Why are most buildings rectangular? - UCL Discoverydiscovery.ucl.ac.uk/13217/1/13217.pdf · Why are most buildings rectangular? This is a fundamental question that is rarely asked

The paper offers evidence to show that the geometryof the majority of buildings is predominantlyrectangular, and asks why this should be. Varioushypotheses are reviewed. It is clear that, in thevertical direction, rectangularity is at least in part todo with the force of gravity. In the horizontal plane,departures from rectangularity tend to be found inbuildings consisting of single spaces, and around theperipheries of plans made up of many spaces. Thissuggests strongly that the causes of rectangularity inmulti-room plans lie in the constraints of packingthose rooms closely together. A geometricaldemonstration comparing room shapes and roomarrangements on square, triangular and hexagonalgrids indicates that it is the superior flexibility ofdimensioning allowed by rectangular packings thatleads to their predominance.

Introduction: is it the case?Why are most buildings rectangular? This is afundamental question that is rarely asked. Perhapsvisiting Martians – assuming their interests tendedto the geometrical – might want to raise the issue.Certainly from the evidence of science fiction filmsthe dwellings of aliens seem to be non-rectangular –presumably to signal their exoticism andunEarthliness. The question is worth pursuing allthe same, I believe, because of its implications for atheory of built form. By ‘buildings’ I do not justmean considered and prestigious works ofarchitecture, which possibly tend more often thanothers to the non-orthogonal and curvilinear, forreasons that may become clear. I mean the totality ofbuildings of all types, the whole of the stock, botharchitect-designed and ‘without architects’, presentand past. And to be slightly more specific about thequestion itself: I mean to ask ‘Why is the geometry ofthe majority of buildings predominantly rectangular?’There will, certainly, be many small departures fromrectangularity in otherwise rectangular buildings,and it would be rare indeed to find a building inwhich every single space and component wasperfectly rectangular. We are talking about generaltendencies here.

Is it actually true? Everyday experience would

indicate that it is: but we do not have to rely just onsubjective impressions. At least two surveys havebeen made, to my knowledge, of the extent ofrectangularity in the building stock. The first was asurvey of houses carried out in the 1930s by theAmerican architect Albert Farwell Bemis, reported inhis book The Evolving House.1 Bemis was interested inthe potential for prefabrication in house building.Taking for granted that components making up anyprefabricated system would have to be rectangular,he measured a sample of 217 conventionallyconstructed houses and apartments in Boston, todetermine what percentage of their total volume –conceiving the building as a ‘solid block’ in each case– was organised according to a rectangular geometry.He found that proportion to be 83%, the remaining17% being largely attributed to pitched roofs.2 Thismeasurement gave Bemis an indication of the extentto which current house designs could be replicatedwith standardised rectangular components.

The second survey was made by M. J. T. Krüger inthe 1970s, in the course of a study of urbanmorphology and the connectivities between streets,plots and buildings.3 Krüger took as his sample theentire city of Reading (Berkshire), and includedbuildings of all types. He inspected just the outlinesof their perimeters in plan, working from OrdnanceSurvey maps. (Of course these outlines would havebeen somewhat simplified in the maps.) Hedistinguished plan shapes that were completelyrectangular in their geometry (all external wallswere straight line segments, and the angles betweenthem right angles) from plans with curved walls, orwith straight walls but set at acute or obtuse angles.He found that 98% of the plan shapes wererectangular on this definition. On the evidence ofthese two studies then – and acknowledging theirlimited historical and geographical scope – it doesseem fair to say that the majority of buildings arepredominantly rectangular.

HypothesesAt least part of the answer to our question in relationto the vertical direction is no great mystery:rectangularity in buildings has much to do with the

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theoryThe constraints of packing rooms together, and the flexibility of

dimensioning allowed by rectangular arrangements, explain the

predominance of the right angle in architectural plans.

Why are most buildings rectangular?Philip Steadman

force of gravity. Floors are flat so that we, and pieces offurniture, can stand easily on them. Walls and piersand columns are made vertical so that they arestructurally stable and the loads they carry aretransferred directly to the ground – although thereare obviously many exceptions. Larger buildings as aconsequence tend to be made up, as geometricalobjects, from the horizontal layers of successivefloors. In trying to understand general rules ortendencies it is often informative to look atexceptions, at ‘pathological cases’. When are floors notflat? The most obvious examples are vehicle andtrolley ramps – but we humans seem to prefer to riseor descend, ourselves, on the flat treads of stairs orescalators. Theatres and lecture rooms have rakedfloors; but these, like staircases, are not locally sloped,and consist of the shallow steps on which the rows ofseats are placed. Truly sloping floors on which theoccupants of buildings are expected to walk are rare;and where they do occur – as in the helical galleries ofFrank Lloyd Wright’s Guggenheim Museum in NewYork – can be a little disturbing and uncomfortable.

