Lionel MarchProfessor of Design and Computation Center for Medieval and Renaissance Studies University of California Los Angeles CA 90095-1615 USA
Thomas Gray, 'Ode on the Death of a Favourite Cat'
I dieci libri dell'archittetura di M. Vitruvio Pollionis traduitti
et commentati ... by Daniele Barbaro was published by Francesco
Marcolini in Venice and the collaboration of Palladio acknowledged.
In the later Latin edition [Barbaro 1567], there are geometrical
diagrams of the equilateral triangle, square and hexagon which
evoke ratios involving
and , but there are
no drawings of pentagons, or decagons, which might explicitly
alert the perceptive reader to the extreme and mean proportion,Architectural examples employing 2 and 3 include the Roman theater and Greek theater, respectively. A figure designed to illustrate Vitruvius's written description of a peripeteral circular temple shows one with columns spaced at 20 equal points around a circumference. Another figure shows arrangements for tetrastyle and hexastyle porticoes and, here following Vitruvius, a 20-gon sets out the position of the flutes around a column's cross-section. In both these examples, a pentagon, or decagon, will have been used in the geometric construction. Elsewhere, a pentagonal bastion is illustrated, but this plan is definitely not based on an equilateral pentagon. In the archaeological Book IV of How would Barbaro—and perhaps his illustrator, Palladio—have constructed a pentagon or decagon? In the mid-fifteenth century, Alberti had described in words an exact construction for the decagon.[2] Albrecht Dürer, 1525, illustrates two distinct constructions for the pentagon, one according to geometric theory, and another traditionally used by masons and craftsmen which is only approximate [Dürer 1977: 144-147]. By the 1540s, Serlio shows Dürer's exact construction [Serlio 1996: 29]; yet as late as 1569, Barbaro shows only Dürer's approximate construction [Barbaro 1569: 27]. Whereas the exact construction leads to the extreme and mean ratio, the approximate construction does not. Someone seriously aware of the relationship of the extreme and mean ratio to the pentagon, or decagon, would surely use the exact method, especially if that relationship was seen to have aesthetic value. But there really is no evidence that any of these authors had strong commitments to the extreme and mean ratio for aesthetic purposes. While Luca Pacioli enthuses over the extreme
and mean ratio in the first book of The one errant work is the Villa Mocenigo at Marocco, built
at the same time as Villa Emo, but since destroyed [Palladio
1997: 55]. It has four 13:8 rooms, four 8:5 rooms. It also has
four, square rooms and a square atrium with four columns, 1:1.
The atrium is part of a large double square space, 2:1, containing
the grand stairs. The remaining part of this double square space,
between the entrance loggia and the stairs, is proportioned 8:5.
Palladio ranges the lengths of two rooms, 10 Palladio does use ratios which better converge towards the finitely unreachable extreme and mean ratio. These lie between the underestimate 8:5 and the overestimate 5:3 . The ratio 13:8 (=1.625) is among these, but 21:13 is not one of them. Palladio uses the dimensions 26 ½ In a project for Count Barbarano [Palladio 1997: Book II,
22], the vaulted entrance has dimensions 41 ½ by 25 In his reconstruction of a private house for the ancient Romans
[Palladio 1997: Book II, 35], the atrium is shown with dimensions
83 1/3 by 50 The ratio 28:17 (H1.647) is found in the two largest rooms in the Palazzo Antonini [Palladio 1997: Book II, 5]. The ratios of consecutive terms in the Nicomachus X sequence converge, as do all such ratios, on the extreme and mean ratio in the long run: In a previous analysis [March 1998: 236-239], it has been
suggested that the proportional design of this building is an
occult play on Plato's Timaean theme of world-making elements:
the equilateral triangle related to the faces of the tetrahedron
(fire), the octahedron (air), and the icosahedron (water); the
square related to the faces of the cube (earth); and the equilateral
pentagon to the faces of the decahedron (cosmos). The large rooms
are proportioned by the equilateral pentagon : the width to the
side, 17 The rooms that lie behind these great rooms
are proportioned quadrato, or where "the length will equal
the diagonal of the square". The ground-floor hall is 32
To arrive at such proportional design, it seems that Palladio would have made use of rational estimates for square roots of non-square numbers, such as 2, 3 and 5. There were several techniques for computing the numerical values at the time, but once such computations were made it would probably have been convenient to look them up in tables, or simply to remember at least the most commonly used values. Typical values in the generative process which converge on these square roots are given in Table 1.[6] Note that the ratio 5:3 may stand for , and is not to be read uniquely as an early term in a Fibonacci approximation to .
