CORNELL UNIVERSITY LIBRARY GIFT OF THE PUBLISHER FINEi.ARTS NK4645.Hir"""""""^'-"'""^ Dynamic symmetry; the Greek vase, by Jay H 3 1924 019 526 882 DATE DUE M Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924019526882 DYNAMIC SYMMETRY PUBLISHED UNDER THE AUSPICES OF THE SCHOOL OF THE FINE ARTS, YALE UNIVERSITY, ON THE FOUNDATION ESTABLISHED IN MEMORY OF RUTHERFORD TROWBRIDGE AN UNUSUALLY HANDSOME NOLAN AMPHORA, FOGG MUSEUM AT HARVARD A theme in root-two DYNAMIC SYMMETRY THE GREEK VASE BY JAY HAMBIDGE MCMXX-YALE UNIVERSITY PRESS NEW HAVEN CONNECTICUT AND NEW YORK CITY LONDON • HUMPHREY MILFORD" ■ OXFORD UNIVERSITY PRESS tV, \ao5 COPYRIGHT, 1920, BY YALE UNIVERSITY PRESS f^COLLEGE >> THE RUTHERFORD TROWBRIDGE MEMORIAL PUBLICATION FUND ~^HE present volume is the first work published on the Ruther- ford Trowbridge Memorial Publication Fund. This Founda- tion was established in May, 1920, through a gift to Yale University by his widow in memory of Rutherford Trowbridge, Esq., of New Haven, who died December 18, 191 8, and who in 1899 had established the Thomas Rutherford Trowbridge Memorial Lectureship Fund in the School of the Fine Arts at Yale. It was in a series of lectures delivered on this Foundation that the material comprised in this volume was first given to the public. By the establishment of the Rutherford Trowbridge Memorial Publication Fund, the Univer- sity has been enabled to make available for a ' much wider audience the work growing out of lectures given at Yale thrdugh the generosity of one who sought always to render service to. the comhiunity in which he lived; and, through its university, to the world. , ACKNOWLEDGMENT r ■~^0 the School of the Fine Arts of Yale University credit is due for this book, on the shape of the Greek vase. When the discovery was made that the design forms of this pottery were strictly dynamic and it became apparent that an analysis of a sufficient number of vase examples would be equivalent to the recovery of the technical methods of Greek designers of the classic age, William Sergeant Kendall, Dean of the Yale School of the Fine Arts, imme- diately recognized its importance and offered his personal service and that of the University to help in the arduous task of gathering reUable material for a volume. After the investigation of the shape of the Greek vase was begun the two great American Museums, the Museum of Fine Arts of Boston and the Metropolitan Museum of New York City, through the curators of their departments of Greek art, Dr. L. D. Caskey and Miss G. M. A. Richter, volunteered their services in furthering the work. Most of the vase examples in the book were measured and drawn by the staffs of these Museums, which readily gave per- mission for their publication. Dr. Caskey, during the past year, has devoted almost all his time to a critical examination of the entire collection of the Greek vases in the Museum of Fine Arts with the result that he now has a book nearly ready for publication. He is especially equipped for a research of this character, because of the fact that in addition to his attainments as a Greek scholar, he has had much archaeological experience in the field at Athens and elsewhere. JAY HAMBIDGE TO M. L. C. CONTENTS VITRUVIUS ON GREEK SYMMETRY 9 PREDICTION BY EDMOND POTTIER IN 1906 RELATIVE TO GREEK SYMMETRY. ■. 10 CHAPTER ONE: THE BASIS OF DESIGN IN NATURE 11 CHAPTER TWO: THE ROOT RECTANGLES 19 CHAPTER THREE: THE LEAF 30 CHAPTER FOUR: ROOT RECTANGLES AND SOME VASE FORMS 44 CHAPTER FIVE: PLATO'S MOST BEAUTIFUL SHAPE 59 CHAPTER SIX: A BRYGOS KANTHAROS AND OTHER POTTERY EX- AMPLES OF SIMILAR RECTANGLE SHAPES 65 CHAPTER SEVEN: A HYDRIA, A STAMNOS, A PYXIS AND OTHER VASE FORMS 75 CHAPTER EIGHT: FURTHER ANALYSES OF VASE FORMS 91 CHAPTER NINE: SKYPHOI 105 CHAPTER TEN: KYLIKES 114 CHAPTER ELEVEN: VASE ANALYSES, CONTINUED 123 CHAPTER TWELVE: STATIC SYMMETRY 138 LIST OF PLATES AN UNUSUALLY HANDSOME NOLAN AMPHORA, FOGG MUSEUM AT HARVARD A WHITE-GROUND PYXIS, MUSEUM OF FINE ARTS, BOSTON A WHITE-GROUND PYXIS, METROPOLITAN MUSEUM, NEW YORK AN EARLY BLACK-FIGURED KYLIX OF UNUSUAL DISTINCTION, BOS- TON MUSEUM OF FINE ARTS A BLACK-FIGURED HYDRIA, MUSEUM OF FINE ARTS, BOSTON KANTHAROS, CONSIDERED BY THE WRITER AS ONE OF THE FINEST OF GREEK CUPS LARGE BRONZE HYDRIA, METROPOLITAN MUSEUM, NEW YORK A LARGE STAMNOS, METROPOLITAN MUSEUM, NEW YORK A DINOS AND STAND, MUSEUM OF FINE ARTS, BOSTON A BLACK-FIGURED AMPHORA FROM THE BOSTON MUSEUM A LARGE BELL KRATER WITH LUG HANDLES, MUSEUM OF FINE ARTS, BOSTON A RED-FIGURED KALPIS IN THE METROPOLITAN MUSEUM, NEW YORK A BLACK-FIGURED SKYPHOS, METROPOLITAN MUSEUM, NEW YORK A BLACK-FIGURED EYE KYLIX, MUSEUM OF FINE ARTS, BOSTON AN EARLY BLACK-FIGURED LEKYTHOS, STODDARD COLLECTION AT YALE A BLACK GLAZE OINOCHOE FROM THE STODDARD COLLECTION AT YALE FOREWORD JOME twenty years ago, the writer, being impressed by the inco- herence of modern design and convinced that there must exist in nature some correlating principle which could give artists a con- trol of areas, undertook a comparative study of the bases of all design, both in nature and in art. This labor resulted in the de- termination of two types of symmetry or proportion, one of which possessed qualities of activity, the other of passivity. For convenience, the active type was termed dynamic symmetry, the other, static symmetry. It was found that the passive was the type which was employed most naturally by artists, either consciously or unconsciously; in fact, no design which would be recognized as such — unless, indeed, it were dynamic — would be possible without the use, in some degree, of this passive or static type. It is apparent in nature in certain crystal forms, radiolaria, diatopis, flowers and seed pods, and has been used consciously in art at several periods. The principle of dynamic symmetry is manifest in shell growth and in leaf distribution in plants. A study of the basis of design in art shows that this active symmetry was known to but two peoples, the Egyptians and the Greeks; the latter only having developed its full possibilities for purposes of art. The writer believes that he has now recovered, through study of natural form and shapes in Greek and Egyptian art, this principle for the proportioning of areas. As static symmetry is more or less known and its principles easily under- stood, its explanation will be reserved for a chapter at the end of this book. Dy- namic symmetry, on the contrary, is entirely unrecognized in modern times. It is more subtle and more vital than static symmetry and is pre-eminently the form to be employed by the artist, architect and craftsman. After an explanation of the fundamental principles of this method of proportioning spaces, the writer will attempt a complete exposition of its application in art through analyses of specific examples of Greek design. He believes that nothing better can be found for this purpose than Greek pottery, inasmuch as it is the only pottery which is absolutely architectural in all its elements. There is no essential difference between the plan of a Greek vase and the plan of a Greek temple or theater, either in general aspect, or in detail. The curves found in Greek pottery are identical with the curves of mouldings found in Greek temples. There are com- paratively few temples and theaters, while there are many thousands of vases, many of these being perfectly preserved. Other reliable material for study is furnished by the bas-reliefs of Egypt, many of which, like the vases of Greece, are still intact. The history of dynamic symmetry may be given in a few words: at a very 8 DYNAMIC SYMMETRY early date, possibly three or four thousand years B. C, the Egyptians devel- oped an empirical scheme for surveying land. This primitive scheme was born of necessity, because the annual overflow of the Nile destroyed property bound- aries. To avoid disputes and to insure an equitable taxation, these had to be re- established; and of necessity, also, the method of surveying had to be practica- ble and simple. It required but two men and a knotted rope. When temple and tomb building began, it became necessary to establish a right angle and lay out a full sized plan on the ground. The right angle was determined by marking off twelve units on the rope, four of these units forming one side, three the other, and five the hypotenuse of the triangle, a method which has persisted to our day. This was the origin of the historic "cording of the temple.'"" From this the step to the formation of rectangular plans was simple. From the larger operation of surveying, and fixing the ground plans of buildings by the power which the right angle gave toward the defining of ratio- relationship, it was a simple matter to extend and adapt this method to the elevation plan and the detail of ornament, in short, to design in general, to the end that the architect, the artist or the craftsman might be able to control the proportioning and the spacing problems involved in the construction of build- ings as well as those of pictorial composition, hieroglyphic writing and decora- tion. At some time during the Sixth or Seventh Century B. C. the Greeks ob- tained from Egypt knowledge of this manner of correlating elements of design. ' In their hands it was highly perfected as a practical geometry, and for about three hundred years it provided the basic principle of design for what the writer considers the finest art of the Classic