The rectangularity of buildings in the horizontalplane is more mysterious. I have solicitedexplanations from a number of colleagues whocombine an interest in architecture with amathematical turn of mind, and they have offereddifferent ideas.1. The cause, they suggest, lies in the use by architects

of instruments – specifically drawing boards with T-squares and set-squares – that make it easier todraw rectangles than other shapes. (Techniques forsurveying and laying out plans on the groundmight also favour right angles.) We know thatdrawing apparatus of this general kind is veryancient. The earliest known examples are fromBabylonia and are dated to around 2130 BC [1].However, we also know that very many, in factalmost certainly the majority of buildings inhistory, were built without drawings of any kind;and that many of these were neverthelessrectangular. So this explanation is clearlyinadequate.

2. The cause is to be found deeper in our culture andintellectual make-up, and has to do with Westernmathematical conceptions of three-dimensionalspace – with the geometry of Euclid, and with thesuperimposition on to mental space of theorthogonal coordinate systems of Descartes.(Architects’ drawing equipment would then be justone symptom of this wider conceptualisation.)However this argument is subject to the sameobjections as the first. What about all thoserectangular buildings produced in non-Westerncultures, or in the West but before Greek geometry,or erected by builders who had absolutely noknowledge of Western geometrical theory?

3. The cause is to be found yet deeper still in ourpsychology, and has to do with the way in which weconceptualise space in relation to the layout,mental image and functioning of our own bodies.Our awareness of gravity and the earth’s surfacecreates the basic distinction between ‘up’ and‘down’. The design of the body for locomotion, and

the placing of the eyes relative to the direction ofthis movement, creates the distinction – in thisargument – between ‘forward’ and ‘backward’. Wenow have two axes at right angles. A thirdorthogonal axis, distinguishing ‘left’ from ‘right’,reflects the bilateral symmetry, in this direction, ofarms, legs, eyes and ears. When we walk, we steerand turn by reference to this sense of left and right.We organise our buildings accordingly.4

This third argument for a general rectangularity inbuildings is subtler if even more speculative than theprevious two. If true, it would clearly apply tohumans and their buildings in all times and places.Such a mental and bodily disposition to see andmove in the world relative to an orthogonal systemof ‘body coordinates’ might conceivably – althoughthis is very hypothetical – account for the extent towhich we humans are comfortable in buildingswhose geometry is itself rectangular. But this is not inmy view the explanation – or at least not the soleexplanation – for the occurrence of thatrectangularity in the first place.

Where do departures from rectangularity occur in plans?Once again it is instructive to examine some counter-examples. In what circumstances do we tend to findbuildings, or parts of buildings, which are notrectangular in plan? Many buildings that comprisejust one room – or one large room plus a few muchsmaller attached spaces, such as porches or lobbies –have plan perimeters whose shapes are circles,ellipses, hexagons or octagons. Primitive andvernacular houses provide many familiar examplesof circular one-room plans: igloos, Mongolian yurts,tepees, Dogon and Tallensi huts [2]. Temples, chapelsand other small places of worship are often singlespaces, and again their plan shapes are frequentlynon-rectangular. You could cite circular temple plansfrom many cultures and periods. Figure 3 reproducesa detail of a plate from J.-B. Séroux d’Agincourt’sHistoire de l’Art, Architecture showing circular religious

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Philip Steadman Why are most buildings rectangular?

1

1 Carved representationof a drawing boardwith scribinginstrument and scale,from a statue of Gudeaof Ur, c. 2130 BC(Musée du Louvre)

buildings dating from the fourthto the sixteenthcentury [3]. The space of the auditorium is dominantin the forms of some theatres, and here too we findthat where the plan of the auditorium is a semi-circle, a horseshoe, or a trapezium, then it can givethis shape to the perimeter of the building [4].

These are buildings whose plans consist of, or aredominated by, one single space. A second type ofsituation in which curvilinear or other non-orthogonal elements are often found – in the plansof buildings that may consist of many rooms – isaround the building’s outer edges. Many of theotherwise rectangular churches in Figure 3 havesemi-circular apses. In simple rectangular modernhouses we find semi-circular or angled bay windows.These provided one of the more frequent departuresfrom rectangularity in Bemis’s survey (the mostobvious non-orthogonality in many buildings in thevertical direction – as confirmed by Bemis – is intheir pitched roofs; again on the outer surfaces ofthe built forms, by definition).