Recall that the Fibonacci sequence was unknown as such during the Renaissance, and was most probably burrowed away as the solution to the rabbit breeding problem in some dusty, unstudied manuscript. How then do the numbers, 89, 144, 233, from the sequence F(2) 1, 2, 3, 5, 8,
13, 21, 34, 55, 89, 144, 233, ....arrive, as a handwritten note, on a copy of a 1509 edition of Euclid by Luca Pacioli [Herz-Fischler 1987: 157-158]? The value of was certainly known from Euclid as .[7] Substituting the rational convergents for , shown in Table 1, into this formula gives the answers set out in Table 2. It will be noted that the first row gives the usual Fibonacci sequence, F(2), but the second row gives ratios from the sequence F(3) 1, 3, 4, 7, 11,
18, 29, 47, 76, 123, 199, 322 ...
Earlier, the discussion of pentagonal proportional design
in two
This construction is no whim. It derives from an arithmetical
interpretation of Euclid, Proposition 10, Book XIII [Heath 1956:
455-457]. Unquestionably to be counted among the most aesthetically
pleasing of all the propositions in The Elements, Proposition
10 reads: "If an equilateral pentagon be inscribed in a
circle, the square on the side of the pentagon is equal to the
squares on the side of the hexagon and on that of the decagon
inscribed in the same circle" ( Let the side of the pentagon be whence, the side of the pentagon Using the defining relation and the value the expression for the side can be reduced to Computationally, this is what Euclid's proposition implies; and, without the advantages of modern algebraic notation, this is very much the kind of procedure that Piero della Francesca would have had to follow in his fifteenth-century programme for the arithmeticization of Euclidean geometry. How would such an expression, albeit in different notation, be evaluated? It would be necessary to substitute a rational value for . But what value? It would be convenient if the remaining square root after the substitution was of a square, or near-square, number. Scanning through Table 1, the values 9/4 and 20/9 show promise since the numerators are square numbers and their roots can be brought outside the main square root sign. The value 9/4 leads to but 22 is not a near-square number, whereas the value 20/9 gives and is very close to . Thus, a good rational solution is s : r :: 7: 6.( This is precisely the ratio that Palladio employs in his triangles
AEC and CBE. Now, each is seen to be the isosceles triangle on
the side of an equilateral pentagon with apex at the center of
the circumscribing circle. Referring to Table 2, it will be found
that an equilateral pentagon of side 3.7 = 21, will have a chord
length of 34, and proportionately will have a radius of 3.6 =
18. This example is typical of the wit required to find integral
values to fit the numerical irrationality of most geometrical
objects, especially before the arrival of decimal notation in
the seventeenth century ( The value 20/9 used here to arrive at Palladio's construction, gives added credence to the interpretation that the 20 x 9 rooms in Villa Barbaro at Maser were designed conceptually to the ratio , or the diagonal of a double square to its width [March 1998: 267-271]. Table 3 summarizes the proportional design at Maser.
The rooms and spaces now are seen to be a play on the theme
of the Pythagorean 3-4-5 triangle, but using root forms that
Alberti had advocated in the 1450s.[8] The appearance of the ratio 7:6 here
finds a different interpretation from the previous pentagonal
one. Now it is seen as the base of an equilateral triangle to
its height—a relationship that allows for the proportional
design of the Star of David to be placed at the crossing, without,
it should be noted, even the faintest whiff of today's political
connotations (
Figure 8).Essentially, the analysis plays on the well-known property
that when either a square is added to the short side of a golden
rectangle, or a square is deducted from a golden rectangle, the
new issue is itself a golden rectangle (
Figure
12)?A simple model that compares Rachel Fletcher's
analysis with Palladio's declared dimensions can be established
with two unknowns: If the wall thickness is an unknown must hold. This requires that the equation be true. The equation reduces to the parabola Set The solutions to this are and . These values do not correspond to the hypothesis that , the golden section. The first value falls short of the golden section value of 1.618... by almost 12%. The second solution is too way out even to contend. Suppose that the wall thickness is larger. Set This is worse than the previous result, and since the function is monotonic, any increase of wall thickness beyond 1 piede will never make things better. Try the assumption that Palladio has used centerline dimensions.
