HEXAGRAM AND EQUILATER

rts and unity, the simplicity latent in the infinitely complex, the potential complexity of that which is simple. Proceeding to things visible and tangible, this indwelling harmony, rhythm, proportion, which has its basis in geometry and number, is seen to exist in crystals, flower forms, leaf groups, and t

52: PROPORTION

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apprehended. A knowledge of it formed part of the equipment of the painters who made glorious the golden noon of pictorial art in Italy during the Renaissance. The problem which preoccupied them was, as Symonds says of Leonardo, "to submit

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disposing figures and combining them in masses with almost mathematical precision. It would have been indeed surprising if Leonardo da Vinci, in whom the artist and the man of science were so wonderfully united, had not been greatly preoccupied with the mathematics of the art of painting. His Madonna of the Rocks, and Virgin on the Lap of Saint Anne, in the Louvre, exhibit the very perfection of pyramidal composition. It is however in his masterpiece, The Last Supper, that he combines geometrical symmetry and precision with perfect naturalness and freedom in the grouping of individually interes

EMPLOYMENT OF THE EQ

ANCE PA

TRICAL BASIS OF THE SIS

on 57: ASSY

GEOMETRICAL BASIS OF T

SI

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7). Whenever sculpture suffered divorce from architecture the geometrical element became less prominent, doubtless because of all the arts architecture is the most clearly and closely related to geometry. Indeed, it may be said that architecture is geometry made visible, in the same sense that music is number made audible. A building is an aggregation of the commo

EGYPTIAN; GREEK;

FERSON'S PEN SKETCH F

ITY OF

LICATION OF THE EQUIL

EUM AT

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EQUILATERAL TRIANGLE

e from the Egyptian pyramids to the cathedrals of Ile-de-France-are harmoniously proportioned, their principal and subsidiary masses being related, sometimes obviously, more often obscurely, to certain symmetrical figures of geometry, which though invisible to the sight and not consciously present in the mind of the beholder,

EQUILATERAL TRIANGLE I

THE HEXAGRAM IN GO

aissance, knew and followed certain rules, but though this theory be denied or even disproved-if after all these men obtained their results unconsciously-their

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the circle, the equilateral triangle and the square? Because, aside from the fact that they are of all plane figures the most elementary, they are intimately related to the body of man, as has been shown (Illustration 45), and the body of man is as it were the architectural archetype. But this simply removes the inquiry to a different field, it is not an answer. Why is the body of man so constructed an

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re and the triangle. In Thomas Jefferson's Rotunda for the University of Virginia, a single great circle was the determining figure, as his original pen sketch of the building shows (Illustration 61). Some of the best Roman triumphal arches submit themselves to a circular synop

with its third and fifth) is the musical equivalent of the equilateral triangle. It is scarcely necessary to dwell upon the properties and unique perfection of this figure. Of all regular polygons it is the simplest: its three equal sides subtend equal angles, each of 60 degrees; it trisects the circumference of a circle; it is the graphic

The same angle, erected from the central axis of a column at the point where it intersects the architrave, determines both the projection of the cornice and the height of the architrave in many of the finest Greek and Roman temples (Illustrations 67-70). The equilateral triangle used in conjunction with the circle and the square was employed by the Romans in determining the proportions of triumphal arches, basilicas and baths. That the same figure was a factor in the designing of Gothic cathedrals is sufficiently indicated in the accompanying facsimile reproductions of an illustration from the Como Vitruvius, published in Milan in 1521, which shows a vertical section of the Milan cathed

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l the formul? in the world-if it be really happy, the artist will probably find that he has "followed the rules without knowing them." Even while formulating concepts of art, the author must reiterate Schopenhauer's dictum that the concept is unfruitful in art. The mathematical analysis of spatial b