Thoughts on Proportions in Architecture
(A work in progress: revised, 19 February 2006)
Christopher K. Egan,
Architect
“ar-chi-tec-ture
1: the art or science of building; specif: the art or practice of
designing and building structures and esp. habitable ones.
2 a:
formation or construction as or as if as the result of conscious act. b:
a unifying or coherent form or structure.”
…Webster’s Collegiate
Dictionary
“Architecture
is the thoughtful making of spaces.”
…Louis Kahn, The
Sketchbooks
Introduction: Uses of
mathematics in architecture
There are at least three
obvious ways that an architect uses mathematics: to measure, to dimension
and to calculate.
1.
We measure the sites for our buildings to see how they will fit in
the landscape. We measure the spaces needed by humans, their activities and
their belongings, and then we design buildings to accommodate these.
2.
We tell the builders how big to make the building and its components
by assigning dimensions to our designs.
3.
We calculate many aspects of a design, from the structural loads on
building components to the number of bathrooms that are needed to the size
of a restaurant kitchen to the quantities of materials and their costs.
However, the architect
also uses mathematics in ways that reach beyond the pragmatic and
functional. For at least 2,500 years architects have used mathematics in an
attempt to bring order, beauty and meaning to their works.
We know that the
architect’s primary task is to design spaces for use by humans, and then to
join these spaces with other spaces to make a building. This nearly always
means that the architect must design complex structures comprised of many
different elements and many different spaces. The obvious danger is that the
end result could be a chaotic and confusing mess; this is why much of the
art of architecture is a search for ways to give order and unity to the
complex reality that is a building. In fact, the use of ordering systems is
so much a part of architecture that the word “architecture” is often used in
other fields, as when people talk about the “architecture of a computer
system” or the “architecture of a novel.” In this paper we will look at two
separate but related mathematical tools that the architect uses to bring
unity to complexity: proportion and scale.
What is
proportion?
“pro-por-tion fr.
L. por (for) + portion (portion…related to part) 1: the relation of
one part to another or to the whole with respect to magnitude, quantity, or
degree: ratio. 2: the harmonious relation of parts to each other or
the whole.”
….Webster’s New
Collegiate Dictionary
“Proportion” refers to
the relative sizes between two elements of a design, and among many elements
of a design. For instance, if a window in a room is 2 feet wide by 4 feet
high, we would say that is has a proportion of 1 to 2, usually described as
a ratio using the notation “1:2.” If we then notice that the room is 10
feet wide by 20 feet long, we will see that it also has a proportion of 1:2.
Then if we move to an internal patio in the same house, we may find that it
has dimensions of 20 feet by 40 feet, which would also produce a 1:2
proportion. Finally if we move outside the house to an urban plaza, we may
find that it has dimensions of 120 feet by 240 feet, which would also
produce a 1:2 proportion. Notice that the proportion does not indicate the
size of a thing…only the relationship between two or more things. In the
example, the window, a room, a patio and a plaza all have the same
proportions, even if they are dramatically different in size. In drawing and
developing a system of proportions, one of the most useful tools is the
diagonal of the rectangle involved. As long as two forms have the same angle
for their diagonals, they also have the same proportions.
Notice that I only
described two dimensions when discussing the window, room, etc. However, we
can also talk about the proportions of a three-dimensional volume. If we go
back to the 10 foot by 20 foot room, we may find that it is also 10 feet
high. We could then talk about a 1:2:1 proportion of width, length, height.
Then if we go to the patio and find that it is 20 feet tall, we would find a
volume that is 20 feet wide x 40 feet long by 20 feet wide, which produces
the same 1:2:1 proportion. However, if we find that the patio is only 10
feet high, then we would find a proportion of 2:4:1…so that the patio would
have a different proportion than the interior room. Similarly, if there is
another room in the house that is also 10 feet wide by 20 feet long, but
instead of 10 feet high it is 30 feet high, we would have one room that has
a 1:2:1 proportion and another with a 1:2:3 proportion.
If you were to start
drawing rooms with different proportions, you would find that some
proportions make a space feel vertical, others would make a space feel long,
others would make a space feel square, so that the feel of a space can be
understood by examining its proportions.
