Roger Herz-FischlerDepartment of Mathematics Carleton University Ottawa, Ontario, Canada, K1S 5B6 Example of reckoning
of a pyramid, Height 250, base 360 cubits.sekedWhat is its ?......(Answer:
5+1/25 hands). From the Rhind Papyrus.[1]
Space, Shape and Form /An Algorithmic
Approach.[2]
The material in this paper is a revised version of a chapter
entitled "Proportions" that appeared in the problems
part of the book.So that the reader can better understand the raison d'être of what follows, and also to decide if they agree with my views on the teaching of mathematics in the architectural curriculum, I shall first describe how the chapter, and the rest of the book, came to be written. In 1972 I was asked to teach the mathematics course for first-year students of architecture. Since I was essentially free to choose the topics for the course I decided to introduce some material dealing with the use of mathematical proportions in architecture. After a few years I discovered that much of what I had read in the literature was at best not completely evident and at the worst out and out nonsense. This in turn led to an apology to my students and a change in my own research from theoretical probability into the fields of the history of mathematics, proportions—in particular the "golden number"—and what I refer to as the sociology of mathematical myths. [3] Indeed parts of what I present below are based on my own research. On the other hand the course itself came more and more oriented to what we might call applications of geometry to architecture. Together with this change in approach, I developed a book containing theory in the first part and a large collection set in the second part. [4] I felt that analysing some historical examples of proportions in the first problem chapter would not only be a good mathematical "warm-up" for the more complicated problem chapters, but would also be of particular interest to the students. As a secondary agenda, I hoped to put the students on guard against many of the dubious statements concerning proportions that appeared in the architecture literature.
The first example—this is explored further in Case Study
1—involves the Great Pyramid. As indicated by the example
cited in the opening quotation, we know that the ancient Egyptians
had a well-defined theoretical method of indicating the slope
of a pyramid. Now consider the drawing of the excavated base
of the mastaba at Medum, Figure 1.The horizontal lines are well defined
as being one cubit apart, but there is not the slighest mark
to indicate how the slope was determined. In view of the ancient
Egyptian mathematical texts one would suppose that the The second example—explored further in Case Study 10—
involves Le Corbusier's 1927 villa at Garches. What graduating
architecture student—not to mention the numerous books and
articles which use it to explain Le Corbusier's infatuation with
the golden number—has not seen the view in Figure
2 that was published in Le Corbusier's
These two examples show that even when we have pictorial and written evidence, we must act with extreme caution in deciding upon the theoretical basis of an architectural structure. Then there is of course the ever present difference between theory and practice. [5] In my opinion, these examples of difficulties—and the others that will come up in the course of the problems—suggest how we should present proportions to students of architecture. We should present proportions as the theoretical thoughts of the architects of the past and we should insist that it is much more important to think about the theoretical basis, and the mathematical techniques involved, than to engage in an endless, and ultimately futile, search for "hidden" proportions. We can now turn to the Case Studies themselves. Note that the presentation, and the questions asked, vary according to problem. The presentation depends upon whether I presenting the method as something that we definitely know something about or an example of speculation.
*Seked*theory: In ancient Egypt the basic linear unit of architectural measurement was a cubit of about 52.3 cm. Each cubit was divided into 7 hands of 4 fingers each. The opening quotation of this article cites a pyramid example from the Rhind Papyrus. From these examples we can ascertain that the*seked*of an inclined line was the horizontal run for a rise of 1 cubit. In the diagram shown in Figure 1 of the excavation of the mastaba at Medum the horizontal lines are one cubit apart. The*seked*theory of the Great Pyramid assumes that the Pyramid was built so that the*seked*was 5 hands and 2 fingers.- The arris had a rise over run ratio of 9:10.
- The angle of inclination corresponds to that of a triangle inscribed in a regular heptagon.
- "Area" theory: The Great Pyramid was built so that a square with sides equal to the height would have the same area as one of the faces.
- "Pi" theory: The Great Pyramid was built so that
the base would have the same perimeter as a circle whose radius
was the height. For what well-known estimate for the value of
pi does the angle given by the pi-theory exactly equal to that
given by the
*seked*theory?
To understand the meaning of this question and answer, it
is necesary to consider the earlier history of the construction.
