Theories approaching architecture from the point of view of its quantifiable properties often assum e that its intellectual basis is mathematical and geom etrical. W ittkow er's conviction that the M odulor is founded on an urge for intelligibility or Sartoris’ description of a building as a recognisable geometrical shape express a belief that architecture appeals to the intellect through a numerical or geometrical order.
Theories o f m athem atical order originate from P ythogoras’ discovery o f the relationship betw een numerical ratios and musical consonance^^. Theories of geometrical order are based on Plato’s ideas of the relation of regular solids to natural elem ents^^. The m ost influential text these theories use is Vitruvius’ ‘Ten Books of Architecture’^^ . Vitruvius defines proportions using the term sym m etria as ‘a proper agreement between the members of the work itself, and relation between the different parts and the whole general scheme, in accordance with a certain part selected as standard’
The Renaissance theorists and artist sought quality in architectural form in a manner sim ilar to Vitruvius. Alberti suggested that natural excellence resides in the form of a building in a way that ‘it excites the mind and is immediately recognised by it’. He defined beauty as ‘...that reasoned harmony o f all the parts within a body, so that nothing may be added, taken away, or altered but for the worse. ’ He also suggested that beauty is ‘... a form of sympathy and consonance of the parts within the body, according
P y th a g o ra s d is c o v e re d th a t th e th re e sim p le c o n c o n a n c e s , th e o c ta v e , th e fifth a n d th e fo u rth a n d th e tw o co m p o u n d c o n so n a n ce s, the d o u b le o ctav e an d th e o c ta v e p lu s a fifth , a re a rith m e tic a lly e x p re s s e d b y th e ra tio s o f th e fo u r in teg ers, 1 : 2 : 3 ; 4, R u d o lf W ittk o w e r. ‘A rc h ite ctu ra l P rin c ip le s in th e A g e o f H u m a n is m ’. A c ad e m y E d itio n s 1988, p. 148.
P lato a ssig n e d each o f th e basic fiv e so lid s, th e tetrah e d ro n , th e o c ta h e d ro n , th e h e x a h e d ro n , th e d o d e c a h e d ro n a n d th e ic o s a h e d ro n to e ac h o f th e fiv e e le m e n ts , i.e. th e c u b e to th e e a rth , th e te tr a h e d r o n to fire , th e o c ta h ed ro n to a ir th e ico sah ed ro n to w ater, an d th e d o d e ca h ed ro n to th e e n c lo s in g sky.
V itru v iu s . ‘T h e T e n B o o k s o f A rc h ite c tu re ’, trans. b y M ick y M o rg an , D o v e r P u b lic a tio n s , N e w Y o rk 1960. ^ Ibid. B o o k I, p. 14. T h e m odern v ersio n o f the w o rd sy m m etry re fe rs to b ila te ra l sy m m etry in w h ic h a fo rm has
id en tica l sid es o r p a rts on each sid e o f an axis. V itru v iu s ’ symmetria c o m e s fro m th e G re e k w o rd analogia
m ea n in g p ro p o rtio n . V itru v iu s d e fin e s sy m m etria in his th ird b o o k a lso , (th e d e fin itio n s ta te d in th e tex t c o m e s from th e first b o o k ), as the ‘co rre sp o n d e n c e am o n g th e m easu res o f th e m em b e rs o f an e n tire w o rk , an d o f th e w h o le to a certain part selected as sta n d a rd ’, ibid., B o o k 111, p. 73.
^ ^ * L eon B a ttista A lberti 'O n the A rt o f B u ild in g in T en B o o k s’, trans. by J. R y k w e rt, N. L ea ch an d R. T av e rn o r,
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to definite number, outline, and position, as dictated by concinnitas^^^, the absolute and fundam ental rule in Nature’ 103. Similarly to Alberti, Palladio defined beauty also as depending on proportions :
‘Beauty will result from the beautiful form and from the correspondence of the whole to the parts, o f the parts amongst themselves, and of these again to the whole; so that the structure may appear an entire and com plete body, wherein each m em ber agrees with the other and all m em bers are necessary for the accomplishment of the building’
The notion of proportional form was complemented by a notion o f geometrical form l^^. Alberti defined geometrical form using the term ‘a rea ’^^^ as an entity enclosed by a perim eter constructed by lines and anglesl07. Using examples like the earth, the stars the animals and their nests he proposed that ‘Nature delights prim arily in the circle’ This shape helps to generate the square, the hexagonon, the octagonon, the decagonon and the dodecagonon.
