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not immediately suggest algorithms, quantifiable properties, or objective formulas.
But some do and it is with those that our discussion begins.
10.2.1 Formulaic and Geometric Theories
The mathematician George David Birkhoff published a mostly speculative topic in
1933 titled “Aesthetic Measure”. Birkhoff limits his theory to aspects of pure form
(the “formal”) and doesn't address symbolic meaning (the “connotative”). He then
proposes the formula
M
O/C
where
M
is the measure of aesthetic effective-
ness,
O
is the degree of order, and
C
is the degree of complexity. Birkhoff (
1933
)
notes, “The well known aesthetic demand for 'unity in variety' is evidently closely
connected with this formula.”
Birkhoff warns that his measure can only be applied within a group of similar
objects and not across types such as a mix of oil and watercolour paintings. He
also finesses variation in experience and taste intending
M
to be a measure for an
“idealised 'normal observer' ” as a sort of mean of the population.
While most of the topic is presented from a mathematical point of view, it is
sometimes forgotten that Birkhoff begins with an explicit psychoneurological hy-
pothesis. He describes complexity (
C
) as the degree to which unconscious psycho-
logical and physiological effort must be made in perceiving the object. Order (
O
)is
the degree of unconscious tension released as the perception is realised. This release
mostly comes from the consonance of perceived features such as “repetition, sim-
ilarity, contrast, equality, symmetry, balance, and sequence.” While Birkhoff views
complexity and order as ultimately psychological phenomena, for analysis he op-
erationalises those concepts using mathematical representations. He then goes on
to analyse examples such as polygons, vases, and harmonic structures in music to
illustrate his theory.
Birkhoff's theory has been disputed from its first publication. For example, in
1939 Wilson published experimental results showing that Birkhoff's measure did
not correlate with actual subjects' stated aesthetic preferences regarding polygons
(Wilson
1939
). Alternate formulas have been offered that seem to correlate more
closely with the judgements of subjects (Boselie and Leeuwenberg
1985
, Staudek
1999
). And for some, Birkhoff's formula seems to measure orderliness rather than
beauty, and penalises complexity in a rather unqualified way (Scha and Bod
1993
).
But there are at least two aspects of Birkhoff's work that remain in legitimate
play today. First is the intuition that aesthetic value has something to do with com-
plexity and order relationships. Second is the idea that modelling brain function
can illuminate discussions of aesthetics. Indeed, both of these reappear as themes
throughout this chapter.
The positing of mathematical bases for aesthetics long predate Birkhoff.
Pythagoras is traditionally credited with the discovery that dividing a vibrating
string following simple consecutive integer ratios such as 1:2, 2:3, and 3:4 yields
pleasing harmony relationships. The Golden Ratio
φ
, an irrational constant approx-
imately equal to 1.618, and the related Fibonacci series have been said to generate
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