Dirk Huylebrouck
and Patrick LabarqueSint-Lucas Institute for Architecture. Paleizenstraat 65, 1030 Brussels BELGIUM
Nevertheless, colour theory already forced the authors to recognise a new true application of the metallic means. A partitive mixing of 2 complementary colours is "well-balanced and pure" if rotation of a disk covered for 50% each by these colours yields a 50% grey. Yet, the colour or grey promised by the software is rarely the one of the final output. It turns out that proportions of grey tones related to the metallic means play interesting roles in partitive mixing. This will be described in Section 2, and Section 3 will show extensions of these results to "golden colours", whose Red, Green, and Blue components are each of the proportion 1/f. Some related computational recreations illustrate this colourful theory. More recently, the authors were again surprised to come across the golden number, in the completely different field of the determination of gears and the addition of speeds. This will be described in Section 4. Together with the colour theory applications, it shows that the appearance of the golden number is not always the result of a far-fetched imagination, like many statistical and historical 'studies' about the golden number. For instance, it has often been repeated, with dubious justification, that the rectangle of width 1 and length is considered as the most elegant one. Yet, in Section 5 we will answer a question concerning the optimality of the golden rectangle in artistic paintings or buildings' facades. This could reboot the mathematical career of the metallic means.
x (>1) is divided into two
pieces of lengths 1 and x-1, such that the whole length
is to 1, as 1 is to the remaining piece, x-1. Thus, the
x / 1 ratio must equal the ratio 1/(x-1). This
produces the equation x^{2}
- x -1 = 0, of which f = (1+Ö5)/2 »1.6180
is the positive solution; note that f-1=
1/f »0.6180.
More generally, the positive roots of x^{2}
- nx -1 = 0 yield the family of metallic means, for n=1,
2, .... For n=2, the positive root is the silver mean
s_{Ag}
= 1+Ö2; for n=3, it is
the bronze mean s_{Br}
= (3+Ö13)/2, etc.Literature often describes the golden section as a cookbook-like
formula. The historical justification is but a recipe: the (unequal)
division of a line such that the whole length stands to the larger
piece, as the larger is to the smaller. The authority of Euclid
may be a good reason for using this recipe, which was termed
simply "division in extreme and mean ratio". What importance
Greek mathematicians attached to this number is unknown, and
after all, the adjective "golden" dates from only 1844
[Markowsky 1992: 4]. The properties of these numbers have been
described extensively (for a comprehensive survey, see [Spinadel
1998]). Of course, half the diagonal of a rectangle with length
2 and width 1 corresponds to the 'irrational part', Ö5/2,
of the golden section, but this is so obvious that it is hardly
a "mathematical application". The only related, straightforward
yet surprising geometric properties are perhaps those of the
pentagon. Therefore, the related angle of 36° is perhaps
appropriately called the "golden angle" ( Some authors contend that the golden number is "the most
irrational of all irrational numbers", with the other metallic
means following in order of decreasing irrationality, because
their representations in continued fraction form are the slowest
converging. However, this justification does not use the conventional
mathematical approach, as it does not specify when a number is
more or less irrational than another number. To understand this,
note that since and similarly, Continued fractions converge more quickly when the successive denominators are large, but since the continued fraction expansion for f always involves adding only a 1 (the smallest possible positive integer) to the denominator, it converges the slowest (cf. [Spinadel 1998: 3-8]). While this may be important in some contexts, it is by no means standard to judge the "irrationality" of a number by the rate of convergence of its continued fraction expansion. The lack of straightforward mathematical reasons to justify
a prominent status of the golden section forces some to evoke
aesthetical, historical, or natural considerations. The golden
rectangle of width 1 and length f
would be the most elegant one, as various designs would show.