period. Euclidean geometry gives us the Greek development of the idea in pure mathematics; but the secret of its artistic application completely disappeared. Its recovery has given us dy- namic symmetry— a method of establishing the relationship of areas in design- composition. VITRUVIUS ON GREEK SYMMETRY' r "^HE several parts which constitute a temple ought to be sub- ject to the laws of symmetry; the principles of which should be familiar to all who profess the science of architecture. Symmetry results from proportion, which, in the Greek lan- guage, is termed analogy. Proportion is the commensuration of the various constituent parts with the whole, in the existence of which symmetry is found to consist. For no building can possess the attributes of composition in which symmetry and proportion are disregarded; nor unless there exists that perfect conformation of parts which may be observed in a well-formed human being. . . . Since, therefore, the human frame appears to have been formed with such propriety that the several members are commensurate with the whole, the artists of antiquity must be allowed to have followed the dictates of a judgment the most rational, when, trans- . ferring to the works of art, principles derived from nature, every part was so regulated as to bear a just proportion to the whole. Now, although these principles were universally acted upon, yet they were more particularly at- tended to in the construction of temples and sacred edifices — the beauties or defects of which were destined to remain as a perpetual testimony of their skill or of their inability." PREDICTION BY EDMOND POTTIER IN 1906 RELATIVE TO GREEK SYMMETRY WILL add that the proportions of the vases, the relations of dimen- sions between the different parts of the vessel, seem among the Greeks to have been the object of minute and delicate researches. We know of cups from the same factory, which, while similar in appearance, are none the less different in slight, but appreciable, variations of structure [cf., for example, Furtwangler and Reichhold, "Griechische Vasenmalerei" p. 250). One might perhaps find in them, if one made a profound study of the. subject, a system of measurement analogous to that of statuary. We have, in fact, seen that at its origin the vase is not to be separated from the figurine (p. 78); down to the classical period it retains points of similarity with the structure of the human body (Salle H). As M^ Froehner has well shown in an ingenious article {Revue des Deux Mondes 1873, ^- CIV, p. 223), we our- selves speak of the foot, the neck, the body, the lip of a vase, assimilating the pottery to the human figure. What, then, would be more natural than to sub- mit it to a sort of plastic canon, which, while modified in the course of time, would be based on simple and logical rules? I have remarked' ("Mo««»?:'' Fig. iia. Fig. 11^. Fig. I la shows two perpendiculars in the rectangle, and rectangular, spirals wrapping around two poles or eyes. If, as in Fig. iil>, four perpendiculars are drawn to the two diagonals, and then lines at right angles to the sides and ends through the intersections, the area of the rectangle will be divided into similar figures to the whole, the ratio of division being two. / J c Fig. 1 2a. Fig. i2i>. If, instead of lines coinciding vi^h. the spiral wrapping, as in Fig. i la, lines are drawn through the eyes, and at right angles to the sides and ends, the rec- DYNAMIC SYMMETRY 15 tangle will be divided into similar shapes to the whole, with a ratio of three. (See Fig. 12.) AB is one third of AC, while AD is one third of AE. A rectangle whose side is divided into three equal parts by horizontal lines drawn through the points of intersection of the perpendiculars and the sides of the rectangle has a ratio between its end and its side of i, or unity, to 1.732 or the square root of 3. This is a root-three rectangle and has characteristics simi- lar to those of a root-two rectangle, except that it divides itself into similar shapes to the whole with a ratio of 3. AB, BC and CD are equal. (Fig. 13.) Lines drawn through the eyes of the spiral divide this rectangle into four equal parts. The square on the end of this rectangle is one-third the area of the square on the side. r A rectangle whose side is divided into four equal parts by a perpendicular has a ratio between its end and its side of one to two, or unity to the square root of four. This rectangle has properties similar to those of a root-two or a root-three rectangle, except that it divides itself into similar rectangles by a ratio of four, and the area of the square on the end is one-fourth the area of Fig. i4«. Fig. 14^. i6 DYNAMIC SYMMETRY the square on the side. This is a root-four rectangle. Lines drawn through the eyes of the spirals of a root-four rectangle divide the area into five equal parts similar to the whole. (Fig. 14^.) A rectangle whose side is divided into five equal parts by a perpendicular has a ratio between its end and its side of one to 2.236, or the square root of five. This area is a root-five rectangle and it possesses properties sirnilar to those of the other rectangles described, except that it divides itself into rectangles similar to the whole with ratios of five and six. A square on the end is to a square on the side as one is to five, that is, the smaller square is exactly one-fifth the area of the larger square. There is an infinite succession of such rectangles, but the Greeks seldom employed a root rectangle higher than the square-root of five. m Fig. i5«. Fig. 153. The root-five rectangle, moreover, possesses a curious and interesting prop- erty which intimately connects it with another rectangle, perhaps the most ex- traordinary of all. To understand this strange rectangle, we must consider the phenomena of leaf distribution. This root-five rectangle may be regarded as the base of dynamic symmetry.* Closely linked with the scheme which nature appears to use in its construc- tion of form in the plant world is a curious system of numbers known as a sum- mation series. It is so called because the succeeding terms of the system are obtained by the sum of two preceding terms, beginning with the lowest whole number; thus, i, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. This converging series of numbers is also known as a Fibonacci series, because it was first noted by Leonardo da Pisa, called Fibonacci. Leonardo was distinguished as an arith- metician and also as the man who introduced in Europe the Arabic system of DYNAMIC SYMMETRY 17 notation. Gerard, a Flemish mathematician of the 17th century, also drew attention to this strange systerri of numbers because of its connection with a celebrated problem of antiquity, namely, the eleventh proposition of the second book of Euclid. Its relation to the phenomena of plant growth is admirably brought out by Church,^ who uses a sunflower head to explain the phenomena. What is called normal phyllotaxis or leaf distribution in plants is represented or expressed by this summation series of numbers. The sunflower is generally accepted as the most convenient illustration of this law of leaf distribution. An average head of this flower possesses a phyllotaxis ratio of 34 x 55. These numbers are two terms of the converging summation series. The present inquiry is concerned with only two aspects of the phyllotaxis phenomena: the character of the curve, and the summation series of numbers which represents the growth fact approximately.' The actual' ratio can be ex- pressed only by an indeterminate fraction. The plant, in the distribution of its form elements, produces a certain ratio, 1.6 18, which is obtained by dividing any one term of the summation series by its predecessor. This ratio of 1.6 18 is used with unity to form a rectangle which is divided by a diagonal and a perpendicular to the diagonal, as in the root rectangles. (Fig. 19.) Xo/'"''^ ^ D '\ /. > i .6/e ; ^^ ^ ^<3 S' \ 2 \ Fig. ig. Fig. 20. "A fairly large head, 5 to 6 inches in diameter in the fruiting condition, will show ex- actly 55 long curves crossing 89 shorter ones. A head slightly smaller, 3 to 5 inches across the disk, exactly 34 long and 55 short; very large 1 1 inch heads give 89 long and 144 short; the smallest tertiary heads reduce to 21 and 34 and ultimately 13 and 21 may be found; but these being developed late in the season are frequently distorted and do not set fruit well. A record head grown at Oxford in 1 899 measured 22 inches in diam- eter, and, though it was not counted, there is every reason to believe that it belonged to a still higher series (144 and 233). "Under normal conditions of growth the ratio of the curves is practically constant. Out of 140 plants counted by Weisse, 6 only were anomalous, the error thus being only 4 per cent." A. H. Church, "On the Relation of Phyllotaxis to Mechanical Law.''^ i8 DYNAMIC SYMMETRY Thus, we may call this "the rectangle of the whirling squares," because its continued reciprocals cut off squares. The line AB in Fig. 19 is a perpendicular cutting the diagonal at a right angle at the point O, and BD is the square so created. BC is the line which creates a similar figure to the whole. One or unity- should be considered as meaning a square. The number 2 means two squares, 3, three squares, and so on. In Fig. 19 we have the defined square BD, which is unity. The fraction .618 represents a shape similar to the original, or is its reciprocal. Fig. 20 shows the reason for the name "rectangle of the whirling squares." 