Set The solutions to this are and . These are still totally inadequate estimates for . What values of and set . The equation now reduces to the linear equation The solution gives ,
or a negative wall thickness of over 5
E(2, 3, 5) {2, 3, 9,
12, 15, 16, 20, 24, 27, 48}.The subset, E(2, 3) {2, 3, 9, 12,
16, 24, 27, 48},
The set is rich in classical proportionalities:[10]
are geometric proportionalities in which the ratio of the difference between the greater extreme and the mean and the difference between the mean and the lesser extreme to is equal to the ratio between the mean and the lesser extreme. The proportionality is arithmetic in which the difference between the mean and the lesser extreme is equal to the difference between the greater extreme and the mean. There are two harmonic proportionalities in which the ratio of the difference between the greater extreme and the mean to the difference between the mean and the lesser extreme is equal to the ratio of the two extremes,
There is also one example of the Nicomachus X mean:
where the difference of the greater extreme and lesser extreme is equal to the mean. This will be recognized as being in proportion to the first three terms in what is now known as the Fibonacci sequence, F(3). The full dimensional set for Villa Emo, E(2, 3, 5), can be
arranged on a three-dimensional version of the lambda ( In addition to the proportionalities in the first lambda,
are arithmetic proportionalities. The proportionality is harmonic. There are also five less familiar classical proportionalities in the full Villa Emo set, E(2, 3, 5). The proportionality
is Nicomachus IV, subcontrary to harmonic, in which the ratio of the difference between the greater extreme and the mean to the difference between the mean and the lesser mean is equal to the ratio of the lesser extreme to the greater. The proportionality is Nicomachus VII in which the ratio of the difference of extremes, 16 - 12 = 4, to the difference of the first two terms, 15 - 12 = 3, is in the same ratio as the extreme terms, 4:3. The proportionality is also a Nicomachus VII in which the ratio of the difference of extremes, 27 - 9 = 18, to the difference of the first two terms, 15 - 9 = 6, is in the same ratio as the extreme terms, 3:1. The proportionality
is Nicomachus X, which in modern terms comes from the Fibonacci sequence, F(2) 1, 2, 3, 5, 8,
13, 21, 34, ....The proportionalities
are Nicomachus X, which corresponds to the Fibonacci sequence, F(4) 1, 4, 5, 9, 14,
23, 37, 60, .... .Finally, the proportionality does not figure among Nicomachus's ten means, but is an instance
of Pappus 8 [11]
in which the difference of extremes, 20 - 15 = 5, to the difference
of the last two terms, 20 - 16 = 4, is in the same ratio as the
last terms, 20 Arithmetically, the Villa Emo set may be said to be rich in classical proportionalities. Such a set is the architect's proportional palette. He may not need all the colors, or even be aware of all the mixes, but Palladio, like his contemporary the music theorist Giosoffo Zarlino, undoubtedly appreciated the potentialities of a set based on the factors 2, 3, and 5. Its more recent architectural significance was investigated by Ezra Ehrenkrantz [1956] as a contribution to modular coordination for industrialized building.[12] The Pythagorean lambda is the base for Plato's theory of universal
harmony. Pythagorean harmony is entirely based on a cycle of
perfect fifths, 3:2. From the unison, 1:1, the fifth 3:2, is
reached by ascending the scale. The fifth beyond this is 3
On either side of the The most direct application of the 3-4-5 triangle is in the
16 x 12 Look again at the Villa Emo, and look at the colonnade in
front of the agricultural buildings ( Using just five surveyor's ropes, of lengths 9, 12, 15, 16,
20
Villa Emo "glisters" among Palladio's works., but it is not cloaked in the gold of the golden proportion. If the cloak doesn't fit, you must acquit. Palladio is not guilty. But there is plenty of guilt to spread around. The author of the paper which "saw" the golden section in Villa Emo is an innocent adherent of a morphological church that has flourished since the advent of Zeising's work [1854]. But she is no Selima, that "Presumptious Maid!". On the contrary, Rachel Fletcher has presented a diligent and exemplary study of its kind. Her misfortune, in casting a cloak of golden proportion over the Villa Emo, is that, unlike the quintessential studies of, say, M Borissavlievitch [1958], or R A Schwaller de Lubicz [1977], Palladio has given the actual measurements. In other studies, and in the absence of the architect's specification, the investigator is at liberty to choose where to take measurements and with what precision.: "With a little precision in taking measurements, it [the golden section] is easily found" [Schwaller de Lubicz 1977: 66]. But the cloak cannot be checked.[14] What is surprising is that a visually gratifying result is
so very wrong when tested by the numbers. It suggests that there
is enormous opportunity for visual error in the search for the
golden 'whatever', an error that computation exposes ruthlessly.