Figure 5 illustrates the plans of an apartmentbuilding at Sausset-les-Pins in France, designed byAndré Bruyère and completed in 1964. From theexterior the block gives the impression of beingdesigned according to an entirely ‘free-form’curvilinear geometry. Closer inspection of theinterior, however, shows the majority of the rooms inthe flats to be simple rectangles, or nearapproximations to rectangles. The curved profile ofthe exterior is created by curving some of theexterior-facing walls of the living rooms, and byadding balconies with curved outlines [5]. Many latetwentieth-century office buildings that have bulgingfacades are similarly curved only on these exteriorsurfaces, with conventional rectangular layoutsconcealed behind.

Naval architecture offers some interesting parallelshere. The hulls of ships are doubly curved, forobvious hydrodynamic reasons. In smaller boatswith undivided interiors the plan shape of the singlecabin follows directly the internal lines of the hull.But in ocean liners with many spaces, the interiorlayout tends to consist mostly of rectangular rooms,with only the curved walls of those cabins that lie onthe two outer sides of the ship taking up thecurvature of the hull [6].

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Why are most buildings rectangular? Philip Steadman

2a

2b

2c

2 Traditional andvernacular houses withcircular plans: a Mongolian yurtb Mandan earth lodgec Neolithic northernJapanese shelter (reprinted withpermission fromShelter © 1973, ShelterPublications, California,p. 16, p. 18 and p. 21)

3 Detail of plate‘Summary andGeneral Catalogue ofthe Buildings ThatConstitute theHistory of theDecadence ofArchitecture’showing religiousbuildings from thefourth to thesixteenth century

with circular plans, aswell as somerectangular buildingswith semi-circularapses, from J.-B.Séroux d’Agincourt,Histoire de l’Art par lesMonuments Depuis saDécadence au IVe

Siècle jusqu’à sonRenouvellement auXVIe (Paris, 1811–23)

3

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4a 4b 4c

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4 Plans of twentieth-century theatres: a Festival Theatre,Chichesterb ShakespeareMemorial Theatre,Stratford-on-Avonc Belgrade Theatre,Coventry(from R. Ham,Theatre Planning(1972), p. 256 and

p. 267, reproducedwith permission fromElsevier)

5 Plans of apartmentbuilding by AndréBruyère, Sausset-les-Pins, France, 1964(published inL’Architectured’Aujourd’hui, 130(1967), pp. 92–93)

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7

6

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6 Part plan of B Deck ofthe ‘Queen Mary’

7 Third floor plan of theGuggenheimMuseum, Bilbao byFrank Gehry, 1997(from C. vanBruggen,GuggenheimMuseum Bilbao(1998), p. 151). Notice

the rectangularplanning of thesmaller galleries andthe office wing

8 Plan of Altes Museum, Berlin by K. F. Schinkel, 1823–30,to show the centralcircular hall createdwithin an otherwiserectangular plan

If one were to ask the general public to name onecontemporary building, above all others, whoseform is definitively ‘free’ and non-rectangular, themost frequent answer might well be Frank Gehry’sGuggenheim Museum in Bilbao. Without questionthe external titanium shell is geometrically complex.But if one looks deeper into the building one findsthat in some places this shell is used to enclose large‘free-form’ galleries, practically detached from theremainder of the structure and so very nearly ‘single-room buildings’ in themselves [7]. Meanwhile, inthose parts of the Museum where many rooms ofcomparable dimensions are located together, such asthe multiple smaller galleries, and theadministrative offices, the planning reverts to anorthogonal geometry.

It is true that in certain classically plannedbuildings with many rectangular rooms, as forexample villas or country houses, there can be spacesdeep in the interior, such as central halls, whoseplans are circular, polygonal or elliptical [8].However, these are produced by filling out thecorners of rectangular spaces – in Beaux Arts termsthe poché – with solid masonry, cupboards, lobbies orspiral stairs; and the overall planning disciplineremains rectangular.

What all this evidence indicates, I would suggest, isthat rectangularity in buildings in the horizontalplane has to do, crucially, with the packing together ofrooms in plan. When many rooms of similar orvarying sizes are fitted together so as to pack withoutinterstices, it is there that rectangularity is found.Non-rectangular shapes occur on the edges of plans,or in single-room buildings, since in both cases theexigencies of close packing do not apply.

Packings of squares, triangles and hexagonsOne reasonable objection to this line of argumentwould seem to be that it is possible to pack other two-dimensional shapes besides rectangles to fill theplane. Among regular figures (equal length sides,equal angles), there are just three shapes thattessellate in this way: squares, equilateral trianglesand hexagons. Some architects have laid out plansvery successfully on regular grids of triangles orhexagons. Figure 9 shows Frank Lloyd Wright’s SundtHouse project of 1941, and Figure 10 shows BruceGoff’s house for Joe Price of 1956. Use of a triangularorganising grid does not force the designer intomaking all rooms simple triangles. The triangularunits can be aggregated into parallelograms,trapezia and other shapes (including hexagons), asevidenced in the Wright and Goff plans [9 & 10]. Thetriangular grid, that is, offers a certain flexibility ofroom shape to the architect. Similarly, unit hexagonsin a hexagonal grid can be joined together to makeother more complex shapes.

