The Order of Proportion

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The Order of Proportion
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For the harmony of the world is made manifest in
Form and Number, and the heart and soul and all
the poetry of Natural Philosophy are embodied in
the concept of mathematical beauty. DArcy
Wentworth Thomson, 1917
God as the Architect of the Universe
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Le Corbusier Le Modulor, 1949
Nature is ruled by mathematics and the
masterpieces of art express the laws of nature,
and themselves proceed from those laws. It
makes the bad difficult and the good easy,
Albert Einstein on Le Corbusiers Modulor, 1946
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Though there were never a circle or a triangle
in nature, the truths demonstrated by Euclid
would for ever retain their certainty and
evidence. David Hume, Enquiry concerning Human
Understanding As far as the propositions of
mathematics refer to reality, they are not
certain and as far as they are certain, they do
not refer to reality. Albert Einstein, Geometry
and Experience, 1921 The order and regularity
in objects, which we entitle nature, we ourselves
introduce. Immanuel Kant We actively try to
impose regularities on the world Karl Popper,
1965
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He begins with the rigid line, which is
essentially abstract and alien to life . He
seeks further geometrical possibilities of line,
creates triangles, squares, circles, places
similarities together, discovers the advantages
of regularity, in short, creates a primitive
ornament which provides him not only with a mere
delight in decoration and play, but with a table
of symbolic absolute values, and therefore with
the appeasement of his condition of deep
spiritual distress.Wilhelm Worringer, Form in
Gothic, 1912
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The naturalistic art of empathy proves to owe
far more to the human, and abstract art far more
to nature, than the associations of naturalism
and abstraction lead us to assume. Inside every
empathist there is an abstractionist fighting to
get out, and vice-versa.Richard Padovan,
Proportion, 1999, p.26
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The unit is conceived as something chosen but
not chosen at random. It must be large enough to
count with respect to the thing to be measured,
while on the other hand, if the unit is to be of
any use as a measure, it naturally cannot be of a
similar size to the thing to be measured. In
relation to the thing, the unit must be not too
little and not too big. Padovan, p.43
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Geometric progression adding each number to the
preceding term.Addition of any term to any other
becomes too complex. True order being a balanced
combination of unity and complexity an effective
system of proportions must combine the
multiplicativeness of the geometric progression
with the additiveness of the arithmetic
progression.Padovan p.44
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Square root of 2 1.4142Square root of 3
1.732Square root of 4 2.0Square root of 5
2.236Golden section 1.618Plastic number
ratio 1.325
Interwoven grid (Pythagorean-Platonic
system1 y y2
y3x xy xy2 x y3
x2 x2y x2y2 x2y3x3
x3y x3y2 x3y3
1 3 9 27 2 6 18 54 4 12 36 108 8 24 72 216
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The fundamental unity of geometry and numberWe
must finally construct (6) a second unit square
ABMN adjacent to the first. The purpose of this
is to reveal the additional lengths EN, GN, IN,
and KN, and thus the proportionally all-important
complementary numbers 1 v2, 1 v3, 1 v4, and
1 v5, or approximately 2.414, 2.723, 3.0 and
3.236. In this way, we provide each of the four
square systems with the weft series it needs to
become completely effective. Moreover, bisecting
the 1 v5 rectangle (7) we obtain two Golden
Section rectangles, since Golden Section
(1v5)/2
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The Parthenon, Athens
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When one already has a theory one tends to find
confirmation of it wherever one looks. Padovan
p.81Simple whole numbers?Egyptian triangles
base8height5sidesv41, just over 6.4What
determines the positioning of the underside of
the entablature?
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The Parthenon as Whole number relationshipsCore
rectangle of 15x6 standard column spaces.
Stylobate of 94

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The human body is so designed by nature that the
face, from the chin to the top of the forehead
and the lowest roots of the hair, is a tenth part
of the whole height the open hand from the wrist
to the tip of the middle finger is the same
Vitruvius c.25BC
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Leonardo da Vinci Vitruvian Man c.1500
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Chartres Cathedral1194-1224
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(No Transcript)
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The total length of the nave and choir .. .
comprises seven double bay units, marked out by
the alternation of round and octagonal piers We
have found a series of whole number foot
dimensions that would have allowed the work to be
measured out with a yardstick - perhaps literally
a three foot measure, since all the vertical
dimensions are multiples of three feet. Cords
and pegs would still have been needed as they
are today, to determine right angles.Padovan
p.196
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For us, the outline is a certain correspondence
between the lines that define the dimensions one
dimension being length, another breadth, and the
third height I affirm again with Pythagoras it
is absolutely certain that Nature is wholly
consistent The very same numbers that cause
sounds to have concinnnitas, pleasing to the
ears, can also fill the eyes and mind with
wondrous delight. From musicians therefore or
from those objects in which Nature has displayed
some evidence and noble quality, the whole method
of outlining is derivedLeon Battista Alberti
(1404-72), On the Art of Building in Ten Books
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Andrea Palladio 1508-80
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Villa Pisani, Bagnolo, from Andrea Palladio, Four
Books on Architecture
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The Veneto villa as working farm and place of
leisure
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Villa Pisani, Bagnolo
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Villa Pisani, Bagnolo - rear elevation
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Villa Pisani, BagnoloCentral Hall
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Typical windows of London houses, 18th
centuryfrom Dan Cruickshank and Peter Wyld,
London, the art of Georgian Building, 1975
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Doorway from Gentleman and Builders Repository,
by Edward Hoppus, 1737
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Proportions of typical London house, 18th
century, from Cruickshank and Wyld
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Reading List
Leon Battista Alberti, On the art of building in
ten books, 1988, 720/ALB Hans van der Laan,
Architectonic Space, 1977, 720.1/LAA Le
Corbusier, The modulor, 1954/61 701.8/LEC Richard
Padovan, ProportionScience, Philosophy,
Architecture, 1999, 720.1/PAD Rudolf Wittkower,
Architectural Principles in the Age of Humanism,
1949/73 724.1/WIT