Yet, strong arguments show that the Greeks did not use it deliberately
in their Parthenon ( Still, the Fibonacci sequence is linked to f, since the quotient sequence, formed by taking the quotient of each term divided by the previous one, tends to f. Painters and sculptors often subdivide their (rectangular) canvas or stone using specific proportions, and 3 : 5 or 5 : 8 arrangements are frequent. This provides a connection to the Fibonacci numbers. Yet, 2 : 4 : 8 schemes are popular too and even used by great masters such as Seurat, Monet and Cézanne. It would be hard to prove that Fibonacci subdivisions lead to the "most beautiful" artworks. Furthermore, some contradiction is involved in the statement that golden section subdivisions are the key to fine art creations - why then would artists and their creative skills still be needed instead of a golden section computer program? In the last resort, if no mathematical, aesthetical or historical reasons help, some turn to statistics. Yet, Markowsky asserts that the statistical experiments about the golden rectangle do not seem to confirm a preference for the golden rectangle. It is not indisputable that one can pick out the "most elegant" (golden) rectangle from an arbitrary list. Other examples seem mere repetitions of the same unreliable statistical studies. Matila Ghyka was one of the prominent advocates of the golden section myth, and his popular books influenced many readers, such as Le Corbusier. Of course, there is nothing wrong with the fact that architects use mathematics or arithmetical proportions as inspiration, instead of some romantic, ecological, ergonomic or other kind of muse to create their works of art. It even puts mathematics in a position of great honour that someone like Le Corbusier uses it. Nevertheless, these claims are artistic ones, without scientific consequences. In spite of the vigorous considerations in disfavour of the golden section, many mathematicians continue to spread the myth. Mathematicians may be somewhat biased about the subject, maybe because they appreciate the attentions artists want to pay them, for once. A moderate conclusion comes from a psychologist, Christopher D. Green:
Nevertheless, we believe two cases can be added to the (not so lengthy) list of true applications of the golden section. The first is about colour theory, while the second stems from still another field, i.e., the study of planetary gears. In addition, the last section below concerns a classical mathematical justification for the interpretation of the golden section as an optimal solution, despite the previous critical considerations. Thus, maybe there is something more to be discovered about that (in)famous golden section.
_{1}
is the W-value of the first layer, and W_{2}
the value of the second, their product provides the resulting
W-value of the double layer. (Given two independent events A
and B, the probability of both A and B occurring
is given by their product: Pr(A and B)=Pr(A)Pr(B).)
The illustration shows such a subtractive mixture (see the upper
portion of Figure 3).Rapidly rotating a disk with two different grey tones is another
technique to judge the final impression of a black and white
mixture. At each moment, a pixel from one grey tone OR from the
other grey tone appears. Thus, the probability of seeing one
of the two pixels in the rotating result will correspond to their
weighted average, i.e., if the W These definitions are not of mere theoretical importance.
Partitive mixture, as a reference, can be used to determine the
W value of a transparent layer. For the study of colour mixing
in the next section, combinations of transparent coverings of
a single and a double layer are used. In preparation to this,
we first work some similar operations with grey tones. On a rotating
disk, suppose a single layer of W grey covers a (1-r)
fraction of the disk, and a double layer of W grey covers a r fraction. Then their mixture corresponds
to a grey tone with value The following natural question arises: given a single layer
covering a given share (fraction) of a disk, is there a corresponding
grey value W^{2}-(1-n)W-1=0.For The first of these values, 0.618, is the inverse of the golden number and the second, -1.618, its opposite. Rejecting the negative value, an approximate 61.8% grey provides the solution. Summarising, if a disk is half covered by a 61.8% grey and the other half by a double layer of this grey, then the rotation yields a resulting grey that is 1/2 white and 1/2 black. Similarly, if a disk is 2/3 covered by a Ö2-1»41.4% white grey and the other 1/3 by a double layer of this grey, then the rotation yields a 1/3 white result. For 3/4 coverage by a (Ö13-2)/2 »30.3% grey and the other 1/4 by a double layer of this grey, then the rotation yields a 1/4 white result.