When I started to think about this particular issueof flexibility in planning, I imagined that a regularsquare grid perhaps offered more possibilities forroom shapes, and more possibilities for arrangingthose shapes, than did a triangular or a hexagonalgrid – and that here might lie part of an explanationfor the prevalence of rectangularity in buildings. The

following demonstration, though very far fromproviding any kind of mathematical proof, serveshowever to indicate that my intuition was wrong.

Consider three fragmentary square, triangularand hexagonal grids, each comprising nine grid cells[11]. We can imagine that the plans of simplebuildings – perhaps small houses – are to be laid outon the grids with their walls following selected gridlines. How many distinct shapes for rooms can bemade by joining adjacent grid cells together, in eachcase? Let us confine our attention, since we arethinking about rooms in buildings, to convex shapes,convexity being a characteristic of most smallarchitectural spaces. Let us also count shapes that aregeometrically similar, but are of different sizes (madeup of different numbers of grid cells) as beingdistinct. Thus on the square grid we can make threedifferent square shapes, with one cell, four cells ornine cells. Our criterion of convexity means thatthere is only one shape that can be made on thehexagonal grid: the unit hexagonal cell itself. Allshapes made by aggregations of hexagonal cells arenon-convex. On the square grid by contrast it ispossible to make six distinct shapes [12]; but on thetriangular grid it is possible to make 10 distinctshapes [13]. The triangular grid, contrary to what Ihad expected, offers a greater range of shapes forrooms than does the square grid.

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9

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Perhaps these shapes made by aggregatingtriangular units, although more numerous, cannotbe packed together without gaps in as many distinctarrangements as can the shapes made from unitsquares? For square grids, this question of possiblearrangements was intensively studied, in effect, byseveral authors, during the 1970s and ’80s. Thepurpose of that work was more general, as I shallexplain. But one incidental result was to show for a 3x 3 square grid, how many different arrangementsare possible, of rectangular and square shapes madeup from different numbers of grid cells, packed tofill the grid completely. These possibilities, of whichthere are 53, are all illustrated [14]. (The enumerationis derived, with modifications and additions, fromBloch.)5 No account is taken, in arriving at this total,of the particular orientation in which anarrangement is set on the page. That is to say, thesame arrangement, simply rotated through 90º or180º or 270º, is not regarded as ‘different’. Certainarrangements can exist in distinct left-handed andright-handed versions. Just one representative istaken in each case. The count includes the twoextreme cases in which the packing consists of nineunit squares, and of one single 3 x 3 square.

The enumeration of these square grid packingswas made by computer. Essentially the same resultswere achieved by Mitchell, Steadman, Bloch,

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9 Plan of Sundt houseproject by FrankLloyd Wright, 1941,laid out on atriangular grid

10 Plan of house for JoePrice by Bruce Goff,Bartlesville,Oklahoma, 1956-,also laid out on atriangular grid

11 Fragments of grids,each comprised ofnine cells whose

shapes are squares,equilateral triangles,and regularhexagons

12 The six possibleconvex shapes madeby aggregating unitcells in the squaregrid of Figure 11

13 The 10 possibleconvex shapes madeby aggregating unitcells in the triangulargrid of Figure 11

14 All 53 possiblearrangements inwhich combinationsof the rectangularshapes shown inFigure 12 can bepacked, withoutinterstices, into thesquare grid of Figure11. Differences solelyby rotation andreflection are ignored

15 All 68 possiblearrangements inwhich combinationsof the shapes shownin Figure 13 can bepacked, withoutinterstices, into thetriangular grid ofFigure 11. Differencessolely by rotationand reflection areignored

Krishnamurti, Earl and Flemming using severaldifferent algorithms.6 I have carried out a similarexercise by hand for the grid of nine triangular cells,to count the number of arrangements in whichcombinations of the shapes illustrated in Figure 13can be packed to fill this grid. Once again, differencesby rotation and reflection are ignored. My results areillustrated [15]. The number of possibilities is 68.Here once again, and contrary to my expectation, thetriangular grid offers rather more possiblearrangements of shapes than does the square grid.