1,2,3, 4, 5, 6, etc., are the squares whirling around the pole O. .&IS Fig. 21. If the ratio 1.6 18 is subtracted from 2.236, the square root of 5, the remainder will be the decimal fraction .618. This shows that the area of a root-five recT tangle is equal to the area of a whirling square rectangle plus its reciprocal, that is, it equals the area of a whirling square rectangle horizontal plus one perpendicular, as in Fig. 21. The writer believes that the rectangles above described form the basis of Egyptian and Greek design. In the succeeding chapters will be explained the technique or method of employment of these rectangles and their application to specific examples of design analysis. CHAPTER TWO: THE ROOT RECTANGLES r ""^HE determination of the root rectangles seems to have been one of the earliest accomplishments of Greek geometers.' In fact, geometry did not become a science until developed by the Greeks from the Egyptian method of planning and sur- veying. The development of the two branches of the same idea went together. Greek artists, working upon this basis to elaborate and perfect a scheme of design, labored side by side with Greek philosophers, who examined the idea to the end that its basic principles might be understood and applied to the solution of problems of science. How well this work was done, Greek art and Greek geometry testify. As early as the Sixth Century B. C. Greek geometers were able to "deter- mine a square which would be any multiple' of a square on a linear unit." It is evident that in order to construct such squares the root rectangle must be em- ployed. We find the Greek point of view essentially different from ours, in con- sidering areas of all kinds. We regard a rectangular area as a space inclosed by lines, and the ends and sides of the majority of root rectangles, because these lines are incommensurable, would now be called irrational. The Greeks, how- ever, put them in the rational class, because these lines are commensurable in square.* This conception leads directly to another Greek viewpoint which resulted in the evolution of a. method employed by them for the solution of geometric problems, to wit, "the application of areas."" Analysis of Greek design shoWs a similar idea was used in art when rectangular areas were exhausted by the application of other areas, for example, the exhaustion of a rectangle by the application of the squares on the end and the side, in order that the area receiving the application might be clearly understood and its pro- portional parts used as elements of design. If the square on the end of a root- two rectangle be applied to the area of the rectangle, it "falls short," is "elliptic," and the part left over is composed of a square and a root-two rectangle. (See Fig. la.) If the same square be applied to the other end, so as to overlap the first applied square, the area of the rectangle is divided into three squares and three root-two rectangles. (See Fig. iL) And, if the square on the side of a root- two rectangle be applied, it "exceeds," is "hyperbolic," and the excess is com- posed of two squares and one root- two rectangle." (See Fig. ic.) This idea is quite unknown to modern art, but that it is of the utmost im- portance will be shown in this book by the analyses of the Greek vases. Let us now consider various methods of construction of the root rectangles. 20 DYNAMIC SYMMETRY and, of course, the whirling square rectangle. We will commence with the latter^ which is intimately connected with extreme and mean ratio, a geometrical con- ception of great artistic and scientific interest to the early Greeks. Using dy- namic symmetry, this problem of cutting a line in extreme and mean ratio may be solved through subtracting unity from the diagonal of a root-four rectangle; the Greek method was not essentially different. To the early geometers it was the cutting of a line so that the rectangle formed by the whole line and the lesser segment would equal the area of the square described on the greater segment.' J t^z S n s 1^2 s 7^2 S Fig. \a. Fig. \b. Fig. \c. Euclidean construction furnishes an easy method for describing not only the whirling square, but also the root-five rectangle, after the following man- ner: A square is drawn and one side bisected at A. The line AB is used as a radius and the semi-circle CBFD described. DE is a root-five rectangle. BC and DF are rectangles of the whirling square, as are also CF and BD. (Fig. 2.) Fig. 1. The relation of the rectangles, which have been described, to certain com- pound shapes derived from them will now be shown. If, in a rectangle of the whirling squares mapped out as in Fig. 3, a line parallel to the sides be drawn through the eyes A and B, it cuts from the major shape a root-five rectangle, /. e., a square and two whirling square rectangles, C, D, and E, — D being the square. Fig. 4 shows how a line drawn through the eyes F and G, parallel to DYNAMIC SYMMETRY 21 the end, defines also a root-five rectangle, C being the square. Obviously this may be done at either end and side, resulting in the determination of four root-five rectangles overlapping each other within the major shape. In a whirl- ing square rectangle (Fig. ^a), if lines be drawn through the eyes A, B, C, D parallel to the ends, and A and B connected by another line, an area will be Fig. 3- Fig. defined, composed of the square E and the rectangle F. This shape, composed of E and F, is numerically described as the rectangle 1.382. The square E is unity. The rectangle F is the fraction .382, this being the reciprocal of 2.618, 1. e., it is a whirling square rectangle, 1.618 plus i. (Fig. 5^.) If this 1.382 rectangle is divided by 2, the shapes G, H (Fig. 5c), result and each is composed of a square and a root-five rectangle. 1.382 divided by 2 equals .691, which, divided into unity, proves to be the reciprocal of 1.4472, and .4472 is the recip- rocal of root-five and is itself a root-five rectangle. Many Greek vases were constructed according to the principles inherent in this 1.382 shape. 'VS S c ■ rs s Fig. 5«. Fig. 5^. Fig. 5c If a whirling square rectangle is subtracted from, or applied to, a square, the defect is .382 or a whirling square rectangle plus a square. (See Fig. 6.) .618 subtracted from i. equals .382. If, as in Fig. 7, a whirling square rectangl e is 22 DYNAMIC SYMMETRY placed in the center of the shape 1.382, the "defect" area on either side is com- posed of a square and a whirling square rectangle. 5 ^j ; s rvs S wa ; S Fig. 6. Fig. 7- The reciprocal of 1.382 is .7236; .4472 multiplied by 2 equals .8944, and this result added to .7236 equals 1.618. (See Fig. 8.) The areaof Fig.-8 is com- posed of two root-five rectangles, .4472 x 2, plus a .7236 shape. Fig. 8. All of these shapes are found in abundance in both Egyptian and Greek art. The square is considered the unit form or monad. "lamblicus (fl. circa 300 A. D.) tells us that . . . 'an unit is the boundary between number and parts because from it, as from a seed and eternal root, ratios increase recip- rocally on either side,', i. e., on one side we have multiple ratios continually increasing, and on the other (if the unit be subdivided), submultiple ratios with denominators continually increasing." ("The Thirteen Books of Euclid's Ele- ments," by T. F. Heath, Def. Book VII.) The Reciprocal Ratios Within a Square The root rectangles are constructed within a square by the simple geometri- cal method shown in Fig. 9. AB is a quadrant arc with center D and radius DB. DC is a diagonal to a square and it cuts the quadrant arc at F. A line, parallel to a side of the square, is drawn through F. This line determines a root-two DYNAMIC SYMMETRY 23 rectangle and DE is its diagonal. A diagonal to a root-two rectangle cuts the quadrant arc at H. GD is a root-three rectangle, the diagonal of which cuts the quadrant arc at J. DI is a root-four rectangle and its diagonal cuts the quad- rant arc at L. DK is a root-five rectangle and so on. All the root rectangles may- be thus obtained within a square. Fig. 9. The root ratios outside of a square are obtained from diagonals, Fig. 10. AB, the diagonal of the unit form or square, determines the point C, the side of a root-two rectangle. The diagonal of a root-two rectangle, as AD, becomes the side of a root-three rectangle, as AE. AF, the diagonal of a root-three rec- tangle, becomes the side of a root-four rectangle, as AG. AH, the diagonal of a root-four rectangle, becomes the side of a root-five rectangle, as AI. AJ, the diagonal of a root-five rectangle becomes the side of a root-six rectangle, and so on to infinity. In any of these rectangles a square on the end is some even multiple of a square on the side. The square constructed on the line AC is dou- ble the square on AK; the square on AE is three times the area of the square on AK; the square on AG is four times the square on AK; the square on AI is five times the square on AK, etc. This was the Greek method of describing squares which would be any multiple of a square on a given linear unit.