By George David Birkhoff
Each reader of this booklet will locate it thought-provoking. Its first allure is to the analytically-minded aesthetician; yet there are questions and implications some distance past any specialist's domain.
In bankruptcy 1, Professor Birkhoff offers the subsequent mathematical formula of the elemental challenge: "Within each one type of aesthetic gadgets, to outline the order O and the complexity C in order that their ratio M=O/C yields the classy degree of any item within the class." In next chapters, this challenge is solved for varied basic periods of aesthetic items in nice aspect. the writer distinguishes among "formal" and "connotative" institutions, and explains that "our awareness can be directed virtually solely towards the formal part of paintings, to which by myself the fundamental formulation of aesthetic degree will be quantitatively applied," yet with "no purpose of denying the transcendent value of the connotative part in all inventive art."
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Extra resources for Aesthetic Measure
On the other hand, the order 0 must take account of all types of associations induced by the objects, whether formal or connotative; and a suitable index is to be assigned to each. Unfortunately the connotative elements of order cannot be so treated, since they are of inconceivable variety and lie beyond the range of precise analysis. It is clear then that complete quantitative application of the basic formula can only be effected when the elements of order are mainly formal. Of course it is always possible to consider the formula only in so far as the formal elements of order are concerned, and to arrive in this way at a partial application.
Likewise, when the triangle is very nearly but not quite isosceles, or very nearly but not quite equilateral, there is produced a feeling of ambiguity. We are now in a position to list the aesthetic factors that have been so far encountered: vertical symmetry ( + ) , symmetry about an inclined axis (o), equilibrium ( + ) , rotational symmetry ( + ) , diversity of direc- (21 ) A E S T H E T I C M E A S U R E tions (—), small side (—), small angle or angle nearly i8o° ( — ) , other ambiguity (—).
An isosceles triangle illustrates the fact that a polygon may possess symmetry about an axis without having rotational symmetry; and the case of the parallelogram shows that it may possess rotational symmetry without having symmetry about an axis. B y the angle of rotation we mean of course the least such angle. Suppose that q successive rotations through this least angle effect one complete revolution. T h e least angle of rotation is then 360°/$'· B u t twice this angle, three times this angle, and, more generally, any multiple of it, are admissible angles also.