One more comment may be
helpful…..that proportions can be applied to forms that are not rectangular.
A circular room would have a 1:1 proportion because its width is the same as
its length, whereas an elliptical room could have different proportions
depending on the relationship between length and width. Finally, a circular
volume whose height is the same as its diameter would have a 1:1:1
proportion, whereas a circular room that is twice as high as its diameter
would have a 1:1:2 proportion, and would feel different to an occupant.
What
is so important about proportions?
There seem to be at
least three basic attitudes that architects have with reference to
proportions:
1.
Some architects find the study of proportions useful in general terms
without caring too much about specific mathematical formulae. They feel it
is important to know whether a space is tall or short, long or wide, a
little long or very long; but they don’t give much attention to the specific
numbers involved.
2.
Other architects consider the study of proportional systems to be
overly formalistic or academic. These architects are scornful of the study
of proportions which they feel is rooted in ancient mythologies and
theoretical nonsense.
3.
A third group of architects consider the use of precise proportional
systems to be profoundly important. Some of these are driven by theological
or philosophical beliefs, others by a commitment to specific formal
disciplines, and others may simply enjoy the esthetic results that come from
the disciplined application of a rigorous system of proportions. Whatever
their motivation, this third group of architects, committed to the
disciplined use of proportional systems, have their roots in the ancient
legacy of architecture.
Based on the evidence
of their built work, it is safe to assume that architects have been
interested in proportions at least since the time of the Egyptian architect
Imhotep, designer of the stepped pyramid of Saqqara, a few thousand years
before the time of Christ. However, the sophisticated study and use of
proportional systems apparently began with the work of the Greek
mathematician, philosopher and mystic Pythagoras, who lived around six
centuries before Christ (he is best known as the author of the “Pythagorean
theorem” that describes the geometries of a right triangle). The best
description is provided by Jacob Bronowski in his book The Ascent of Man:
Samos
is a magical island. The air is full of sea and trees and music. Perhaps
Pythagoras was a kind of magician to his followers, because he taught them
that nature is commanded by numbers. There is a harmony in nature, he said,
a unity in her variety, and it has a language: numbers are the language of
nature.
Pythagoras found a basic
relation between musical harmony and mathematics. …what he discovered was
precise. A single stretched string vibrating as a whole produces a ground
note. The notes that are harmonious with it are produced by dividing the
string into an exact number of parts: into exactly two parts, into exactly
three parts, into exactly four parts, and so on.
Pythagoras had found
that the chords which sound pleasing to the ear – the western ear –
correspond to exact divisions of the string by whole numbers. To the
Pythagoreans that discovery had a mystic force. The agreement between nature
and number was so cogent that it persuaded them that not only the sounds of
nature, but all her characteristic dimensions, must be simple numbers that
express harmonies. For example, Pythagoras or his followers believed that we
should be able to calculate the orbits of the heavenly bodies (which the
Greeks pictured as carried around the earth on crystal spheres) by relating
them to the musical intervals. They felt that all the regularities in nature
are musical; the movements of the heavens were, for them, the music of the
spheres.
This helps to explain
why, for most of the history of architecture, the study of proportional
systems was treated with great reverence. Pythagoras had discovered, and
proven, that beauty in music could be understood through numbers, and in
doing so it seems that he had unlocked the secrets of divine harmony and
creation. For several thousand years no one could be considered an architect
unless they had mastered the use of proportional systems, and a building
that was not based on careful proportions was considered flawed. It is
important to note that this idea is not limited to ancient Greek belief
systems. Most religions believe that the universe was designed by a divine
Creator who used specific mathematical systems to guide the work. This is as
true of the ancient Maya as it was of the Zoroastrians of Persia or the
Buddhists of India or the Christians of Northern Europe. This profound
respect for precise geometry in architecture can be found wherever humans
developed advanced civilizations, because such societies were based on
mathematics, whether is was the sophisticated analysis of astronomical
patterns, or the precise measurement of the calendar or the need to plan
cities for thousands of inhabitants. For architects in such a civilization,
it was obvious that the greatest works of architecture would be those that
were most carefully based on the proportions used by the Master Architect.
