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By George David Birkhoff

ISBN-10: 0674730224

ISBN-13: 9780674730229

Each reader of this booklet will locate it thought-provoking. Its first charm 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 every one category of aesthetic items, to outline the order O and the complexity C in order that their ratio M=O/C yields the cultured degree of any item within the class." In next chapters, this challenge is solved for varied basic periods of aesthetic gadgets in nice aspect. the writer distinguishes among "formal" and "connotative" institutions, and explains that "our cognizance can be directed virtually completely towards the formal aspect of artwork, to which by myself the elemental 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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Here the axis of rotation is a line perpendicular to the plane of the polygon at the center of symmetry. T h e square possesses such rotational symmetry about its center, and the angle of rotation is clearly 90 o or one quarter of a complete revolution. Similarly the rectangle and parallelogram have rotational symmetry with 180 0 as angle of rotation. 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.

With this granted, we conclude at once that the ratios 0'/C and 0/C are the same. In consequence the aesthetic measure only depends upon the ratio O to C: The final step can now be taken. Since it is not the actual numerical magnitude of / that is important but only the relative magnitude when we order according to aesthetic measure, and since M must increase with 0/C, we can properly define M as equal to the ratio of 0 to C. M = o) when the tone of feeling due to the associated ideas is indifferent.

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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Aesthetic Measure by George David Birkhoff


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