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By George David Birkhoff
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 provides the subsequent mathematical formula of the basic challenge: "Within every one type 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 numerous uncomplicated periods of aesthetic gadgets in nice element. the writer distinguishes among "formal" and "connotative" institutions, and explains that "our cognizance should be directed virtually solely towards the formal part of paintings, to which on my own the elemental formulation of aesthetic degree should be quantitatively applied," yet with "no purpose of denying the transcendent significance of the connotative part in all artistic art."
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Additional info for Aesthetic Measure
Of the two general cases represented by the quadrilaterals (c) and (d) it is clear that the convex type (c) is definitely superior to the alternative re-entrant quadrilateral (d). The latter quadrilateral suggests a triangle from which a triangular niche has been removed. C^4) POLYGONAL FORMS I t is not the mere fact that the quadrilateral is re-entrant which is decisively unfavorable. Consider, for example, the ordinary six-pointed star having the same outline as the familiar hexagram. This star is evidently highly pleasing in form, and yet is re-entrant.
We shall not attempt at this juncture to compare quadrilaterals and triangles. Thus there are two kinds of aesthetic factors brought to light by our examination of quadrilaterals. The first is of negative type and is occasioned by the fact that the quadrilateral is re-entrant. Further insight into the nature of this factor will be obtained as we proceed. The second factor is connected with the parallelism of sides. As we have observed, it is more convenient to regard this second factor in its negative aspect, when it appears as that of diversity of directions.
But for a rectangular tile, such as we are considering, this is certainly not the case, since tiles with a considerably greater ratio are very pleasing and are used frequently for decorative purposes. T o avoid all suspicion of either ambiguity or lack of utility, we must therefore restrict ourselves further to rectangles whose characteristic ratio is plainly between ι and 1. 414 . . in the figure above. All such rectangular forms are both pleasing and useful. 25), but does not take account of the usefulness of rectangular forms.