By George David Birkhoff

Each reader of this booklet will locate it thought-provoking. Its first attraction is to the analytically-minded aesthetician; yet there are questions and implications a ways past any specialist's domain.

In bankruptcy 1, Professor Birkhoff offers the subsequent mathematical formula of the basic challenge: "Within each one category of aesthetic items, 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 numerous uncomplicated sessions of aesthetic gadgets in nice element. the writer distinguishes among "formal" and "connotative" institutions, and explains that "our awareness can be directed virtually completely towards the formal aspect of artwork, to which on my own the fundamental formulation of aesthetic degree will be quantitatively applied," yet with "no purpose of denying the transcendent significance of the connotative facet in all inventive art."

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B y a 'niche' of a polygon is meant any outer area lying within the minimum enclosing convex polygon. Another important factor becomes obvious when the polygon is closely related to some uniform network of horizontal and vertical lines. The relationship may be direct, as in the case of the Greek cross (see (a) in the following figure) ; or it may be indirect, in that the polygon is directly related to a uniform diamond network with its sides equally inclined to the vertical (see (b)), while this network in turn suggests a uniform horizontal-vertical network.

Evidently a further actual aesthetic factor in many cases is some accidental connotation, such as is present in the case of crosses and the swastika. Our theory leaves such connotative factors out of account. 19. ON THE STRUCTURE OF T H E A E S T H E T I C FORMULA According to the general theory proposed in the first chapter, we seek an aesthetic formula of the type M = 0/C where M is the aesthetic measure, 0 is the order, and C is the complexity. In the case of polygonal form, 0 will be separated into five elements of order: O = F + E + R + HF — F.

As far as I can analyze my own impressions, I am led to the following explanation of this aesthetic uncertainty: For me and many other persons the orientation of the square with sides vertical and horizontal is superior to the orientation with inclined sides. 25). Hence there arises a feeling of 'unnecessary imperfection' (Chapter I , section 1 1 ) when the square is in the inclined orientation, just because it would be so easy to alter it for the better. As soon as this association, which is really irrelevant, is abstracted from, the inclined square (a) will be found, I believe, to be superior to the diamond (b).

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