In particular, suppose that the three colours are tones of the colours Cyan, Magenta and Yellow (the additive primary colours). Their RGB values refer to the amounts of Red, Green and Blue (the subtractive primary colours) light in them, as was previously done for the amount of white. That is, a colour with an R-value of .30 has 30% of Red light. If all RGB values are 0, black is obtained, while 3 values of 1 yield "maximum white light". Three arbitrary but identical RGB values yield grey. The RGB values of Cyan, Magenta and Yellow can be placed in vectors: C=(R_{c},
G_{c}, B_{c}), M=(R_{m}, G_{m},
B_{m}) and Y=(R_{y}, G_{y},
B_{y}).Firstly, we compare the Red value _{1})R_{c}+r_{1}R_{m}R_{y}=w_{1}.Proceeding similarly for the Red components of Magenta and
Yellow produces a system of 9 equations, and 9 variables. The
equations of the system can be solved in groups of three, comparing
the amount of _{1})R_{c}+r_{1}R_{m}R_{y}=w_{1}(1-r _{2})R_{m}+r_{2}R_{y}R_{c}=w_{2}(1-r _{3})R_{y}+r_{3}R_{c}R_{m}=w_{3}This yields a fifth-degree equation in tR_{c}+t-1)^{2} ((t^{2}-t)R_{c}+2t+st-t^{2}-1)(-tR_{c}^{2}+(t-1)R_{c}+s)=0The third factor corresponds to the quadratic equation R_{c}^{2}+(1/t-1)R_{c}-s/t=0Thus, if The proposed mathematical tools have interesting or at least
amusing applications on the well-known problem of getting the
same colour in a print-out and on screen. Other applications
are the question of finding the subdivision to be used as an
approximation of a given colour using a pattern (partitive mixture)
of a set of given colours, in discrete amounts, such that there
is no combination that yields the desired colour (see [Huylebrouck
and Labarque 2001];
We indicate the central sun wheel by the letter A. Supposing
its radius is 1, the length of the arc on its circumference corresponds
to the change of the angle a (in radians)
in its centre. The middle ring with three planetary wheels on
it is represented by B, and its angular change by b.
Finally, g will be the angular change
of the outside wheel C, with radius For instance, suppose that wheel A is fixed, B the drive wheel,
and C the wheel that yields the result. The resulting arc g A comparison to the other gears yielded the ranking given
in
The geometric construction concerns a rectangle of arbitrary
length This produces a function: Its extreme value is computed by setting both the partial
derivative with respect to Solving these equations simultaneously yields the positive solution corresponding to a saddle point. The The silver section can be obtained through such an optimisation
procedure, but we summarise it in a somewhat simplified construction,
starting with a square ( The derivative corresponds to and the silver mean is the positive solution of ( The given formulation corresponds to what is most probably intended by the expression of "optimal solution", as given by so many authors, and refuted by others. It proposes a rectangle that is neither too small nor too large, with respect to a given verifiable objective criterion. It seems but a rephrasing exercise, using mathematical vocabulary, but it could help to reboot the mathematical career of the golden section. It is another matter to state that the given criterion produces
the most elegant rectangle. Nevertheless, psychological golden
section tests may be reformulated as "finding the rectangle
such that the added area is maximal when compared to two squares
constructed on the diagonal". People already have difficulties
in choosing the right shape of a rectangle simply because estimating
sizes is not easy. This obstacle may be important when making
judgements about elegance of shapes and sizes. Campbell, S. L. and C. D. Meyer, Jr. 1979.
Green, Christopher D. 1995. All that Glitters:
A Review of Psychological Research on the Aesthetics of the Golden
Section. Huylebrouck, D. 2001. Similarities in irrationality
proofs for p, ln2, z(2) and z(3), Pp. 222-231 in Huylebrouck, D. and P. Labarque. 2000. Symmetric
partition of colours on a rotating disk and the golden number.
Pp. 217-218 in Huylebrouck, D. and P. Labarque. 2001. Colourful Errors. Paper presented at symposium, Mathematics & Design 2001. Third International Conference, July 3-7, Melbourne, Australia. Huylebrouck, D. and P. Labarque. Golden grey. Submitted for publication. Markowsky, George. 1992. Misconceptions about
the Golden Ratio. Néroman, D. 1983. Neveux, Marguerite. 1995. Le Mythe du Nombre
d'Or. Sharp, John. 2002. Spirals
and the Golden Section, Spinadel, Vera W. de. 1998. Sturmey-Archer internal hub gearing (web site), http://www.sturmey-archer.com/p20.htm. Williams, Kim. 2001. The Shape of Divinity.
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