In
his website “harmony and proportion” John Boyd-Brent of Scotland’s Royal
College of Art describes the meaningfulness of proportions in this way:
“The idea is that the harmoniously interdependent relationship of parts,
within the whole, and to the whole, symbolises the relationship of ourselves
and our universe of space and time, with the ‘single’ whole; our
metaphysical reality. Thus, to be concerned with harmonious creation, be it
architectural, artistic, musical, or even agricultural, came to be seen as a
natural consequence of awareness of our harmonious relationship with God.”
For
this reason, proportional systems were taught to architects as an essential
part of their education from the Golden Age of Periclean Athens around 500
B.C. all the way into the 20th century. Admittedly, starting
around 1600 A.D. the reasons used to explain the importance of proportions
began to shift from a theological desire to emulate the Divine Geometry to a
more “scientific” and secular belief that “good” proportions would provide a
beauty that was “true” as opposed to a more ‘decadent” beauty based on
emotions and intuition. Claude Perrault, founder of the Royal Academy of
France and architect of the Louvre additions in the 1600s, introduced his
essay on proportions in this way: “One must suppose that there are two
kinds of beauties in architecture, those that are founded on convincing
reasons and those that depend on prejudice.”
However, the study of
proportions suffered two major blows in the form of the discoveries of
Galileo and Newton. Galileo shattered the 2000-year old belief that the
Earth was the center of the universe, offering a more dynamic view of the
cosmos, although it still assumed the importance of the circle and the
square. It was Newton who demonstrated that the movements in nature are not
based on the relatively stable circle, but on the more dynamic curves of
ellipses, parabolas, and spirals. Since the entire validity of
Pythagorean-Platonic proportional systems was based on the belief that they
described the mathematics of the universe, Newton’s discovery stripped these
systems of their authority. They continued to be valued by some architects,
but more as historical antiques, or for their own internal logic, rather
than for any cosmic or theological significance. Wittkower describes the
change: “With the rise of the new science the synthesis which had held
microcosm and macrocosm together, that all-pervading order and harmony in
which thinkers had believed from Pythagoras’ days to the 16th and
17th centuries, began to disintegrate. This process of
‘atomization’ led, of course, to a re-orientation in the field of
aesthetics, and implicitly, of proportion.”
The teaching of
rigorous, mathematically-based, proportional systems became less common in
architecture schools of the mid-20th century as a result of the
modernists rejection of “historically based” architecture. This was a
symptom of a general cultural revolution that swept through the United
States and Western Europe in the period just before and after the First
World War (approx. 1914 to 1918). During this period, which was the birth of
the Modernist movement in architecture, anything that was based in classical
systems or historical architecture was considered decadent because it
represented the aristocratic culture that had led the world into a pointless
and murderous war that left Europe devastated. For the architects,
engineers and scientists who followed during the 1920s, anything that was
based in theology or cultural tradition instead of science was considered
decadent and corrupt. This included the study of the Pythagorean-Platonic
proportional systems.
Ironically, the
emergence of modern science did not permanently eliminate the architect’s
fascination with geometry and proportions. On the contrary, modern science
has delighted in discovering the rich and beautiful mathematics that direct
the actions of nature, whether it is the form of a cloud or the dance of
galaxies. Perhaps this is what gave rise to a new interest in proportional
systems among some early modernists. The rigorous application of
proportional systems in architecture was given a major rebirth through the
work of the ultimate modernist, Le Corbusier. Perhaps his interest in
mathematics was due to his early training as a watchmaker in Switzerland, or
perhaps because of his philosophical roots in the French intellectual
tradition, based on the mathematically rationalistic work of Descartes
(author of the “Cartesian grid” and the leading philosopher of the modern
world). Whatever the motivation, Le Corbusier gave new impetus to the
application of rigorous proportional systems through his development and use
of the system “Modulor.”