^ The given linear unit is the line AK. The rectangles inside the square are the reciprocals of the rectangles outside the square. A root-two rectangle inside the square, for ex- ample,is one-half the area of the root-two rectangle outside the same square; a root-three inside, one- third of a root- three outside; a root-four inside, one- 24 DYNAMIC SYMMETRY fourth of a root-four outside and a root-five inside, one-fifth of a root-five out- side. And a reciprocal to any rectangle is obtained by drawing a perpendicular from one corner. The whirling square rectangle and the root-five rectangle are placed within a square thus: ^"-•^^ i J.^-_ — ___, --^ ^. ^ 9 ^^ Fig. \\a. Fig. 11^. The square is first bisected by the line AB, to obtain a root-four rectangle or two squares. From the diagonal of this rectangle CB, unity, or BE, is subtracted to determine the point D, and CD, furnishes the side of the whirl- ing square rectangle FE. See Fig. i \a. A line drawn through the point D, parallel to a side of the square, determines the root-five rectangle GH. Fig. \\b. In a whirling square rectangle inscribed in a square, if lines be drawn through the eyes and produced to the opposite side of the square, a root-five rectangle is DYNAMIC SYMMETRY 25 constructed in the center of the square, see Fig. 12a. The area AB is this area, and if these lines be made to terminate at their intersection with the diagonals of the square, the whirling square rectangle CD, is defined as in Figs. 12^' and 12c. That this construction was used by the Egyptians in design is shown by the bas-relief in the form of a square herewith reproduced: a Wl Fig. 11a. Fig. lib. Fig. lie. When, as in Fig. 13, a whirling square rectangle is comprehended within a square, CD, the small square, AB, has a common center with the large square, CK, and if the sides of this small square, AB, are produced to the sides of the large square, CK, four whirling square rectangles, overlapping each other to the extent of the small square, AB, are comprehended in the major square. They are HK, EF, CD, and CJ, and the major square becomes a nest of squares and whirling square rectangles. \ I 1 / \ 1 1 1 e / / .^. 1 1 / \ 1 / "— -re-- 4-- ^-. ! V 1 -^tf — / 1 ^^^ 1 \ / y" "-; ^ -■' '"''1 ' \ \ [ ^ 1 Fig. 13. Analysis of the Egyptian bas-relief composition (Fig. 14) shows that its designer not only proportioned the picture but also the groups of hieroglyphs by the application of whirling square rectangles to a square. The outlines of 26 DYNAMIC SYMMETRY the major square are carefully incised in the stone by four bars, two of which have slight pointed projections on either end. The general construction was that of a in Fig. 12. Spacing for additional elements of the design is shown in c. Fig. 12, while b. Fig. 12, exhibits the grouping of the hieroglyphic writing. Fig. 14. Another bas-relief from Egypt shows also how a square which is defined by bars cut in the stone at the top and bottom of the composition has its area dynamically divided for a pictorial composition. In this example the designer ^ has used a root-five rectangle in the center of a square, Fig. 12a. The plan of this arrangement is obvious, Fig. 15. A simple theme in root- two is exhibited in Fig. 16. A goddess is pictured supporting a formalized sky in the shape of a bar. The spaces between the bars on either side of the figure were filled with hieroglyphic writing. These have been omitted in this reproduction. The overall shape of this composition is a DYNAMIC SYMMETRY 27 Fig. 15. Fig. 16. 28 DYNAMIC SYMMETRY root-two rectangle and the simple method of construction is shown in Fig. 17. BC is a square and the side of the rectangle is equal in length to the diagonal of this square: c . o f Fig. 17. AB equals BC. DB and EF are root- two rectangles, the side of each being equal to half the diagonal of the major square, or the line BG. Diagonals to the whole intersect the side of the major square at the points D F. Another theme in root-two is disclosed in Fig. 18. The general shape is a square, carefully defined by incised lines, as in the other examples. Fig. 18. DYNAMIC SYMMETRY 29 v/1 re. y( / \ 4 \ J- H i K m c M L & Fig. i9fl. Fig. 193. The plan scheme of this design is shown in Fig. iga. AB, CD, AE and FG, are four root-two rectangles overlapping each other in the major square, and the side of each, as CG, is equal to half the diagonal of the major shape. These rectangles subdivide the area of the major square into five squares and four root-two rectangles. In Fig. 19^, the use of this spacing, in its direct applica- tion to the design, is shown. The central portion of the major square, composed of the square HG and the root-two rectangle HL, is divided by the diagonals and perpendiculars of this rectangle. B is the center of the semicircle and BC is made equal to BA. This fixes the proportion of space to be occupied by the hawk and the field of formalized lotus flowers. MJ is composed of the two squares MD, DI and the root-two rectangle IJ. The square MD is divided into three parts and one of these parts forms the platform on which stands the hippopotamus god. This god is placed within the space KI. The sarhe con- struction applies to the other side of the composition. The examples of Egyptian bas-relief compositions described are, with one exception, arrangements within a square. These are used because of their obvious character. Like Greek temples and vase designs, the best Egyptian bas-relief plans are composed within the figures of dynamic symmetry, both simple and compound. The Egyptians were regarded by the Greeks as masters of figure dissection. The rational combinations of form, which we may recover from their designs, confirms this and sheds some light on the significance of the ceremonial when "the king, with the golden hammer," drove the pins at the points established by the harpedonaptae, the surveyors or "rope-stretchers," who "corded the temple" and related the four corners of the building with the four corners of the universe.^ CHAPTER THREE: THE LEAF r "^HE rectangles of dynamic symmetry consist of the root rec- ^ tangles, the rectangle of the whirling squares, and compound shapes derived from subdivision or multiplication of either the square root forms or the rectangle of the whirling squares. In both Greek and Egyptian design the compound shapes derived from the rectangle of the whirling squares and the root-five shape greatly preponderate. The rectangle of the whirling squares, as a separate design shape, appears, but seldom. This fact suggests that extreme and mean ratio, per se, has little aesthetic significance. Its chief feature appears to be its power as a coordinating faetor-when"usedTvitircertairrof the compound rectangles. There is unquestionable documentary evidence that the use of the compound rectangles, found so plentifully in Greek art, was not arbitrary. Their bases exist in nature and it is historical that the Greeks thoroughly understood the source from which they are derived. (See the Thirteenth Book of Euclid's Elements.) Their discovery in nature by the writer resulted from examination of the trussing of a maple leaf. The shape of this leaf strikingly resembles a regular pentagon. Fig. la. Fig. lb. The leaf is shown above in Fig. la, and the resemblance of the shape itself and of its trussing to the regular pentagon and its diagonals, is apparent in Fig. ib.lm. regular pentagon inscribed in a circle the relation of the radius of the escribed circle to the radius of the inscribed circle is i : .809. The fraction .809 multiplied by 2 equals 1.618, or the ratio of the whirling square rectangle. This means that if we escribe a square to the circle escribing a regular penta- gon (Fig. 2), the area shown by the heavy lines is represented by the ratio 1.809. A is a square and B two whirling square rectangles. This is a ratio often found in Greek design, among amphorae and skyphoi especially. The division of the pentagon with its escribed square produces two such areas, as in Fig. 3. DYNAMIC SYMMETRY 31 1 Fig. 2. Fig. 3. In Fig. 4, the point B in reference to the center A, is eighteen degrees and the natural sine of eighteen degrees or the line AC, is .309. This fraction multiplied by 2 equals .618. The rectangle AB, therefore, is composed of two whirling square rectangles, placed end to end, a common shape in Greek design. The entire area shown by the heavy lines in Fig. 5, is composed of four whirling square rectangles, two perpendicular side by side, and two horizontal end to end. Fig. 4. Fig. 5. A root-five rectangle is composed of a whirling square rectangle, plus its reciprocal, or 1.618 plus .618. Consequently the area shown by the heavy lines in Fig. 6a is composed of two root-five rectangles, and the area in i, defined by heavy lines, is equal to four root-five rectangles. ^ ^\ \ r 3 ■ ^^ Fig. 6fl. Fig. 6^. 32 DYNAMIC SYMMETRY The total distance AB in Fig. 7, is 1.809. BC is .809, CD is .309, AC is i or unity, and AD is unity minus .309, or .691. This fraction .691, is the reciprocal of 1.4472, or a square plus a root-five rectangle. ED is this shape, the key to the Parthenon plan and many other Greek designs. It is a favorite shape for many vases. ^- "^ < { /\ \ /J \--_ ^^.