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16a 16b

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16 a A packing ofrectangles within thesquare grid of Figure11, set on an x, ycoordinate system.The spacing of thegrid lines is given by the dimensions x1, x2, x3, y1, y2, y3b The same packingenlarged, by asimilaritytransformationc The same packingstretched, by a sheartransformation

d The same packingtransformed bychanging the spacingof the grid lines indifferent ratios

17 a A packing of shapes(compare Figure 13)within the triangulargrid of Figure 11b The same packingenlarged, by asimilaritytransformationc The same packingstretched, by a shear

transformation d The same packingtransformed bychanging thespacing of the gridlines in differentratios

18 Plan of the JonesHouse by BruceGoff, Bartlesville,Oklahoma 1958, laidout as a semi-regular tessellationof octagons andsquares

17a 17b

16c 16d 17c 17d

The flexibility of dimensioning offered by rectangular packingsThis demonstration however ignores one veryimportant aspect of flexibility in the possible packingsof shapes to fill the plane. The wider aim of thecomputer work mentioned above was to enumeratepossibilities for packing rectangles of any dimensionswhatsoever – not just rectangular shapes made byaggregating square cells of some given unit size.

Consider the packing of shapes on the square gridin Figure 16a. Here the arrangement is set on asystem of x, y coordinates, as shown. The spacing ofthe grid lines in x and y is given by a series ofdimensions x1, x2, x3, y1, y2, y3. In the packing of unitsquares as shown in the figure these dimensions areof course all equal [16a]. Suppose, however, that the xand y values are changed. If they are all multiplied, ordivided, in the same ratio, then the packing as awhole will simply be enlarged or shrunk in size, butwill remain otherwise unchanged [16b]. (Inmathematical terms this is a similarity transformation.)If the x values are all multiplied in the same ratio,while the y values are all multiplied in a differentratio, the arrangement will be stretched or shrunk inone direction or the other [16c]. (This is a sheartransformation.) Finally, individual x and y values canbe altered in different ratios, so as to cause localshrinking or stretching of different componentrectangles [16d]. All the shapes in the packingnevertheless remain rectangular throughout.

In this way any dimensioned version whatever of thebasic arrangement of Figure 16a can in principle begenerated. These versions are infinitely numerous,since any of the x and y grid dimensions can bealtered by increments that can be as small as we wish.The same process of assigning dimensions can becarried out for all the different arrangements on 3 x3 square grids in Figure 14; and indeed for allarrangements on 1 x 3, or 2 x 2, or 2 x 3, or 3 x 4 gridsand so on. Such an approach makes it possible togenerate absolutely any packing of rectangles, ofwhatever dimensions, within a simple rectangularboundary. This was the goal of the computerresearch referred to earlier (see note 6).

Now imagine a similar process of dimensioningbeing applied to the grid of nine equilateraltriangles [17a] (or to the grid of nine unit hexagons).The entire pattern can be subjected to a similaritytransformation, enlarging or reducing it in size,without difficulty [17b]. But any overall sheartransformation results in changes in the internalangles of all the triangles [17c]. And any change in thespacing of some of the grid lines results in changes inthe internal angles of the triangles between thoselines [17d] – indeed some ‘grid lines’ now becomebent, and the very idea of a grid ceases to apply. Ingiving different dimensions to the grid spacing insquare grids, we generate shapes that are alwaysrectangles. In giving different dimensions to thespacings of lines in grids of equilateral triangles, bycontrast, we generate shapes that are alwaystriangles, certainly, but they are no longerequilateral triangles. A similar argument applies tothe hexagonal grid.

Here, I would propose, we are approaching theheart of the issue, the key reason for the superiorflexibility of rectangular packing over other shapesthat fill the plane. That flexibility lies in part in thevariety of possibilities for configurations of rectangles,irrespective of their sizes; but much more, in theflexibility of assigning different dimensions to thoseconfigurations, while preserving their rectangularity. Anyrectangular packing can be dimensioned as desiredin the general way illustrated in Figure 16 – inprinciple in an infinity of ways – and the componentshapes will all still be rectangles.

Looking at this flexibility from another point ofview: it is always possible to divide any rectanglewithin a packing into two rectangles, and to divideeach of these rectangles into two further rectangles,and so on. In the context of plan layout, the designercan decide to turn one proposed rectangular roominto two. More generally, he or she can squeezegroups of rectangular spaces together, or can pullthem apart and slide others in between. In buildingsonce constructed, a new partition wall can divide anyrectangular room into two rectangular parts. (It isequally possible to divide any triangle into twotriangles – but those component triangles will havedifferent internal angles from the first.)

I have confined discussion so far to packings ofregular figures – squares, equilateral triangles,hexagons – and how these may be dimensioned.Beyond these, there are packings of other shapes,notably the semi-regular or Archimedeantessellations, which are made up of combinations ofdifferent regular figures. Some of these have servedon occasion as the basis for architectural plans, as forexample Bruce Goff’s Jones House of 1958, whoserooms are alternate squares and octagons [18]. Andthere is an infinite variety of irregular shapes thatcan fill the plane, examples of which form the basisof many of M. C. Escher’s irritating puzzle pictures.But the argument about inflexibility ofdimensioning applies, I would suggest, with evengreater force in all these cases. Goff’s planning of theJones House is ingenious. But he picked ageometrical straitjacket for himself, and it is notsurprising that few others have followed his lead.