Artists like Leonardo Da Vinci, Francesco Di Giorgio and Serlio explored geometrically ordered layouts. They produced a series of drawings ranging from simple to complex configurations. These were translated to circular, square, hexagonal or octagonal plans of churches with small chapels, (illustration 1.31). As W ittkow er suggests, the notion of commensurable number found perfect geometric expression in these layouts where every point in the circumference has the same relationship to the centre
Concinnitas is th e su ccessfu l co m b in a tio n o f n u m b er, m ea su re an d fo rm , (numerous, finitio, and collocatio),
L eo n B a ttista A lb e r ti. Ib id ., G lo ssary , p. 4 24. L eo n B a ttista A lb e rti. Ibid., B ook IX , p. 303.
A n d re a P a lla d io . ‘T h e F o u r B ooks o f A rc h ite ctu re ’. T rans, b y Isaak W are. N e w Y ork: D o v er, 1965, B o o k I. R u d o lf W ittk o w e r sug g ests th at th eo ries o f p ro p o rtio n s and g e o m e try are d e m o n s tra tio n s o f tw o d iffe re n t k in d s o f m a th e m a tic a l c o n ce p ts. T h e fo rm e r a re c o n c e p ts b a se d o n n u m b e rs , w h ile th e la tte r a re b a se d o n b a s ic g e o m e tric a l shapes. T h ese can n o t b e e asily c o n v erted to a rith m e tic a l ra tio s b e c a u se th ey a re in c o m m en s u rab le. T h e first o n e w as fav o u red d u rin g th e R e n aissan c e, w h ile th e se c o n d o n e d u rin g th e m id d le ag es. T h e p re fe re n ce th e R e n a is s a n c e a rch ite c ts sh o w ed in c o m m e n su ra b le ra tio s w a s b a s e d o n a n o rg a n ic a p p ro a c h to n a tu re in w h ic h ev ery th in g w as related to ev ery th in g e lse by num ber. R u d o lf W ittk o w e r. L e C o rb u s ie r’s M o d u lo r’. in th e b o o k . In th e F o o tstep s o f L e C o rb u s ie r’. R izz io li In te rn a tio n a l P u b lic a tio n s ’, 1991, p. 13.
L e o n B a ttis ta A lb e r ti. Ibid., B ook I, p. 20.
1 0 7 R y k w e rt, L each and T av e rn o r su g g est th at area refers to th e n o tio n o f th e arch ite c tu ra l plan. A s such it could b e se e n as a p p ro x im a tin g the n o tio n o f g e o m e tric a l fo rm b e c a u s e a r c h ite c tu r a l p la n s a re tw o d im e n s io n a l p ro je c tio n s o f g eo m etrical e n tities. Ibid., G lo ss ary p. 420.
Ib id ., B o o k seven, p. 196.
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These theories em phasised the notion o f bilateral sym m etry also. A lberti w rote that the parts o f a building should be disposed ‘with an exact correspondence as to the number, form and appearance so that the right may answ er to the left, the height to the low, the sim ilar to the sim ilar...’ ^ Palladio also sought symmetry in the organisation of the building as the m ost memorable form o f order.
Theories of proportions and geometry followed V itruvius’ axiomatic idea that architecture m ust m irror the human figure. A unit of measurem ent such as the head o f the human body was transposed to the diam eter o f the colum n in a building establishing the m etrical relationships am ongst the parts and am ongst the parts and the w hole^^k The geometrical analogy between the human body and architecture was used again in a series of drawings and writings exploring the relations am ongst the various parts of the body and the church (illustration 1.32).