^ Fig- 7- The intersection of two diagonals to the pentagon, in Fig. 8, determines the area shown by the heavy lines, which is composed of two squares and two root-five rectangles or the ratio 1.38a. Fig. 8. The distance AB in Fig. 9, is the difference between 1.809 ^'^d 2, or .191, and this fraction multiplied by 2 equals .382, the reciprocal of 2.618. Therefore the area AD is composed of four shapes, two squares and two whirling square rectangles. Fig. 9. DYNAMIC SYMMETRY 33 The radius of a circle escribing a pentagon is i, and the radius of the inscribed circle is .809. Therefore the area AB, in Fig. 10, is composed of two whirling square rectangles. The area BC plus AD is composed of eight squares and Fig. 10. eight whirling square rectangles. If these areas BC, AD, are placed one over the other, the area is then expressed as 5.236, /. e., 1.236 plus four squares. The reciprocal of 5.236 is .191. (Fig. 11.) + + 4 Fig. II. The area ';,.i2fi. The relation of the diameter of the inscribed circle of a pentagon to the diam- eter of the escribed circle is the ratio 1.236, /. e., root five, 2.236, minus i, or .618 multiplied by 2 (the reciprocal of 1.236 is .809). When the squares escribing these circles are placed in position, it will be apparent that the larger square is greater than the smaller square by sixteen whirling square rectangles and twelve squares. (Fig. 12.) ^"^ ~--^ / V 1 Fig. 12. When four squares are placed in the pentagonal construction, as AB in Fig. 13, the area shown by the heavy lines is composed of two rectangles, each of 34 DYNAMIC SYMMETRY which consists of a square and two whirling square rectangles or the ratio 1.309. ^ \ '^^ 7 1.309 / ^^^ ^-^ 1 Fig. 13. The area AB, in Fig. 14, is composed of a square and a root-five rectangle, as is also the area BC. The areas BD, BE, are 1.309 rectangles. Fig. 14. The ratio 1.382 is obtained by dividing 1.309 into 1.809. It is represented by the area AB in Fig. 15, and consists of a square plus .382 and this fraction is the reciprocal of 2.618, i. e., a square plus a whirling square rectangle. Also, if this ratio of 1.382 is divided by two, it will be noticed that the area could be expressed by two .691 shapes, each of which is the reciprocal of 1.4472 or a square plus a root-five rectangle. The area BC is a whirling square rectangle, .691 divided into I.I 18 producing 1.618. The area CD is a square. Fig. 15. DYNAMIC SYMMETRY 35 A line drawn through the intersection of two diagonals of a pentagon divides the area of the major square as in Fig. i6, into three shapes, two of which are rectangles of the whirling squares and one is composed of a square and a root- five rectangle. 4 o * \\w*/^ x// ^ Fig. 1 6. When the area of a major square is subdivided, as in Fig. 17, four very in- teresting shapes result. AB is a rectangle of the whirling squares. BC is rep- resented by the ratio 1.1708, this being composed of .618 plus .5528, the latter ratio being the reciprocal of 1.809 or a square plus two whirling square rec- tangles. The ratio 1.1708 could be expressed by .4472 plus .7236. The rectangle BD is the ratio 1.7236, a square plus .7236, this fraction being the reciprocal of 1.382. The area BE, representing the ratio 1.099, i^ ^ complicated but very important shape. That it was used by the Greeks with tilling effect is evi- denced by a bronze wine container of the Fifth Century B. C, now in the Museum of Fine Arts in Boston. //?/ t .d07 ^"y/7a^\ ^\ 7S?-'-W a \ E ^^-^ . ^-^ i Fig. 17. Two of the four rectangles in Fig. 18 have been described. The area BD, being 1.0652, consists of a whirling square rectangle plus a root-five rectangle, .618 plus .4472. BE is a 1.382 rectangle. 36 DYNAMIC SYMMETRY , .^ ^1 \ T^ ■ lokSi l.3Bt 1 1/ I ^^^--^ ^--^ 1 Fig. 1 8. Fig. 19. The area of the major square in Fig. 19 is divided into twelve shapes. Two are squares. AB consists of two .382 or 2.618 rectangles, CD four such figures, while BC consists of four .854 shapes. This .854 shape is valuable. It consists of .618 plus .236; the latter being the reciprocal of 4.236 or root-five plus two. The ratio .854 is the reciprocal of 1.1708. Seven of the thirteen subdivisional figures in Fig. 20 are squares. AB is a square and BC consists of a square plus two root-five rectangles, the ratio being 1.8944, and its reciprocal .528. The area BD is represented by the ratio 2. 118, i. e., root five, 2.236 divided by two, 1.118, plus one. Fig. 20. The rectangle AB in Fig. 21, has a ratio of 1.4472; a square plus a root-five rectangle, .4472 being one-fifth of 2.236 and a reciprocal of that number. Fig. 21. DYNAMIC SYMMETRY 37 The area AB in Fig. 2'2a is a whirling square rectangle, 1.809 on the side and I.I 1 8 on the end. CB is the major square of this rectangle. The shape DE is the ratio 1.2764, i. e., .691 divided into .882. Of this area .691 by .691 makes a square, and .191, the difference between .691 and .882, divided into .691 fur- nishes 3.618, i. e., a whirling square rectangle plus two squares. The area BD, .882 by 1. 118, supplies the ratio 1.267. This ratio is more easily recognized if we consider its reciprocal .7888. Four root-five rectangle reciprocals equal the ratio 1.7888, .4472 multiplied by four. .7888, therefore, is four root-five rectan- gles minus one. It is .a beautiful shape and may be obtained readily from the whirling square rectangle. This particular ratio was discovered independently by Wm. Sergeant Kendall, in the form of overlapping whirling square rectangles creating a root- five rectangle by their union as in Fig. 22^^. The area AB in Fig. 23 is composed of two squares and two root-five rec- tangles, or the ratio 2.8944, i. e., 1.4472 multiplied by two; .691 divided into 2.000. The fraction is not quite .691, but this number is sufficiently close for all practical purposes. BC and CD are two equal areas each composed of a square and two whirling square rectangles, i. e., each has a ratio of 1.309. Fig. 23. 38 DYNAMIC SYMMETRY In Fig. 24 the area AB, unity on the end and 1. 191 on the side, is a square plus .191 and this fraction represents two squares and two whirling square rectangles. The area BC represents four whirling square rectangles; .618 mul- tiplied by four, 2.472 minus one or 1.472. DE is a root-five rectangle.* , ^^ y< __^ai , K ^^ \ rs / / \\ ,.,,. ""--^ ^^ 1 Fig. 24. The area AB in Fig. 25 is divided into squares, root-five rectangles and rec- tangles of the whirling squares. In Fig. 26 the area AB is composed of two rectangles each consisting of a square plus .382, this fraction being the reciprocal of 2.618. The area AB may be expressed also, as a square and a root-five rectangle, 1.4472. The area BC is composed of two whirling square rectangles. 7" rs- \^ *\ \ V"* S W J / / wa ». ^^ ^^ Fig. 25. Fig. 26. The whirling square rectangle AB in Fig. 27 may also be expressed as two squares and two root-five rectangles. The area AB in Fig. 28, consists of six whirling square rectangles. The side of this rectangle is 2.000 and the end .927. * Euclid, XIII, I, in substance proves that a rectangle which is .809 on the end and 1.809 on t^he side is a root-five rectangle. DYNAMIC SYMMETRY 39 Fig. ay. Fig. 28. In the thirteenth book of the Elements, Euclid proves the relationship of the end, side and diagonal of the whirling square rectangle. Proposition 8 is devoted to proving that diagonals to a pentagon cut each other in the propor- tion of the whirling square rectangle. The fact enunciated in this proposition suggests the reason why the Pythagoreans of the Sixth Century B. C. used the Pentagram as a symbol of their school. Fig. 29. The Pythagorean Pentagram Symbol. Fig. 30. Diagram from Euclid XIII, The first six propositions of the 13th book are devoted to the consideration of the relationships of areas described on lines connected with the whirling square rectangle. In the first proposition the geometrical construction brings out the fact that a rectangle, the end of which is .809 and the side 1.809, is a root-five rectangle. In the 9th proposition proof is furnished that the side of a hexagon and the side of a decagon added, form a line which is cut in extreme and mean ratio, and the side of the hexagon is the greater segment. (Fig. 31.) Proposi- tion 10 furnishes the proof that the square on the side of a pentagon inscribed in a circle is equal in area to the squares on the sides of a hexagon and a decagon inscribed in the same circle. Fig. 31a shows this relationship. This figure is of necessity a right-angled triangle. 4° DYNAMIC SYMMETRY Fig. 31. Fig. 31a. Later, in XIII, 18, the rectangular relationship is more clearly shown in a root-five rectangle. The Euclidean diagram of the i8th proposition is peculiarly interesting in the light of dynamic symmetry because it suggests what may have been the Greek method of constructing the dynamic rectangles in a square. The writer's method of describing a root-five rectangle in a square is shown in Fig. 32. Fig. 32. In the square AB, Fig. 32, draw the line CD, dividing the square into two equal parts. Draw ED, the diagonal to two squares. On DG describe a semi- circle. The arc of this cuts the line ED at F. Through the point F draw the line HI parallel to GB. The area HB is a root-five rectangle within the area of the square AB. DYNAMIC SYMMETRY 41 In the 1 8th proposition of the thirteenth book a diagram is furnished which illustrates the setting out of the "five figures" for the purpose of comparison. The "five figures," of course, mean the five regular solids. These solids were of much interest to the Greeks of the Sixth Century B. C, because it was then thought that the atoms of the elements, which made up the universe, were shaped like the tetrahedron, the octahedron, the cube and the icosahedron. The dodecahedron was regarded as the shape which encompassed all the others. The basis of the diagram In the i8th proposition of the 13th book is a semi- circle on a given line. In brief the operation is this : Fig- 33- AB is the given line and ABE is the semicircle. (See Fig. 23-) Euclid in sub- stance says: at A draw a line equal to AB at right angles to that line and call its point of termination G. The point C is midway between A and B. Connect C and G. In Euclid's diagram the point H Is the intersection of the line GC with the arc of the semicircle AEB. From H a line Is drawn parallel Fig. 34- to AG to meet AB at K, BL is made equal to AK. From the point L a line is drawn, parallel to AG to meet the arc of the semicircle at M. It is obvious that HLKM is a square and that HA and MB are rectangles of the whirling squares. In other words, Euclid has here constructed a root-five rectangle and defined the square In the center, as is often necessary in the analysis of Greek design. 