For Le Corbusier the
application of a proportional system was not related to any theological
belief regarding the geometry of God; rather it was based in his search for
an architecture that could be considered “authentic” after the death of
“corrupt historicism.” Any legitimate search for an authentic new
architecture must always include a consideration of what has been proven
true and what has been proven cultural baggage to be discarded. Corbusier
rejected forms whose sources were superficial historicism and sought forms
that could be based in authentic human experience. In the development of his
ideas he decided that rigorous geometry was a fundamental…and therefore
authentic… manifestation of the human creative mind and spirit. For this
reason he embraced the disciplined application of proportional systems as a
foundation for a legitimate new architecture. His proportional system The
Modulor was based on the combination of the Golden Section and the Fibonacci
Series. (The Fibonacci Series is a sequence of whole numbers that
approximate the Golden Section. Each new number is the sum of the previous
two numbers: 1,1,2,3,5,8,13,21,34, etc.). Perhaps what is most interesting
in the Modernist use of proportion is that the old closed, stable ratios of
simple whole numbers, symbolic of a world of static simple order, have been
replaced by the more open-ended dynamic ratios of the ever-increasing Golden
Proportion, the Fibonacci Series and the expanding, dynamic spiral that
forever dances around a center that it never reaches. In this way we can see
that, if the Greeks used proportions to represent their stable world view,
we today use proportions to manifest the dynamic complexities of our times.
Systems of Proportion
Remember that the
interest in proportion began with Pythagoras’ discovery of the relationship
between musical harmonies and simple numbers. This would lead to an interest
in the numbers 1,2,3,4 that produced the proportions 1:2, 2:3, 3:4. Two
hundred years later, Plato developed the idea a bit further, as described by
Rudolf Wittkower:
“In the wake of the
Pythagoreans, Plato in his
Timaeus explained
that cosmic order and harmony are contained in certain numbers. Plato found
this harmony in the squares and cubes of the double (2) and triple (3)
proportion starting from unity (1), which led him to the two geometrical
progressions, 1,2,4,8 and 1,3,9,27. Traditionally represented in the shape
of the [Greek letter] Lambda [see below] the harmony of the world is
expressed in the seven numbers 1,2,3,4,8,9,27 which embrace the secret
rhythm in macrocosm and microcosm alike. For the ratios between these
numbers contain not only all the musical consonances, but also the inaudible
music of the heavens and the structure of the human soul.”
The form “Lambda” in
which these two series is usually represented is this:
1
2 3
4 9
8 27
Those who followed the
Pythagorean-Platonic belief in proportions believed that these numbers and
proportions described all the harmonies in the universe, and that they were
complete within themselves, without need for expansion into larger numbers,
as described by Wittkower: “…in the cube of the one and the other [2 and
3], they said, the work was terminated. For one cannot proceed beyond the
third dimension in length, width and depth.” Thus we can see that
between Pythagoras and Plato, architects of the Golden Age of Athens (600 to
500 B.C.) had a solid theory of proportions that described a static,
permanent world ordered by the relationships among a finite collection of
whole numbers, all beginning with one, then moving on to both 2 and 3, and
then squaring these and cubing them.
As we will see, in some
later systems of proportions, irrational numbers were included, in
particular the square root of 2, the square root of 5 and the number known
as the Golden Section. We will also see that many systems of proportions are
interested in numbers beyond those of the Greek “Lambda,” so that a way of
generating new numbers in a proportional series becomes an important goal.
As we examine different approaches, we can use the terminology “a:b:c” in
which “a” is the first number, “b” is the next larger number and “c” is the
next number larger than “b”. We can use these terms to describe different
systems of proportions.
An “arithmetic” system
is based on the idea of using a constant difference “b-a” is the same as
“c-b”. Examples would be 1:2:3:4 or 4:6:8:10 or 6:9:12:15 etc.
A “geometric” system is
based on keeping a constant proportion between the smaller and the larger,
so that a:b equals b:c. examples would be 1:2:4:8 or 4:6:9. Notice that this
does not always lead to whole numbers, which was the requirement for
“appropriate” proportions.
A “harmonic” system is
based on a simple idea, but it seems complicated. It is based more on
differences between numbers than on the actual numbers. The formula is
(c-b) / (b-a) = c/a.
It is based on the idea
of the “mean” …the number between the other numbers. Examples would include
6:8:12 in which the difference between 6 and 8 (two) is one third of the
smaller number, and the difference between 12 and 8 (four) is one third of
the larger number. This means that a “harmonic” system tries to find the
number (the “mean”) between two extreme numbers, but it is not simply the
number halfway between them. Instead it is the number that seems to reflect
the progressive relationship between the two extremes.