The church had been started and its foundation laid with a width
of 96 Milanese
In fact the original "ad triangulum" plan was not
followed for, as the answer says, the cathedral was to rise up
to a triangle on the triangular figure". What the council
had really decided was to retain the other piers at 28
Figures 8 deals with the method of making templates used by a German master named Lorenz Lechler (1516) [Shelby 1972: 409 and 419]. It was based on a modular unit determined by the choir wall of the church: "Take the wall thickness of the choir, whether it be small or large, then draw squares through one another; therein you will find all templates, just as you will find them drawn in this book".
Figure 9 is taken from the
Figure 10 is taken from the "Vienna Sketchbook" [Shelby 1971: 153].
Shelby givens the following method of finding the circumference
of a circle, taken from Mattias Roriczer's
One place were more advanced techniques were used was in the requirement that the area of a spire be equal to the area of the ground plan, the basic figure of which was the square. Thus the mason was faced with a problem in transformational geometry, i.e., he had to construct one figure equal in area to another figure. Compare the following two methods of constructing a square equal in area to a given triangle (Figures 12 and 13). In Figure 12, taken from an anonymous 15th century
work, On the other hand, Figure 13, taken
from Matthias Roriczer's
An example of a design based on the above system is shown in Figure 15.
To illustrate the above statements, Wittkower, in his
There are several things that should be remarked. First of all, Leonardo is basing his system on the statements of Virtruvius, as mentioned in the introduction, i.e., simple proportions, circle, square. Secondly, although Leonardo's "man in a circle"
is the best-known example, it is far from being the only Renaissance
attempt at making a drawing based on the statement of Vitruvius
[11].
The following examples that connect the "human in a circle"
with letters of the alphabet are taken from Geofroy Tory's
Start with a pyramid such that a section parallel to one of the edges of the square base is an equilateral triangle. Now consider the triangle obtained when we take a section along the diagonal. This triangle is the one that "fits" the Parthenon. But there is absolutely no historical basis to either Viollet le Duc's solution or any of the other exotic ones that have been suggested.[12]
Figure 22
shows the facade of the type C1 houses at Stuttgart (1927) [Le
Corbusier 1929: 12]. [15]
According to It should be noted that nowhere does Le Corbusier ever say what exactly the "place of the right angle" is supposed to do from a mathematical viewpoint. Le Corbusier himself, as he states, understood little of the strictly mathematical details of what he was doing and he even wrote in 1925, "I studied mathematics, but it did not help me later on. Perhaps however it helped form my intellect" [Le Corbusier 1968: 29; Herz-Fischler 1984: note 9]. What is so interesting about Le Corbusier is his interest in using mathematics and science despite his educational handicap. From a technical point of view, the "place of the right angle" may be explained by means of the diagram in Figure 23.
A word of caution is in order concerning the analysis of Le
Corbusier's works involving the "place of the right angle".
Namely, it is not always possible to tell where he started. Sometimes
it is not clear how certain lines were drawn (see the opening
quote with Case Study 11 below), and sometimes it appears that
several points may have been the starting point. An example of
this is Maisons La Roche-Jeanneret (1923) [Stonorov and Boesiger
1937: 68;
If we now return to drawing 1087 shown in Figure
2 and compare it with the preliminary sketch 1086 shown in
Figure 20 we notice that something
has been added, namely the "golden number" relationship
A:B = B:(A+B). By checking the Atelier Record Book at the Foundation
Le Corbusier I learned that this drawing was made at least 1-1/2
years after the plans for Garches were drawn and approximately
a year after the building was completed. Interestingly in his
earlier writings Le Corbusier had shown himself to be strongly
against the use of the "golden number" [Herz-Fischler
1979]. Then under the influence of Ghyka's
In addition to being an architect, Le Corbusier was a painter. Together with Amedée Ozenfant he founded the so-called "purist" school of painting in reaction to certain tendencies in Cubism. It was during this period that Charles–Edouard Jeanneret adopted the name Le Corbusier, "the crow".[17] The purist paintings of Jeanneret and Ozenfant were based
on a well-defined system, which, however, varied somewhat from
one period of time to another. The basic canvas size was 81 cm
x 100 cm. Inside this canvas two equilateral triangles, facing
in opposite directions, were drawn. The two intersections points
of the triangles determined two "places of the right angle"
(see Case Study 10), which in turn determined the vertices of
two right triangles with a second vertex coinciding with a vertex
of the equilateral triangle. The setup is shown in the diagram
in Figure 25 [18]. Figure 26 shows
Note that there is no "golden number" proportion in the analysis, indeed Le Corbusier's own writings show that he was strongly against the use of the golden number at that time (see Case Study 10).