Close packing of components in buildings and otherartefactsUp to this point we have looked only at the problemsof arranging rooms in architectural plans. But thesolid structure of any building – unless it is wholly ofmud or concrete – itself consists of packings of three-dimensional components at a smaller scale: bricks,beams, door and window frames, floorboards, floortiles. Here too, rectangularity generally prevails, forthe same geometrical reasons, I would suggest, asalready outlined. (And even concrete needsformwork, often assembled from rectangularmembers.) Rectangular rooms can readily be formedout of rectangular components of construction. Thisrectangularity of building components was Bemis’sconcern: to what extent would it be possible toconstruct houses from rectangular pre-made partsthat would be larger than standard bricks or

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timbers, but could still be fitted together in manyways? In traditional brickwork, it is where walls meetat angles other than 90º that there is a need fordifferently shaped, hence more expensive ‘specials’.Otherwise, so long as brick-based modulardimensions are generally adhered to, the standardbrick will serve throughout. Pieces of timber in theform of parallelepipeds can always be cut intosmaller paralellepipeds, and rectangular sheets ofboard into smaller rectangles. Building componentsmust pack vertically as well as horizontally, so here isanother reason besides gravity for the appearance ofrectangularity in the vertical direction.

We find rectangularity in other types of artefact,for essentially the same reasons, I believe, offlexibility in the assembly or subdivision of parts.Woven cloth, with its weft and warp, is produced inrectangles because of the basic technology of theloom. Many types of traditional garment – shirts,ponchos, trousers, coats, kimonos – are then sewnfrom square or rectangular pieces cut from theselarger sheets, so as not to waste any of the valuableand laboriously made cloth.7 Such garments, whenold, are picked apart and the pieces reused for otherpurposes. Figure 19 shows a nineteenth-centuryagricultural labourer’s smock from Sussex, with thepattern of rectangular pieces from which it isassembled [19].

Paper too is manufactured in rectangles that canbe cut in many ways without waste to create smallersheets of different sizes. The rectangularity of thepaper fits with the rectangularity of pages of printedtype. In traditional printing methods each letter andspace was represented by a separate rectangular slugof metal, the whole assemblage of letters – possiblyof many different sizes – clamped together in a

rectangular frame or ‘chase’ [20]. The art ofnewspaper and magazine layout lies in the fittingtogether of differently sized pictures, headlines andblocks of type, to fill each page. Much furniture isbuilt of course from essentially rectangular woodencomponents, and the furniture’s rectangularityallows it to fit in the corners of rectangular rooms. Inthe denser parts of cities, complete rectangularbuildings are packed close together on sites that arethemselves rectangular; and these sites pack to fillthe complete area of city blocks. This was part of thereason for Krüger’s interest in plan shapes.

A transition from round to rectangular in vernacular houses?We have noticed how circularity in plan is often acharacteristic of freestanding, widely spaced, single-room houses in pre-industrial societies. (The circular plan may derive in part from a systemof construction where the roof is supported on acentral pole, or forms a self-supporting cone ordome.) One might imagine that with increasingwealth there could be a change, at some point intime, from single-room to multi-room houses.Another possibility is that in circumstances whereland was in short supply – as for example where itwas necessary to confine many houses within adefensive perimeter – then those houses might have needed to be packed tightly together. In bothcases the theory of rectangular packing proposedhere might lead you to expect a transition intraditional houses from circular to rectangular plan shapes. Can you find actual evidence of suchchanges?

The American archaeologist Kent Flannery made acomparative analysis of the forms of villages and

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19

20

cut from a secondrectangle of clothwithout waste.(From D. K.Burnham, Cut MyCote (1973), Figure12, p. 18)

20 Rectangular metalslugs carryingletters or acting asspacers, clamped

together into arectangular frameor ‘chase’, as used intraditional printingmethods. (From P.Gaskell, A NewIntroduction toBibliography (1974),Figure 43, reprintedby permission ofOxford UniversityPress)

19 Man’s smock,probably fromSussex, 1860-80;and the patternfrom which it wasmade. Onerectangular piece ofcloth serves for thebody of thegarment. All otherpieces are alsorectangular and are

village houses in the period when permanentsettlements first appeared after the end of thePleistocene.8 He looked at examples in the Near East,Africa, the Andes and Mesoamerica. He found twobroad types of settlement: compounds consisting ofsmall circular huts, and ‘true villages’ with largerrectangular houses. The round hut wascharacteristic of nomadic or semi-nomadiccommunities, and consisted of a single space,housing one or at most two people. The rectangularhouse typically had several rooms, accommodatedan entire family, tended to be extended over time,and was found in fully sedentary communities. Insome cases these rectangular houses were indeedconcentrated together for the purposes of defence. InFlannery’s own words:

‘Rectangular structures replace circular ones throughtime in many archaeological areas around the world(although reversals of this trend occur).’ 9

Figure 21 shows this process in action. Thephotograph is reproduced from Bernard Rudofsky’sArchitecture Without Architects, and shows an aerialview of Logone-Birni in the Cameroun.10 There arefreestanding circular huts with roofs both inside andoutside the walled compounds. But the contiguousroofed structures within these compounds –together with some roofless enclosures that arepacked closely with those buildings – are for themost part rectangular [21].