Perhaps, the most alarming consequence of obedience to this morphological
faith is that the extraordinary inventiveness, creativity, wit
and playfulness of homo faber is analyzed into some ideal, universal
system, Palladio had no system of proportion. He was a mannerist. Rules were there to be challenged, to be transformed, to surprise in their unexpected application, or unforeseen consequence. In the process of design, as the dimensions of a work gather around the physical and geometric possibilities and constraints, the designer discerns familiar patterns and potential interpretations. For a humanist during the Renaissance, these might include Plato's Timaean myth, the classical orders of number taxonomy, Euclidean geometry, music theory, cosmology, or just plain, practical expediency. It can be assumed that Palladio's work is executed in a polysemic language, foreign to modern eyes: enrichingly ambiguous, despite its enticing presentational lucidity . Before Selima's eyes "betray'd a golden gleam", "She saw: and purr'd applause". Look. Palladio cannot be seen through prescription glasses.
[2] Alberti gives a written description of the exact
method for drawing an equilateral decagon. See [Alberti 1988:
196]. [3] See Thompson [1942: 912-933 ('On Leaf-Arrangement,
or Phyllotaxis')]. Note particularly: "One irrational angle
is as good as another: there is no special merit in any of them,
not even in the ratio divina", p. 933. [4] 1:1, unison; 2:1, diapason; 3:2, diapente; 4:3,
diatesseron; 5:3, major hexad; 8:5, minor hexad [Zarlino 1558]
. Palesca [1985: 235-244] points out that Lodovico Fogliano (Musica
Theoria, Venice, 1529) had already established the musical provenance
of these ratios. [5] A [6] See March [1998: 65-69 ('Inexpressible Proportion')].
[7] "If a straight line be cut in extreme and mean
ratio, the square on the lesser segment added to half the square
of the greater is five times the square on half of the greater
segment" [Heath 1956, 3: Bk. XIII, Prop. 3, 445-447]. [8] Alberti [1988: 307] writes: "In establishing
dimensions, there are certain natural relationships that cannot
be defined as numbers, but that may be obtained through roots
and powers". See [March 1999c]. [9] For a generalization of this, see [March 1999a]
and [March 1999b]. [10] Nicomachus [1938: 264-284] enumerates ten proportionalities.
H L Heath [1921, I: 84-89] summarises these results. See also
[March 1998: 72-77 ('Proportionality')]. [11] Heath [1921, I: 87] shows that Nicomachus missed
this additional mean, but that Pappus had recorded it as his
eighth mean. [12] Note also Appendix 3 by F. St. J. Hetherton on the
musical analogy, pp. 72-74.
[14] Even when measurement can be replaced by plain counting,
it may be unwise to implicate section d'or. O. A. W. Dilke [1987]
asks: "is it only coincidence" that, in Polycletus's
theater at Epidaurus (c350 BCE), the seats below and above the
diazoma count 34 and 21 respectively to make 55 rows in all?
He then relates this to the Fibonacci sequence and thence to
the Golden Number. But there is a simple classical argument for
this arrangement. 55 is the tenth trigonal number, and the decad
is the root of all numbering. In classical Greek 55 is represented
by en, whose pythmen is 5+5=10, and which spells the word One,
the divine. The theater is located at a sanctuary and this dedication
to the One is surely appropriate. The number 21 is the sixth
trigonal number; but 6 is a perfect number, and so important
to the ancient Greeks that it has its own non-alphabetical character
- digamma. Setting out the decad 1, 2, 3, 4, 5, 6, and 7, 8,
9, 10, it will be seen that the sum is 55, the first six sum
to 21, and the remaining trigonal gnomons count to 34. [15] "... all such speculations as these hark back
to a school of mystical idealism" [Thompson 1942: 933].