42 DYNAMIC SYMMETRY Euclid further shows in this proposition that the comprehension of the icosahe- dron in the same sphere with the other four regular solids involves the side of the hexagon, the side of the decagon and the side of a pentagon inscribed in the same circle. AK, BL are two sides of the decagon and KL, KH, LM or HM the side of the hexagon, and MB is the side of a pentagon. The geometrical constructions used by Euclid for the comprehension of the five regular solids in the same sphere, suggest another method of determining the root rectangles of dynamic symmetry in a square. This method is based upon the fact that an angle in a semicircle is necessarily a right angle. ,. - ./.. %. 4 / //^ s \\^ /' 1 Fig. ■iS- Fig- 36. The simplest example of this is shown in Fig. 35, where ABC is a right angled triangle. B is also the center of the square AD. In Fig. 'T^d the line CB is revolved until it coincides with the side of the square, to determine the point E. The area AEFC is a root-two rectangle. It will be noticed that the diagonal of the reciprocal of the root-two rectangle AEFC cuts the diagonal of the whole at G, and that this point lies on the arc of the semicircle. 'If the line GC is revolved until it coincides with CE it will deter- mine the point for, a root-three rectangle. The poles or eyes of all the root rectangles, that is, the points where the diagonals of their reciprocals cut the diagonals of the whole will lie on the arc of this semicircle and in each case the lines similar to GC of the root- two rectangle will determine the points on CE for each successive rectangle. Fig. 37 suggests the construction for this. Fig. 37- DYNAMIC SYMMETRY 43 The geometrical fact established by Euclid that if a circle is described with a side of a whirling square rectangle as radius, this line equals the side of a hexagon, the end of the rectangle, the side of a decagon and the diagonal of the rectangle, the side of a pentagon, all inscribed in this same circle, suggests the construction of Figs. 39 and 40. 1 ''''' p 1 •'] \ ' 1 1 t i \ ■■ "^^4i' \ >i / \ \ \ H .y 1/ 1 B \ I \ 1 1 \! 1 1 I "■f 'f 1 Fig- Z9- Fig. 40. In Fig. 39, AC is the diagonal of a whirling square rectangle, BC the end and AB the side. AD is the side of a pentagon and AE is the side of a decagon. The line DG is a diagonal of a pentagon inscribed in the circle, and it cuts the side of the whirling square rectangle at H. The area BH is equal to two squares and AH is composed of two root-five rectangles, while HM is equal to four such shapes. The line PI passes through the point E of the decagon. AI is equal to two whirling square rectangles, while PC is equal to a 1.309 shape. NL is an area represented by the ratio 2.118 or 1.618 plus .5. This area is also equal to two root-five rectangles plus a square, 1.118 plus i. JK is a square escribing the circle with radius AB and ML is a whirling square rectangle in the center. The areas MJ and LK are each composed of two whirling square rectangles plus two squares. In Fig. 40 AB is the side of a hexagon equal to AC, the radius of the circle. BD, EF are sides of two equilateral triangles. These two lines divide each of the four whirling square rectangles AH, AG, CI and CJ into two equal parts. The area DF is a root-three rectangle. CHAPTER FOUR: ROOT RECTANGLES AND SOME VASE FORMS NALYSES of Greek and Egyptian compositions show that the artist always worked within predetermined areas. The enclosing rectangle was considered the factor which controlled and de- termined the units of the form. A work-of art thus correlated .became an entity, comparable to an organism in nature. It possessed an individual character, instinct with the life of design. Only such rectangles, simple or compound, were used, whose areas and sub- multiple parts were clearly understood. If the design for a vase shape were being t planned the artist would consider the full height of the vessel as the end or side of a certain rectangle, while the full width would be the other end or side. The choice of a rectangle depended upon its suitability for a purpose, both in shape : and property of propo'rtional subdivision. A rough sketch was probably made ; as a preliminary and this formalized by the rectangle. Most Greek pottery shapes, however, were traditional, being slowly developed through a long period of time; consequently, rough sketches of ideas must have been rare. From gener- ation to generation, from father to son, craft ideas were passed along, acquiring refinement gradually. Modern art, as a rule, aims at freshness of idea and individuality in tech nique of handling; Greek art aimed at the perfection of proportion and work- manship in the treatment of old, well-understood and established motifs. That this is true is not only proven by the standardized shapes of Amphora, Kylix, Kalpis, Hydria, Skyphos, Oinochoe and Lekythos, but by the accepted forms of temples, theaters, units of decoration, treatment of drapery, grouping of sculpture forms and even the proportions of the figure. The opportunity for individual expression existed only in superlative workmanship, in refinement, precision and subtlety. To win distinction as an artist it was necessary for the Greek to be a veritable master. The danger of overrefinement is feared by the modern artist, for it has become a tradition that this leads to sweetness and loss of virility, because it invariably ends in overwork of surfaces. But this peril was almost unknown to the ancient Greek, his care and energy were devoted largely to the refinement of the structure of his creations. Analysis of any fine Greek design is sure to disclose an arrangement of area which produces the quality of inevitableness, so conspicuously absent in mod- ern art. An example of such a theme is furnished by a handsome red-figured amphora of the Nolan type, in the Fogg Museum in Boston. Its greatest width divided into its height produces the ratio of 1.7071. This ratio shows that, as DYNAMIC SYMMETRY 45 Fig. I . Nolan Amphora in the Fogg Museum at Harvard. an area, it is composed of a square plus the reciprocal of a root-two rectangle, /. (?., I. plus .7071, the fraction being the square root of two divided by two. The amphora is contained within the area of a root-two rectangle plus a square on its side. The width of the lip, in relation to the overall form, shows that it is a side of a square comprehended in the center of the root-two rectan- gle. When this square is drawn and its sides produced through the major square, an interesting situation exists in area manipulation. The projection of the sides of this square through the major square produces in the center of that square a root-two rectangle so that the shape as defined by the lip is a square plus a root-two rectangle. Fig. 3a, but the square is on the end of the rectangle instead of on the side as it is in the major shape. The method of simple con- 46 DYNAMIC SYMMETRY struction by which the figures so far described were created is the drawing of a square and its diagonals. (Fig. 2^.) Fig. 2. The shaded area shows the rectangle of the Amphora design. The side of the root-two rectangle is equal to half the diagonal of the square. The method of construction by which the secondary square and root two are placed within the major shape, is shown in Fig.. 4, a, b and c. \ / \ y \ / w / \ / \ / \ \ Fig. 3a. Fig. 3^. A root-two rectangle, AB, is cut off within the major shape, its side being made equal to the diagonal of the major square. This applied rectangle is in Fig. 4«. Fig. 4^. Fig. A,c. DYNAMIC SYMMETRY 47 "defect" and the area left over is composed of two squares and one "root-two rectangle, as shown in b. Fig. 4. The same construction is used, working from the other end of the major shape, as shown in Fig. \c. Fig- 5- Through the centers of the small squares on each corner, lines are drawn paral- lel to the sides of the major figure. These lines determine the secondary square and root- two rectangle, shown in Fig. 5. A diagonal to this secondary shape determines the angle pitch of the lip, and its thickness, also the width of its base, and the width of the neck. (See Fig. i.) LK is this line. The foot of the amphora is proportioned by the small root-two figure and two squares at the base. DE is the root-two rectangle. A square is placed in the center of this shape, being CB. The width of the ring above the foot is the side of this square. The width of the top of the foot exhibits an interesting manipu- lation of the square and root-two figures at the base of the design. The line AB in Fig. I brings out the point. AB is a derived root-two rectangle, and its diag- onal is cut at J by a line through the point I. The thickness of the foot and its width at the bottom are determined by the diagonal and perpendicular of the root- two shape DE. (Fig. i.) The thickness of the ring above the foot is established by the line AB, in Fig. 6, a diagonal to a square and a root-two rectangle, intersecting the side of the square at C. I I — n ^^r 1 L__ Fig. 6. 