The system of proportion known as the Modulor was first presented
in the 1948 book of the same name. Although Le Corbusier used
it in his later works [Le Corbusier 1968], and although it is
often talked about, there are in fact few architects who actually
used this system. One example is Sert's Eastwood Project on Roosevelt
Island, New York City, published in the August 1976 issue of
Stripped of all its romantic elements, the modulor system is very straightforward: basic heights a, 2a (113 cm for the "red" series and 226 cm for the "blue" series) are chosen and one then simply multiplies these heights by increasing and decreasing powers of the "golden number" to obtain the values in the series. (The "golden number" = (1+Ö5)/2). It is often denoted by the Greek letter phi (f), but I prefer to denote it by the letter G in order to avoid entering into the controversy of the applications of the "golden number".) Thus, ^{-2}....aG^{-1}.... a .... aG ....
aG^{2}....(likewise for 2a). The actual values are tabulated in Now what makes these numbers into a "modulor system" is the fact that if we add any two numbers in the sequence we obtain the next term.. This is called the Fibonacci property.
Le Corbusier, however, was not satisfied in presenting the Modulor system in just this straightforward form. It was his desire to connect the "golden number"-based sequence with the "place of the right angle" (see Case Study 10). The solution to this may be explained in Figure 27 as follows:
[2] The school year 1983-1984 was the last time that
I taught the course. The new director believed that poetry was
more important than mathematics and that I should limit the mathematical
level to constructing models of dodecahedra etc. I informed the
chairman of the Department of Mathematics that in good conscience
I could no longer teach the course. After another two years the
school dropped the mathematics course and most of the other "scientific"
courses from the curriculum. As the subtitle of the book suggests,
I tried to teach the students how to analyse architectural objects
and situations with a view to actually obtaining numerical values
for the various dimensions involved. I completely eschewed such
abstract topics as regular polyhedra, groups etc. [3] For a complete
list of my articles and books, the reader may consult my
web page. [4] The first part of the book dealt with the techniques
and tools of solving real problems with an emphasis on a decomposition
of the problems into small steps, each of which could solved
by simple formulae (Phythagorean theorem, etc.). This is the
"An Algorithmic Approach" referred to in the title
of the book. The second part of the book was the problem section.
After the first chapter on proportions came planar problems,
spatial problems, true shape problems, conics, curvature and
optimization. As examples, problem II.8 dealt with the overhang
required to provide shade when the sun was at a certain angle;
problem II. 13 dealt with difficulties (placements of columns,
projectors, etc.) that I encountered in one of the Carleton School
of Architecture's classrooms; problem II.28 involved the analysis
of a housing project in Cluj, Rumania; II.33 with the allowable
location of seats in the physics lecture halls at the University
of Colorado; III.11 and III.12 were based on the Wright's Guggenheim
Museum and dealt with spiral ramps, etc. [5] [Howard and Longair 1982] demonstrates in a scientific
and forceful way the difference between theory and practice.
[6] This example was kindly pointed out to me by Helmut
Schade. Some color plates of Baalbek may be found in Laroche
1979: 142]. [7] For an example see [Creswell 1969, I, 1: 73]. [8] The records are of great interest to the modern
architect for they show how little the difficulties and relationships
with clients have changed in 600 years! We also remark that the
pitfalls of the committee system are nothing new. To top it all
off, the committee hurled invectives at the poor architect: "
He served the building badly, and in the end he gave great loss
and damage to the building by reason of his own malfeasance."