The “irrational”
proportions are those that cannot be described as “ratios” between whole
numbers, and in most cases they describe the proportion between one side of
a rectangle and its diagonal. Square root of 2 is the most basic, and was
included by Palladio in his description of ideal room proportions. It is the
length of the diagonal of a square, and at the same time, as demonstrated by
John Boyd-Brent, it is the radius of a circle in which the square is
inscribed if one corner is at the circle’s center. In other words, it is the
radius of one quadrant of a circle as well as the diagonal of the square
that would fit in that quadrant (see his website).
Role of the Golden
Section
Other irrational numbers
that are often given a place in proportional systems include the square
roots of 3 and 5. But the most important irrational number used in
proportional systems is the Golden Section, or Golden Mean. This is
the only ratio in which the proportion of the smaller part to the larger
part is the same as the ratio between the larger part and the whole,
or:
a:b = b:c
It is constructed by
drawing a square, bisecting the square vertically, and then drawing the
diagonal of one of the two halves. Then arc the diagonal as shown in Ching’s
Chapter 6 to produce the length of the Golden Rectangle.
What is scale?
“scale
[these definitions seem closest to our purpose …egan]
From
Old Norse ‘skal’ (shell):
Either pan or tray of a
balance, an instrument or machine for weighing.
From the Latin ‘scala’
(stair/ladder):
3:
something graduated, esp. when used as a measure or rule.
a:
a series of spaces marked by lines and used to measure distances or to
register something.
b:
a divided line on a map or chart indicating the length used to represent a
larger unit of measure
4:
a graduated series or scheme of rank or order.
5:
a proportion between two sets of dimensions (as between those of a drawing
and its original).
….Webster’s New
Collegiate Dictionary
Scale refers to the size
of a thing relative to some unit of measure, whether it is the foot, the
meter or the size of the human body. In this way scale is completely
different from proportion. Using our example above, both the interior room
(10 feet by 20 feet) and the plaza (120 feet by 240 feet) have the same
proportions, but they have very different scales. One has the scale of a
family dining room and the other has the scale of an urban space.
In architecture we also
use the term “scale” to talk about the measurements of a drawing or model
with reference to the physical reality it represents. However, for the
current discussion we will use “scale” to consider the dimensions of a
building and its elements in relation to a unit of measure.
Most measurements in
architecture are based on either the “imperial” system (feet, inches, yards,
miles) or the “metric” system (meters, millimeters, centimeters,
kilometers), which are relatively abstract units. I prefer to use the human
being as our unit of measure, because it is the human body that must be
accommodated in our work, and it is the human who experiences architecture
in relation to his or her own size. In his book Form Function Design,
Paul Grillo describes the essence of scale in this way:
”Man needed a unit of
measure not only to size his own creative work, but also to allow him to
size up nature’s creation around him in comparison with his own size, and in
relation with his own strength. In other words, he needed as a common
measure a unit that could be a bond between the scope of man’s creative
power and the world around him. …It was only natural that he should find the
answer in the dimensions of his own body….”
Grillo offers examples
from ancient measuring systems. “It was only common sense for man to
choose as a basic unit for measuring length that part of his body most
important for him to accomplish his daily work: his fore-arm. This became
the cubit – the length of the fore-arm from the elbow to the tip of the
middle finger, hand outstretched.” Other examples are the “foot” which
is the comfortable height between rungs of a ladder, or “yard” which is the
normal stretch of a girl’s arm when measuring material on the edge of a
table, or the “acre” which was the are of land a man could harvest in one
day. He summarizes by observing that “…every different kind of human
activity needs a different gauge to be measured properly. This unit
characteristic of each program is the module. It sets the pace of life. It
grants scale to the design.”
In architecture, the
scale of a space will depend on its use. A 5,000 seat opera house will have
a different scale than a 100 seat intimate theater. Thus the size of a space
and of the elements within that space will depend on the use, the size, the
importance and the number of occupants. The size of a television screen in a
neighborhood sports bar will be different than the size of a television
screen in the SBC Center.