Other examples can be seen in the vernaculararchitecture of Europe. The typical hut or borie of the Vaucluse region of France is a stone-builtcylindrical one-room structure with a corbelleddomical stone roof.11 There are a few existing multi-room rectangular bories. In the example of Figure 22,the room layout seems to be in some sort of

transitional stage between a squashing together of circles, and an emerging rectangularity [22].12

The trullo of Apulia has a stone structure similar to the borie. According to Fauzia Farneti the trullowas originally a temporary dwelling with a circularplan found in rural areas, and later evolved to have a rectangular exterior and a circular interior.13

These rectangular trulli, in their final form, became the repeated structural units of multi-roomhouses.

A parting shotFinally, why might we expect to find moredepartures from rectangularity in the work of ‘higharchitects’ than in the general run of more everydaybuildings? Many contemporary architects, it seemsto me, find the rectangular discipline imposed by thenecessary constraints of the close packing of rooms –paradoxically, and despite its flexibility – to be anirksome prison; and they try to escape from it. Theygravitate towards building types such as artmuseums and theatres, not just because these areprestigious and well-funded cultural projects withimaginative clients, but also because they canprovide opportunities for spaces that are close tobeing ‘single-room structures’, which can be treatedsculpturally. It is possible that architects mightchoose to adopt a non-orthogonal geometry inorder, precisely, to set their work apart from themajority of the building stock. Rectangularity can beavoided on the external surfaces of buildings as wehave seen: so there is much free play here forarchitectural articulation and elaboration of a non-orthogonal character. But this treatment comes at acost, and in more utilitarian buildings it may bedispensed with.

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21 Aerial view ofLogone-Birni in theCameroun (from B.Rudofsky,ArchitectureWithout Architects(1964), plate 132, p.131). Notice thefreestanding circularhuts, and the

rectangular huts andrectangularunroofed enclosurespacked within thecompounds

22 Plan of a stone borieat Lacoste in theVaucluse region ofFrance, with seven

spaces (from Bories(1994), p. 177). Theplan appears torepresent atransitional stagebetween the circularsingle-room borie,and a packing ofseveral rectangularspaces

Museum of Modern Art, New York /Marcel Griaule, 21 (from B.Rudofsky, Architecture WithoutArchitects (New York: Museum ofModern Art, 1964), plate 132, p. 131)

Oxford University Press, 20, (from P.Gaskell, A New Introduction toBibliography, (Oxford: ClarendonPress, 1972), fig. 43: ‘An Octavoforme on the imposing stone’)

Parc Naturel Régional du Luberon etEdisud, 22 (from Bories (Aix-en-Provence: Parc Naturel Régional duLuberon et Edisud, 1994), p. 177)

Royal Ontario Museum, 19 (from D. K.Burnham, Cut My Cote (Toronto:Textile Department, Royal OntarioMuseum, 1973), fig. 12, p. 18)

Shelter Publications, 2 a,b,c (fromShelter (Bolinas, CA: ShelterPublications, 1973), p. 16, p. 18 andp.27)

<www.sterling.rmplc.co.uk/visions/decks.html> [accessed January2006], 6

AcknowledgementsHypothesis 3 is due to Joe Rooney. BillHillier took me to see the borie ofFigure 22, and told me about thepaper by Kent Flannery. Jini Williamschecked my enumerations in Figures14 and 15 and found many mistakes,now corrected. Mary Harris andPenelope Woolfitt sent meinformation about the rectangulargeometry of ethnic clothing,including xeroxed pages fromBurnham (1973). Philip Tabor, AdrianForty, Sonit Bafna and an anonymousreviewer contributed ideas,information and suggestions. I amgrateful to them all.

BiographyPhilip Steadman is Professor of Urban and Built Form Studies at theBartlett School, University CollegeLondon. He trained as an architect atCambridge University and taughtpreviously at Cambridge and theOpen University. He has publishedtwo books on the forms of buildings:The Geometry of Environment (withLionel March, 1971) and ArchitecturalMorphology (1983). He is working on anew book on Building Types and BuiltForms (2020 approx).

