- lity; and the intermediate circle of place. Much architecture, from prehistoric times to the present, has been concerned with asserting, defining, amplifying, moulding, or controlling circles of presence.

A tree defines one of its circles of presence by the extent of its canopy of branches.

A candle, or a lighthouse, describes its circle of presence by the light that it emits.

A fire, as suggested in the chapter on Primitive Place Types, identifies a place by its sphere of light and warmth.

A standing stone exerts its presence in the landscape, as an assertion of the presence of those who put it there.

It is perhaps in the handling of circles of place that architecture can be at its richest and most subtle.

Circles of presence are rarely perfect circles; they are almost always affected by local conditions and topography. The world is generally so full

A tree defines one of its circles of presence by the extent of its canopy of branches.

A candle or a lighthouse describe their circle of presence by the brightness of the light they emit.

The statue of Athena Promachos asserted the circle of presence of the goddess over the ancient city of Athens.

A standing stone asserts its circle of presence in the landscape, and establishes the place of those who put it there.

The statue of Athena Promachos asserted the circle of presence of the goddess over the ancient city of Athens.

of bodies that their many circles of presence overlap, interfere, or maybe reinforce one another in complex ways which are sometimes too difficult to analyse fully.

Circles of place have been manipulated by architecture since ancient times, for various purposes.

Most of the buildings on the acropolis in Athens were built during the classical age of ancient Greek culture, around the fifth century BC. The top of this rocky hill in the plain of Attica had been a place sacred to the goddess since time imme morial. Such elevated places were sacred partly because they had a clear identity; they were elevated and sanctuaries in times of trouble; they also possessed extensive circles of pres-ence—they could be seen (and from them one could see) for long distances across the landscape. The hill of the acropolis retains this circle of presence over modern Athens.

By their architecture the ancient Greeks manipulated the circles of presence of the sacred place of Athena. The extent of the circle of place around the sacred site was defined partly

The circle of presence of a significant object can be contained and distorted by the enclosure or cell within which it stands.

The circle of presence of a significant object can be contained and distorted by the enclosure or cell within which it stands.

by the reasonably level area of land on top of the hill, but this was extended and established more firmly by the huge retaining walls which still define the sacred precinct—the temenos— around the temples. The shape of this temenos in plan is not circular, but represents an interaction between the circle of presence of the sacred site and the topography of the hill.

There were two important statues of Athena on the Athenian acropolis. The giant Athena Promachos stood in the open air near to the entrance into the temenos, projecting its own circles of presence over the city, even to ships on the sea some miles away. The other statue was enclosed within the main temple, the Parthenon, which had (and maintains) its own circle of visibility across the city, and which amplified the hidden presence of the image whilst controlling its circle of place and protecting its intimate circle of touchability, both of which were probably only ever penetrated by priests.

In these ways the acropolis illustrates some of the ways in which circles of presence play their parts in architecture: the retaining walls of the temenos define the 'circle' of the sacred site; the Parthenon amplifies the presence of the statue it contains, and its cella controls and protects the statue's circles of place and touchability.

### Lines of sight

We human beings seem fascinated by the fact that we see in straight lines. This fascination is evident in the way one might vacantly line up the toe of one's shoe with a spot on the carpet, or more purposefully when one sights a distant object with the end of a finger to point it out. The fascination with lines of sight is evident in architecture too.

An alignment of three or more things, one of which is one's eye, seems to possess some peculiar significance. The precise alignment of the sun, the moon and the earth, at a solar or lunar eclipse, has always been considered a significant event. The builders of Stone-henge appear to have erected the Hele Stone to align the centre of the henge with the sun rising over the horizon on the Summer Solstice. Standing on a pier, we notice when a ship crosses the line projected by the pier out into the sea. Driving through the countryside we remark when a distant feature is exactly aligned with the road along which we are moving.

Alignment imparts significance, to both the distant object

We are intrigued when the landscape appears to contain alignments.

The Hele stone aligned the centre of Stonehenge with the sun rising on the Summer Solstice.

It appears that sometimes buildings were aligned with sacred mountains.

When he was remodelling the Castelvecchio in Vicenza, Carlo Scarpa would draw lines of sight onto his plans. Emanating from particularly important points in the building—the entrance, or a doorway—these would influence his deliberations on the positions of exhibits, or pieces of landscaping.

Reference for Carlo Scarpa at the Castelvecchio:

Richard Murphy—Carlo Scarpa and the Castelvecchio, 1990.

and the viewer. The 'sight'—the finger tip or the Hele Stone—is a medium, a fulcrum between the two, a catalyst which projects a line between the viewer and the object. Alignment implies a line of contact— an axis—between oneself and the distant object, exciting in the viewer a thrill of recognition of the linkage (which is even stronger when 'eyes meet across a crowded room').

Thinking of architecture as identification of place, a line of sight establishes contact between places. In the ancient world it was one of the ways in which architects tied places into the world around them, establishing them as fragments of matrices which centred on particular sacred sites. It is a power which is important in the design of places for performance, where engagement between actors and spectators depends on sight. It can also be important in designing art museums, where lines of sight can influence the positioning of exhibits.

### Lines of passage

In the physical sciences, one of the laws of motion is that a body remains in a state of rest, or moves in a straight line with uniform speed, unless compelled by a force to change that state. This is often a presumption in architecture too.

Lines of passage are usually considered to be straight, unless diverted by some 'force'. A sensible person usually moves in a straight line between a starting point and a goal, unless there is some obstacle which makes this unwise or

Reference for Carlo Scarpa at the Castelvecchio:

Richard Murphy—Carlo Scarpa and the Castelvecchio, 1990.

impossible. In organising the world into places, architecture also establishes lines of passage between those places, using them as ingredients of serial experiences.

The line of a pathway in the landscape is often a result of people's and animals' tendency to move in straight lines being diverted by changes in the surface of the ground.

Reference for ancient Egyptian pyramids:

I.E.S.Edwards—The Pyramids of Egypt, 1971.

The ancient pyramids of Egypt were connected to valley buildings on the river Nile by long causeways. Sometimes these were straight; sometimes they had to take account of local land conditions, or perhaps changes in plans during construction, and deviated from the direct line.

Lines of passage are often related to lines of sight; but they are not necessarily congruent.

A line of passage can set up or reinforce a line of sight, as when a road aligns with a distant feature in the landscape; but they might not coincide. Sometimes architecture can make a play of aligning a line of passage with a line of sight (as in the nave of a church); but sometimes the line of passage deviates from the line of sight, so that a pathway does not take the most direct route between starting point and goal.

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