See [Ackerman 1949: 96, footnote 42]. See also [Frankl 1945];
[Rooseval 1944] (Norwegian Architecture); and a very interesting
recent book, [Padovan 1999], especially p. 181. [9] For the mathematical computation at that time, see
[Frankel 1945: 53 and Appendix]. The figure obtained was rounded
off to 84 [10] For related material see: [Branner 1957]; [Koop
and Jones 1933]; [Shelby 1971]; [Shelby 1972]; [Shelby 1961];
[Shelby 1965]. Note: The papers by Shelby are an excellent source
of additional references. [11] Other examples of a "man in a circle"
are given in [Wittkower 1971: 14, Plates 1-4]. [12] To compare these fanciful flights of imagination
with ancient techniques, see [Dinsmoor 1923-I], [Dinsmore 1923-II]
and [Dinsmoor 1950]. [13] The number 1086 in Figure 20 {old Figure 21} corresponds
to the number in the Atelier Record Book which is preserved in
the Fondation Le Corbusier in Paris, and it is this number which
enables us to date the drawing. This particular reproduction
is taken directly from the original microfilm in the Fondation
Le [14] Figure 21 {old Figure 22} is the only preserved
preliminary sketch by Le Corbusier for Garches that shows any
"regulating lines". This particular reproduction is
taken directly from the original microfilm in the Fondation Le
Corbusier, but some of the lines have been electronically enhanced
for reproduction purposes. In the Archives reproduction not all
of the lines are visible. Note in particular the triangles inside
rectangles on the bottom left. [15] For full details, citations and references to the
various drawings with "regulating lines", see [Herz-Fischler
1984]. [16] It should also be noted that Le Corbusier himself
gives other versions of the discovery of the "principle",
see [Herz-Fischler 1984: note 9]. For recent discussions of Le
Corbusier's introduction to the method, [Vaisse 1997] and [Padovan
1999: 28, 318]. There does not seem to be any evidence that Michelangelo
or anybody else used the "place of the right angle".
In any case, this concept remained an important one for Le Corbusier
and in 1955 he even wrote a poem about it, [17] On Le Corbusier's paintings, see [Wohl 1971], which
contains many plates of Le Corbusier's paintings. For Ozenfant's
version of the name "Le Corbusier", which differs quite
markedly from that of Le Corbusier, see [Ozenfant 1968: 113].
[18] For details on the system, see [Fischler 1979].
Ars sine scientia nihil
est. Gothic Theory of Architecture in the Cathedral of Milan.
Art Bulletin 31 (1949).Branner, R. 1957. Three Problems from the
Villard de Honnecourt Manuscript. Choissy, A. 1899. Creswell, K. 1969. Dinsmoor, W. 1923-I. How the Parthenon was
Planned. Modern Theory and Ancient Practice. Dinsmoor, W. 1923-II. How the Parthenon was
Planned. Modern Theory and Ancient Practice. Dinsmoor, W. 1950. Fischler, Roger. 1979. The Early Relationship
of Le Corbusier to the 'Golden Number'. Frankl, P. 1945. The Secret of the Medieval
Masons. Hersey, George. 1976. Herz-Fischler, Roger. 1984. Le Corbusier's
'Regulating Lines' for the Villa at Garches (1927) and Other
Early Works. Herz-Fischler, Roger. 2000.
Howard, Deborah and Malcolm Longair. 1982.
Harmonic Proportion and Palladio's Koop, D. and G. Jones. 1933. Laroche, R. 1979.
Le Corbusier. 1929. Tracés régulateurs.
Le Corbusier. 1949. Le Corbusier. 1968. MacCurdy, E. 1956. Ozenfant, A. 1968. Padovan, R. 1999. Panofsky, Erwin. 1968. Robertson, D. 1929. Rooseval, J. 1944. Ad triangulum, ad quadratum.
Shelby, L. 1961. Mediaeval Masons Tools I:
the Level and the Plumb Rule. Shelby, : 1965. Mediaeval Masons Tools II:
Compass and Square. Shelby, L. 1971. Mediaeval Mason's Templates.
Shelby, L. 1972. The Geometrical Knowledge
of the Mediaeval Masons. Smithson, A. and P. Smithson. 1970. Stonorov, O and W. Boesiger. 1937. Tory, Geofroy. 1970. Vaisse, Pierre. 1997. Le Corbusier et le gothique.
Viollet-le-Duc, E. 1863. Wittkower, Rudolf. 1971. Wohl, R. 1971. Introduction.
A Mathematical
History of Division in Extreme and Mean Ratio (Wilfrid Laurier,
1987) and The Shape of the Great Pyramid (Wilfrid Laurier,
2000), and the forthcoming The Golden Number.
Copyright ©2001 Kim Williams top of
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