Using mathematics and geometry these theories sought the laws that intervene between nature and acts of creation like music and architecture. They often attempted combined interpretations about the m icrocosm and the m acrocosm , nature and the man m ade structures under the sam e m athem atical form ula. As Barbaro suggested mathematics became the link between the ‘certain truths’ science is concerned with and the ‘uncertain truths’ of the arts^
W ittkow er observing the correspondence between num bers, geom etrical form , architecture, m usic, religious ideas and cosmology in the Renaissance theories suggests that com m ensurable proportions and the notion o f the circle symbolised underlying ideas o f cosmic order and harmony^ ‘W e maintain, in other words, that the forms o f the Renaissance church have symbolical value or, at least, that they are charged with a particular meaning that the pure forms as such do not contain’ ^
Texts like those o f V itruvius, A lberti Palladio and Barbaro seem to show that W ittkow er is right suggesting that the Renaissance theorists were m anipulating certain m athem atical m odels in different
1 1 0 Ib id ., b o o k V I, p. 156
^ ^ ^ W ittk o w e r re fe rs to L e o n a rd o ’s stu d ies o f th e h u m an fig u re as w ell as to th e stu d ies o f a d e ta il o f th e b a sis o f a c o lu m n b y P ie ro d e lla F ra n c esc a ., R u d o lf W ittk o w e r. Ib id ., p. 153.
^ R u d o lf W ittk o w e r. Ib id ., p. 65.
^ ^ ^ ‘W ith th e R e n a is s a n c e re v iv a l o f th e G re e k m a th e m a tic a l in te r p r e ta tio n o f th e G o d a n d th e w o rld an d in v ig o ra te d by th e C h ristia n b e lie f th at M an as th e im a g e o f G o d e m b o d ie s th e h a rm o n ies o f th e U n iv e rse, th e V itru v ian fig u re in scrib ed in a sq u a re and a c ircle b e ca m e the sy m b o l o f the m ath e m a tic al s y m p a th y b e tw ee n the m icro co sm and the m acro co sm . H ow could the relatio n o f M an to G od b e b etter e x p ress ed , w e feel now ju s tifie d in ask in g , than by b u ild in g the ho u se o f G o d in acc o rd an c e w ith th e fu n d am en tal g e o m e try o f th e sq u a re and the c irc le ? ’, R u d o lf W ittk o w e r . Ibid., p. 25.
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areas of knowledge. However, W ittkow er’s approach to these m odels seem s an over-sim plification of both architectural form and the philosophical and religious ideas o f that period^
W ittkow er’s simplification of architectural form is based on its reduction to a simple geom etrical shape and to com m ensurable number^ An illustration o f R enaissance plans provided by Paul Frankl^^^ shows that even the most diagrammatic drawings of this period are not about a simple geom etrical shape like a circle, (illustration 1.33). They seem to be about a geom etrical structure o f shapes the centres of which are co-ordinated by the axes covering the geometrical centre of the largest shape in the middle.
F ran k l’s studies o f the centralised churches show that the developm ent o f geom etrical form was com plem ented by a developm ent o f circulation systems. Som e o f these system s favoured a circular m ovem ent passing through the ancillary spaces apart from the m ovem ent crossing the central space. Others eliminated peripheral movement so that one returns to the focal space in order to m ove from one ancillary space to the other^
In a reaction to W ittkow er’s consideration of centralised churches as sim ple circles or centres, Robin Evans identifies multiple centres along the vertical direction in Sant* Eligio Orefici, (illustration 1.34). Similarly to Frankl who looks at the ways the centres o f the ancillary spaces are visited, Evans looks at the ways the vertically arranged centres are experienced by an observer inside the church. He suggests that, from a geometrical point o f view, certain centres are strong indicators o f centrality. However, from the point o f view of the observer they present an ambiguity concerning their location in height as well as their actual presence.
W ittk o w e r 's d e s c rip tio n sh ifts fro m m a th e m a tic s an d g e o m e try to m u s ic , to a rc h ite c tu re , to re lig io n an d