Author’s addressProf. Philip SteadmanBartlett School of Graduate Studies1–19 Torrington PlaceUniversity College LondonLondon wc1e [email protected]

Notes1. Albert Farwell Bemis and John

Burchard, The Evolving House, 3 vols.,(Cambridge, MA: MIT, TechnologyPress, 1933-36).

2. Ibid., vol. 3, Appendix A, pp. 303-316. Bemis extrapolated these datato the entire housing stock of theUnited States, based on thefrequency of different house types,to give a figure of 88.5%rectangularity. He also measuredthe rectangularity of thestructural components of a sampledetached house with hipped roof,bow windows and dormers. Hefound that 97% of thosecomponents by number wererectangular.

3. Mario J. T. Krüger, ‘An approach tobuilt-form connectivity at anurban scale: system descriptionand its representation’,Environment and Planning B, vol. 6(1979), 67-88.

4. This proposal comes from JoeRooney (personalcommunication).

5. Cecil J. Bloch, A Formal Catalogue ofSmall Rectangular Plans: Generation,Enumeration and Classification (PhDthesis, University of CambridgeSchool of Architecture, 1979).Bloch enumerated packings ofrectangles on ‘minimal gratings’ –that is to say where all grid linescoincide with the edges ofrectangles. The number of distinctpackings on 3 x 3 minimal gratingsgiven by Bloch is 33. To these wemust add – for the presentenumeration – packings thatcorrespond to 2 x 2 and 2 x 3minimal gratings, suitablydimensioned to fill the 3 x 3 grid.Bloch did not count packings inwhich the number of rectanglesequals the number of cells in thegrating. These (on 1 x 1, 1 x 2, 1 x 3,2 x 3 and 3 x 3 gratings) are alsoincluded here, appropriatelydimensioned to the 3 x 3 grid ineach case.

6. William J. Mitchell, J. PhilipSteadman and Robin S. Liggett,‘Synthesis and optimisation ofsmall rectangular floor plans’,Environment and Planning B, vol. 3(1976), 37-70; Christopher F. Earl, ‘Anote on the generation ofrectangular dissections’, inEnvironment and Planning B, vol. 4(1977), 241-246; RameshKrishnamurti and Peter H. O’N.Roe, ‘Algorithmic aspects of plangeneration and enumeration’,Environment and Planning B, vol. 5(1978), 157-177; Cecil J. Bloch and

Ramesh Krishnamurti, ‘Thecounting of rectangulardissections’, Environment andPlanning B, vol. 5 (1978), 207-214;Ulrich Flemming, ‘Wallrepresentations of rectangulardissections and their use inautomated space allocation’,Environment and Planning B, vol. 5(1978), 215-232; Cecil J. Bloch, ‘Analgorithm for the exhaustiveenumeration of rectangulardissections’, Transactions of theMartin Centre for Architectural andUrban Studies, vol. 3 (1978), 5-34. Seealso the general discussion in J.Philip Steadman, ArchitecturalMorphology (London: Pion, 1979).

7. Dorothy K. Burnham, Cut My Cote(Toronto, Canada: TextileDepartment, Royal OntarioMuseum, 1973).

8. Kent V. Flannery, ‘The origins ofthe village as a settlement type inMesoamerica and the Near East: acomparative study’, in Man,Settlement and Urbanism, ed. by P. J.Ucko, R. Tringham and G. W.Dimbleby (London: Duckworth,1972), pp. 23-53.

9. Ibid., pp. 29-30.10.Bernard Rudofsky, Architecture

Without Architects (New York:Museum of Modern Art, 1964.Reprinted, London: AcademyEditions, 1973).

11.Pierre Desaulle, Les Bories deVaucluse: Région de Bonnieux (Paris: A.et J. Picard, 1965).

12.I am grateful to Bill Hillier fordrawing my attention to bories, andfor taking me to see the verybuilding of Figure 22. It should besaid however that a minority offreestanding single-room borieshave rectangular plans; so thisinterpretation could be open tochallenge.

13. Encyclopedia of VernacularArchitecture of the World, ed. by PaulOliver (Cambridge: CambridgeUniversity Press, 3 vols., 1997), p.1568.

Illustration creditsarq gratefully acknowledges:L’Architecture d’Aujourd’hui, 5 (from

L’Architecture d’Aujourd’hui, 130 (1967)pp. 92–93)

Author, 11, 12, 13, 14, 15, 16, 17Elsevier, 4 a,b,c (from R. Ham, Theatre

Planning (London: ArchitecturalPress, 1972), p. 256 and p. 267)

Guggenheim Museum Publications, 7(from C. van Bruggen, GuggenheimMuseum Bilbao (New York:Guggenheim MuseumPublications, 1998), p. 151)

arq . vol 10 . no 2 . 2006 theory130


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