Art Bulletin 31: 211-263.Ackerman, James S. 1991. Alberti, Leon Battista. 1988. Barbaro, Daniele. 1567. Barbaro, Daniele. 1569. Borissavlievitch, M. 1958. Cornford, F.M. 1952. Davis, Margaret Daly. 1977. Dilke, O.A.W. 1987. Dürer, Albrecht. 1977. Ehrenkrantz, Ezra D. 1956. Fletcher, Rachel. 2000. Golden Proportions
in a Great House: Palladio's Villa Emo. Pp. 73-85 in Heath, Thomas L. 1921. Heath, Thomas L., ed. 1956. Herz-Fischler, Roger. 1987. March, Lionel. 1998. March, Lionel. 1999a. Architectonics of proportion:
a shape grammatical depiction of classical theory. Pp. 91-100
in March, Lionel. 1999b. Architectonics of proportion:
historical and mathematical grounds. Pp. 447-454 in March, Lionel. 1999c. Proportional design
in L B Alberti's Tempio Malatestiano, Rimini. Pp. 259-269 in
Nicomachus of Gerasa. 1938. Palesca, Claude V. 1985. Palladio, Andrea. 1997. Schwaller de Lubicz, R.A. 1977. Serlio, Sebastiano. 1966. Thompson, D'Arcy W. 1942. On Growth and Form. 2nd ed. Cambridge UK: Cambridge University Press. Vitruvius. 1999. Zarlino, Gioseffo. 1558. Zeising, A. 1854.
The Palladio Museum An Itinerary of Villas and their Commissioners Life of Palladio
The Golden Section in Art and Architecture The Golden Section Ratio: Phi Proportion and the Golden Ratio (Mathematics and the Liberal Arts)
Lionel March was admitted
to Magdalene College, Cambridge, to read mathematics under Dennis
Babbage. There he gained a first class degree in mathematics
and architecture while taking an active part in Cambridge theater
life. In the early sixties, he was awarded an Harkness Fellowship
of the Commonwealth Fund at the Joint Center for Urban Studies,
Harvard University and Massachusetts Institute of Technology
under the directorships of Martin Meyerson and James Q Wilson.
He returned to Cambridge and joined Sir Leslie Martin and Sir
Colin Buchanan in preparing a plan for a national and government
center for Whitehall. He was the first Director of the Centre
for Land Use and Built Form Studies, now the Martin Centre for
Architectural and Urban Studies, Cambridge University. As founding
Chairman of the Board of the private computer-aided design company,
he and his colleagues were among the first contributors to the
'Cambridge Phenomenon' - the dissemination of Cambridge scholarship
into high-tech industries. In 1978, he was awarded the Doctor
of Science degree for mathematical and computational studies
related to contemporary architectural and urban problems.Before coming to Los Angeles he was Rector and Vice-Provost of the Royal College of Art, London. During his Rectorship he served as a Governor of Imperial College of Science and Technology. He has held full Professorships in Systems Engineering at the University of Waterloo, Ontario; and in Design Technology at The Open University, Milton Keynes. At The Open University, as Chair, he doubled the faculty in Design and established the Centre for Configurational Studies. He came to UCLA in 1984 as a Professor in the Graduate School of Architecture and Urban Planning. He was Chair of Architecture and Urban Design from 1985-91. He is currently Professor in Design and Computation and a member of the Center for Medieval and Renaissance Studies. He was a member of UCLA's Council on Academic Personnel from 1993, and its Chair for 1995/6. He is a General Editor of Cambridge Architectural
and Urban Studies, and Founding Editor of the journal Planning
and Design. The journal is one of four sections of Environment
and Planning, which stands at "the top of the citation
indexes." Among the books he has authored and edited are:
The Geometry of Environment, Urban Space and Structures,
The Architecture of Form, and R. M. Schindler: Composition
and Construction. His most recent research publications include:
"The smallest interesting world?", "Babbage's
miraculous computation revisited," "Rulebound unruliness,"
"Renaissance mathematics and architectural proportion in
Alberti's De re aedificatoria," and "Architectonics
of proportion: a shape grammatical depiction of classical theory."
His book Architectonics of Humanism: Essays on Number in Architecture
before The First Moderns, a companion volume to Rudolf Wittkower's
Architectural Principles in the Age of Humanism was published,
together with a new edition of the Wittkower, in the Fall 1998.
Copyright ©2001 Kim Williams top of page |
NNJ HomepageNNJ Autumn 2001 Index About
the AuthorComment on this articleRelated
Sites on the WWWOrder
books!Research
ArticlesDidacticsGeometer's
AngleConference and Exhibit ReportsBook
ReviewsThe Virtual LibrarySubmission GuidelinesTop
of Page |