One important issue with
regards to scale in architecture is the degree to which a building reveals
or conceals its size relative to our bodies. For instance, a large cube,
made of one seamless material, offers us no clue as to its size relative to
us. Without other clues from the context, we simply cannot tell from a
distance whether it is 10 feet on a side or 100 feet. Of course, when we see
other humans going in and out of the cube we begin to get a sense of the
building’s scale, but without the presence of humans a seamless building
conceals its true size, or scale.
Perhaps the most basic
way that an architect reveals the scale of a building is to emphasize those
elements that are most closely tied to the physical size of the human being.
Windows and doors immediately begin to suggest the size of the person who
looks or enters through them. Similarly, the location of floor levels gives
some clue as to the height of the spaces inside.
Another way that a
building can reveal its scale is through the articulation of its
construction. This is actually a bit complicated, because it varies with the
means of construction. A house made of brick, in which each brick is laid by
hand, will show its scale as the scale of the hand of the builder, because
the elements of construction cannot be too large to be handled by one or two
workers. On the other hand, a highway bridge or a convention center can have
massive elements scaled for the capacity of the trucks and cranes that
handle them. This suggests that different ways of building will have
different scales.
Scale and proportion
We have discussed
proportions, which are concerned with the mathematical ratios between
the sizes of the components of a building, without reference to the
particular size of the building. Then we discussed scale, which
addresses the size of building and its elements with reference to the human
body. This leads us to consider the possibility of combining these two
elements, in a way that uses a harmonious system of proportions while
revealing the scale of the building.
This question of how to
combine proportions and scale has been a concern of the best architects for
most of architectural history, and it is even more pressing in the case of
very large buildings. If you are designing a building that is 20 stories
high, and you make all the windows and elements the scale of the human, you
will produce a design that resembles a massive honeycomb, with no sense of
overall proportions, and a sense of scale in which the human appears a tiny
and insignificant insect. If instead you make the windows and other elements
harmoniously proportional to the building, they will be gigantic in the
presence of the human, and may suggest that the building is designed for
giants instead of humans.
The masters of the
Renaissance wrestled with the question of proportion vs. scale as it
involves the design of large buildings. The issue, as we saw, is this: when
designing a very large building, the architect would like to give it the
same clarity and proportions that give harmony to a smaller building.
However, this results in the use of elements that are huge in order to be
well-proportioned with the whole. Then, these huge elements seem to dwarf
the humans who use the building. The solution, used by Michelangelo in his
buildings for the Capitoline Hill in Rome, is to combine two scales of
elements: gigantic columns that are proportionate with the overall
composition of the building, and smaller columns set inside them that are
scaled more to the size of the human. Palladio used the same method with
great success in the Church of San Giorgio Maggiore in Venice, with multiple
scales of elements, all tied into a composition that is proportional at the
scale of the building with smaller elements that make sense as the human
approaches. One of the best examples of this is the massive Church of St.
Peter’s at the Vatican, also in Rome. Here the scale changes as you approach
from the Piazza, with columns set inside columns set inside columns.
At least two of the
proportional systems we are considering in this project combine proportion
and scale into one system: the Japanese “ken” and Corbusier’s “modulor.”
Both of these systems provide a way for creating a consistent set of
proportions by beginning with a basic unit, and in both of these systems the
basic units are tied to the scale of the human body.
In the Ken, the basic
unit is the tatami mat, which in turn is a rectangle with proportions of 3
“shaku” by 6 “shaku.” The size of the shaku seems to vary from region to
region, but the most common size of this basic module is 30.3 cm. In this
way, a tatami mat is approximately .9 m by 1.8 m, and is the size for one
person to sleep or for two people to sit. The unit of length “ken” is 6
shaku (1.8 m) and the spatial unit “tsuvu” is the square made of 2 tatami
mats (a square which is I ken by 1 ken, or 6 shaku by 6 shaku, or 1.8 m by
1.8 m). Now we may recall that Palladio gave a series of rules for
establishing the height of a room, and it only makes sense that a room with
larger floor area would be higher as well. However, in most formal
Renassance proportional systems, the elements of a room or building simply
get larger as the area gets larger, with no reference to the human scale. In
contrast, the “ken” system uses human scale as a constant, while providing
higher space for larger rooms. It does this by establishing a “frieze board”
that runs around the room at a height of approximately one ken. This line
establishes a human scale reference. Then, the height of the room reflects
the overall size, in that the height of the ceiling above the frieze board
is 1/3 the number of mats in the room. A room with many mats will be higher
than a room with few mats, but all rooms will have a human scale reference
in the form of the frieze.