48 DYNAMIC SYMMETRY Two white pyxides, ladies' toilet boxes, one in the Museum of Fine Arts,, Boston, and one in the Metropolitan Museum, New York, furnish examples of Greek design for comparative study. These two examples of the ancient pot- ter's craft are exactly of the same overall shape; the ratio in each case being 1. 207 1. This is a compound shape composed of the reciprocals of root four or half a square and root two, .5 plus .7071. The reciprocal of 1.2071 is .8284, and this divided by two equals .4I42, or the difference between unity and the square root of two, 1.4I42, /. e., the square root of two minus i. When a square is subtracted from a root-two rectangle the excess area is composed of a square and a root-two rectangle. rz 5 s < .414-2 X /. Fig- 7- ■ig. 8. The containing rectangle of each pyxis design, therefore, is composed of two .4142 figures, /. e., two squares plus a root-two rectangle. (Figs. 7 and 8.) The details of the two designs, however, are proportioned or themed differ- ently. In the Boston example the line AB of the analysis passes through the center of the root-two shape. (Fig. 10.) The line AB is the top of the pyxis. The width of the bowl at its narrowest point is equal to the end of the major root-two rectangle, i. e., it is the side of the square CD constructed in the cen- ter of this rectangle. (Fig. 9.) HI is a diagonal to a .4142 rectangle, i. e., half the composing shape. This line cuts the diagonal of the square CD at J. There- fore the rectangle JK Is a similar shape to the whole, two squares and a root- two rectangle, and is the containing rectangle of the knob. LK is composed of a square and a root-two rectangle. The line MN is a side of the square MNOP. When unity is applied to a 1.2071 rectangle the excess area is composed of two squares and two root-two rectangles. This is the elevation area of the foot. A WHITE-GROUND PYXIS, MUSEUM OF FINE ARTS, BOSTON {Compare with White-Ground Pyxis from New York) A theme in root-two DYNAMIC SYMMETRY 49 Fig. 9. Drawing by Dr. L. D. Caskey of the Pyxis in the Boston Museum of Fine Arts. R is the center of the two squares of the base. S is the center of the square MP. A further refinement in the design is shown by the sinking of the handle below the outer rim of the cover. The only variation from extraordinary exactitude is at the juncture of the lid shown by the line EF. This is worn at the edges so that it is difficult to determine this line precisely. The error, however, is so small that it cannot be shown in the drawing. This pyxis was measured and drawn by Dr. L. D. Caskey, of the Boston Museum of Fine Arts. The analysis of this vase shows a consistent Greek theme in area and it may readily be seen that not only the content of the design itself but the excess area not occupied by the design, may be expressed in terms of the whole and the two composing shapes, namely, the root-four and root-two reciprocals. HQ is a 5° DYNAMIC SYMMETRY square, HL two squares and a root- two rectangle. The application of this area to the square HQ leaves the area CL, a root-two rectangle. HA is a root-two rectangle. The application of the square HQ leaves the area CA, a square and a root-two rectangle. c e \ 1 >v >: ^ , A. \ V / / / / \ \ \ \ \ / ■ y / / I / \ 1 \ 1 \/ V / \ l\ / \c 1 ^. Fig. ID. Fig. II. The design plan of the pyxis in the Metropolitan Museum, New York, de- pends upon a manipulation of the diagonal to the overall shape and to the two composing figures, the root-four and root-two reciprocals. The manner in which this is done discloses an interesting feature of Greek design practice. It seems to have been recognized early that diagonals were the most important ' lines in the determination of both direct and indirect proportions. In the present example diagonals of the whole intersect diagonals of the root-two rectangle i at A and B, Fig. lo. Through these points are drawn the lines HF, EG, IJ and LK, through the points C and D. These lines subdivide the area of the root-two rectangle into squares and root-two shapes. CE, AG are squares, MC, DN, AP and BO are root-two rectangles. AI and BJ are two root-four rectangles, i. e., shapes of two squares each. IJ is the top of the pyxis, DH the square en- closing the handle or knob. AB in Fig. ii, is a square, one side of which is the width of the bowl at the narrowest point. The sides of this square produced, determine the root-two rectangle BC and fix the line of the base by their intersection with the diagonals of the whole at the points D and E. The intersection of the diagonals of the whole with the diagonals of half the major shape, at AB in Fig. 12, determine the thickness of the lid. A WHITE-GROUND PYXIS, METROPOLITAN MUSEUM, NEW YORK {Compare with the Boston White-Ground Pyxis) A theme in root-two DYNAMIC SYMMETRY 51 Fig. 12. Drawing of Pyxis in the Metropolitan Museum, New York. (Measurements checked by member Museum Staff.) The Fifth Century B. C. bronze oinochoe, Fig. 13, 99.485 in the Museum of Fine Arts, Boston, in its plan scheme, is another admirable illustration of the Greekmethod of arranging a theme in area. Thejug was measured and drawn by Dr. Caskey, before an analysis of the shape was made. The containing rectangle is a root-two shape, and all details are determined by a consistent arrange- ment of the elements of this figure. The diagonals and perpendiculars are drawn to the overall shape and a square described in the center of the* root-two figure AB. This square is CD, the side of which is equal to the width of the lip of the vase. The diagonals of the whole cut the sides of this square at E and F. This determines the area CF,. equal to two squares, EG, FH, and the root-two figure HI. A line drawn "from J to C cuts the side of the square GE at K. The line KLM divides the area of this square into two squares, CL, LI, and two root- two figures, GL and LE. The center of the square CL, fixes the top of the lip; 52 DYNAMIC SYMMETRY Fig. 13. Bronze Oinochoe in the Boston Museum. (Measured and drawn by L. D. Caskey.) the base of this square, ML, establishes the bottom of the lip. Diagonals and perpendiculars to the root-two figure HI, determine other proportions of the lip and handle juncture. Aline drawn through the center of the root-two figure BO, establishes the two root-two figures PO, PQ. The width of the vase, at the base, is fixed by the centers of the two squares SO, RQ. The sides of these squares produced, as from T to I, cut the diagonals of the whole and perpen- diculars, as at T and U. This fixes the figure UV, of which TW is a square. Diagonals to half the area of this square, as WX, determine the triangle in which the goats' heads are drawn. The beard, of one of these heads is shorter than that of the other, probably due to the molten bronze not entirely displac- ing the wax in the casting. If a square is applied to the other end of the shape occupied by the heads of the goats, other details are obtained. This design may now be understood as a theme in root-two and square. The drawing was made exactly the size of the original and no other analysis is possible. DYNAMIC SYMMETRY 53 A black-figured kylix, 98.920 in the Boston Museum (Fig. 14), fills an area composed of three root-two rectangles, and the width of the foot is the end of one of these shapes. AB is a root-two rectangle, BC is a square applied to it, CE is a diagonal to the excess area or to a square plus a root- two rectangle. AF is a root-two rectangle and its diagonal intersects CE at D, and fixes the width of the bowl. The depth of the bowl is determined by the point G, the intersec- tion of a diagonal of the square BG with the diagonal of the root-two rectangle AB. (Compare with Yale Skyphos, p. 62.) Fig. 14. (Measured, drawn and analyzed by L. D. Caskey.) The ratio of a black-figured kylix from Yale, Fig. 15, is that of a square plus a root-two figure or 1.4I42 plus i. In this case the square is drawn in the center and a reciprocal root-two figure on either end. AB is the side of the square. C and D are the intersections of diagonals of squares and root-two rectangles. I and J are the intersections of diagonals to two figures, each com- posed of a root-two rectangle plus the large square, with a line drawn through the middle of the large square, and G and H are the intersections of these same diagonals with the diagonals of the major square. The consistency of the proportions of the foot in relation to the width- of the bowl is now apparent. The point K is the intersection of the diagonal of the whole with the diag- onal of a square. An Attic black-figured hydria, 95.62 in the Boston Museum (Fig. 16), is a vase form of unusual distinction. The plan is a theme in root-two. The vessel is a splen- 54 DYNAMIC SYMMETRY Fig. 15. Black-figured Kylix in the Stoddard Collection at Yale. did example of Greek craftsmanship. If the width of the bowl is taken as the end and the total height as a side of a rectangle the ratio is i .207 1, the reciprocal being .8284. This is the same rectangle as that of the pyxides in this chapter. The overall' ratio obtained by including the handles, is 1.0356. This rectangle is simply .8284 plus .2071, a rather ingenious manipulation of shapes. If the fraction .2071 be divided by two .10356 is the result. This means that the area of the overall. rectangle AB is the 1.2071 shape which is composed of the two squares CD and DJ and the root-two rectangle is AJ. The lines IJ and IC are diagonals to the reciprocals of AJ- These diagonals intersect the diagonals of the 1. 