In the Modulor,
Corbusier used the Golden Section proportion as his basic guide, in
recognition of its powerful ability to tie together all the elements of a
building in one system of proportions. But he then shifted towards the more
practical Fibonacci Series, which approximates the Golden Section but
replaces its pure and irrational numbers with more easily buildable rational
whole numbers. But the real genius of the Modulor is that Corbusier ties all
the numbers in his series to the human body, its parts and its space. This
makes the Modulor a useful system for the architect and builder, while at
the same time providing the harmony sought by the Pythagoreans and
Platonists. This, together with its modernist dynamics, helps explain why so
many architects in the second half of the 20th century were so
intrigued by this system.
Conclusion
In subsequent projects
we will apply these principles in greater detail and with greater
creativity, but for now our task is to master the principles we have
inherited to bring order and richness to our designs.
As for the importance of
proportions, it is clear that the classical Pythagorean-Platonic
proportional systems do not accurately reflect the geometric complexity of
the universe and the century in which we live, but they may have profound
value as a way of anchoring us in a world of turbulence. Similarly, it is
clear that the geometry of our own epoch is a non-Euclidean geometry
characterized by dynamic forms and organic curves. It may just be that with
the complex geometries possible to us today, we need the tools of proportion
to give unity to the complexities we create.
As for scale, I believe
that all architecture must at least acknowledge the scale of the human, even
if the scale of the activity is vast. This becomes a delightful challenge in
the design of vast spaces such as airports and convention centers, and it is
equally a challenge in the design of urban plazas and shopping malls.
Finally, I would like to
suggest that theory cannot replace intuition, and that formulas do not make
great design. The Finnish master architect Eliel Saarinen described this
issue in his book “The Search for Form in Art and Architecture,” when he
suggests that the ancient Greeks first made beautiful art, and then
discovered the mathematics within the art …as opposed to making the
mathematical theory and then using it to create art. He suggested that
perhaps we humans have an inherent sense that vibrates musically with the
patterns of our souls….and that this manifests itself in our delight in
discovering number. His thoughts are not unlike the observations of the
Viennese psychologist Carl Jung, who suggested that we have within us
patterns that desire to be expressed. Saarinen writes:
“…the language of
mathematics does not deal only with the great civilizations of man. It goes
down to man’s everyday life, to man himself…. This only shows that the world
of vibration – of ‘number’- in human art, is inherent in man. It shows, in
other words, that there is within man something inexplicable which comes
individually into expression, and in which – when genuine – as we have put
it, ‘ the creative instinct is the sensitive seismograph that records
vibrations of life and transposes them into corresponding vibrations of
art.”
Bibliography
John Brent-Boyd,
Harmony and Proportions,
www.aboutscotland.com/harmony/prop.html)
Bruce Rawles, Sacred
Geometry Home Page
http://www.intent.com/sg/
Rudolf Wittkower,
Architectural Principles in the Age of Humanism, New York: Norton
Library, 1971.
Eliel Saarinen, The
Search for Form in Art and Architecture (Chapter XIV: Form and Theory),
New York: Dover Publications, 1985.
Cornelius van de Ven,
Space in Architecture, The Netherlands: Van Gorcum Assen, 1980.
Le Corbusier, The
Modulor and The Modulor II, (my copies are in Spanish, printed in
Buenos Aires. I don’t know the US publisher, but it is easy to find)
Vitruvius, The Ten
Books on Architecture, New York: Dover.
Andrea Palladio, The
Four Books of Architecture, New York: Dover.
Leon Battista Alberti,
The Ten Books on Architecture, New York: Dover.
Paul Grillo, Form
Function Design, New York: Dover, 1960.
Details on modulor & proportional
systems