207 1 form as atH. The line OM is a side of the root-two rectangle MN. The line ST bisects the areas of the two squares CD, DJ, and the root-two diagonals, as MN, cut this bisecting line of the two squares at S and T. This fixes the proportions of the foot. The width of the lip is the side of a square, PQ, in the center of the root-two rectangle AJ. The handle extends above the lip and the root-two rectangle XT, with its included square XZ, shows the pro- portional relationship. The diagonal GF cuts the side of the square PQ at A'. The area FA' is a 1.2071 shape and H' is its center. FF' equals two squares and G' is the center. The square A'B' is described on the side of A'F; C is its center. B'D' is a root-two rectangle with a square applied to the end to es- tablish the point E'. The base of the pictorial composition is the line CJ, the top of the two squares CD, DJ. The painted rays at the foot terminate at the line L'M'. This line fixes the side of a square applied to AB, i. e., the line L'M' is distant from the top of the containing shape an amount equal to GB. The point K', which marks the line separating the two pictorial compositions, is obtained by diagonal to the shapes PP' and O'N. (U "tib a ■u» a -:v:-<' /■'■":■-<, Fig. 14^. Fig. 14c. 88 DYNAMIC SYMMETRY Fig. 15. Large Bell Krater 10.185 with lug handles in the Boston Museum. (Drawn, measured and analyzed by L. D. Caskey.) (Fig. 15). The height of this krater divided into its width, produces the ratio .882. The ratio .882 is composed of two squares on top of a .382 rectangle. The .382 ratio may have its composing elements arranged in various ways. For example, a square may be placed in the center and double whirling square rectangles on either side as in No. 3 of the group of small diagrams of the vase made by Dr. Caskey. The plan scheme of this krater shows that its maker possessed a high order of design knowledge, particulajrly in de- termining and .arranging similar figures. The lines AB, BC are diagonals to half the overall shape; at D and E they cut the sides of the square FG, this square being obtained in the analysis by the width of the bowl. The rectangle DG, that is, the vessel without its lip, is a similar shape to the whole. At H and I these two diagonals cut a line drawn through the center of the major shape. The area HQ is a similar shape to the whole. HI is also the width of the foot. The area Q] is a shape similar to double the shape of the whole, and the width of the foot is one half the width of the whole, that is, the area QJ is expressed by the ratio 1.764, this being .882 multiplied by two. The square FG bears a ratio relationship to the width of the vase of 1.1708, the reciprocal of this being .854. The line KQ, divided into the height, also produces the ratio 1.1708 and LQ is the square on the end of this shape. If two squares are defined in the } m ^ ■ . ■ . ' '■■'■■ "v^ 'f: ,•, ^"' U't/U;U,ip!g!jjTOfW^ Sgfe . *v/a f |^aEk^\^i;^lBiM//l^/^ \ > ■■^*%**jc;;:s^ " ■;: w A LARGE BELL KRATER WITH LUG HANDLES, MUSEUM OF FINE ARTS, BOSTON This vase is known as the Actaeon Krater DYNAMIC SYMMETRY 89 .882 shape HQ, the base of the meander^band MN is fixed and the area NQ is a .382 rectangle. The points O and P are intersections of the diagonals of the square FG with the sides of the rectangle QJ. This rectangle is also connected '; s ■ S :■' w W w . w Fig. 16. Dr. Caskey's analysis of the Bell Krater, showing the com- posing units of form. with the 1.309 rectangle because the fraction of the ratio 1.764, i. e., ."764., is the reciprocal of 1.309. If a square on the end be applied to the area QJ the excess would be a 1.309 shape. The area of an .882 rectangle, as subdivided by the details of a well-known and high-class Greek vase, is now sufficiently clear for the artist and designer. For the benefit of the reader unskilled in the technique of design the point is stressed that the basic facts pertaining to the area occupied by a composition are paramount. The average layman, when analyzing a design, almost invari- ably looks for an aesthetic motive, some arrangement of elements which creates a pattern or movement for example. This is a fallacy, for the reason that such effects are always due to personal selection or disposition and consequently cannot be taught except superficially. The facts connected with areas and volumes, however, are impersonal, are general, rnay be exhaustively analyzed and successfully taught. ' Red-figured Krater 07.286.81, Fig. 17, Metropolitan Museum, New York, a large vase, furnishes the same ratio as the two pyxides described in Chapter IV, i. e., i.2o7i,thecomprisingfiguresbeingtwosquaresplusa root-two rectangle, .5 90 DYNAMIC SYMMETRY plus .7071. In the root-two rectangle, AB, DC are squares; AC, DB, are rec- tangles, each consisting of a square plus a root-two rectangle. The two small squares and their subdivisions which fix the proportions of both foot and neck, and the dotted line which shows the relationship of the foot to the neck, do not need explanation. I Fig. 17. A Bell Krater 07.286.81 in the Metropolitan Museum, New York. (Measured and drawn by the Museum Staff.) Fig. I. Black-figured Amphora in the Boston Museum. (Measured and drawn by L. D. Caskey.) CHAPTER EIGHT: FURTHER ANALYSES OF VASE FORMS « "'>JW ' ^ <; ' ■% , ,' If k — 'C /J Fig- 7- In Fig. 7 the 1.309 is subdivided into the following series of proportional areas: AB = 1.236 CG = \\ CJ = 1.236 CD = 2. 118 FK = 3-236 IJ = 2.618 ED = \'V GH = WS AK= Sq. BF -\\ AI = i-3°9 HJ = 1.309, etc DYNAMIC SYMMETRY 95 Fig. 8. Pelike 06.1021. 191, Metropolitan Museum, New York. A large simple pelike, 06.1021. 191 in the Metropolitan Museum, New York, Fig. 8, is a theme in the often occurring rectangle 1.382. This vase supplies material which sheds considerable light on Greek design practice. The width of the lip considered as the end. of a rectangle, of which the full height of the vessel is the side, defines the area of a root-five shape. The end of this rectangle is also the width of the bottom of the foot of the vase. At some stage of development the design probably looked like the diagram in Fig. 9. AB is a 1.382 rectangle, CD is a root-five rectangle in the center of the major shape. The short curved lines inside this latter rectangle at the top and outside at the bottom, suggest respectively the lip and foot. The direct subdivision of a 1.382 rectangle is shown in Fig. 10 where AB and CD are the two squares described on the ends of the shape. AD and CB are two .382 shapes and AE is a rectangle of the whirling squares. When a root-five rectangle is applied to the center of this containing shape, as in Fig. 11, the major area is subdivided in an interesting manner. AB, CD are two whirling square rectangles, AE, BF, CG and DH are each composed 96 DYNAMIC SYMMETRY r Fig. 9. iJ. I ^ \ Fig. 10. Fig. II. of two squares, while EI, and the similar shape on the other side of the square BI are each double whirling square rectangles. BI is a square in the center of the whirling square rectangle AE, Fig. 10. Considered arithmetically the major area, as affected by the root-five shape, is as follows: The reciprocal of 1.382 is .7236. If the side FH, Fig. 11, represents unity, then the end HJ represents .7236. In relation to this fraction, the end of the root-five rectangle CK is expressed by .4472, and this fraction subtracted from .7236 leaves .2764. Dividing this by 2 the fraction .1382 is obtained. Thus the areas AJ and KF are each composed of ten similar shapes to the whole, or ten 1.382 rectangles. The ratio of the ground plan of the Parthenon is 2.1382, /. e., it is composed of two squares plus a rectangle similar to AJ or KF of this pelike design. The fraction .1382 may be further identified as the diflFerence between .309 and .4472 or a root-five shape minus two whirling square rectangles. The diagram. Fig. 1 2, shows this relationship. ^- ■|' .1^-: T-^ J. Fig. 12. AB is a root-five rectangle with the square FG in the center. AF, ED are two whirling square rectangles, as are alsoAE, FD. The shape CB is a .1382 rec- DYNAMIC SYMMETRY 97 tangle and represents the difference between the root-five rectangle AB and the double whirling square area AH. The meander bands, which define the limits of the pictorial composition, are related to the general proportion of the 1.382 rectangle. 7^ .^; Fig- 13- Fig. 14. When a 1.382 rectangle is divided into two parts, as in Fig. 14, each half is composed of a sguare plus a root-five figure. The bottom of the meander band at the base of the figure composition passes through the center of this square. The .382 area of a 1.382 rectangle is composed of a square plus a whirling square rectangle, see Fig. 13. AB is the whirling square rectangle. AC is its major square and D is the inter- section of diagonals to these two shapes. This point marks the top of the mean- r / ~7\ F \ .M_ \ \/ ^i ^" Fig. 15. 98 DYNAMIC SYMMETRY der band above the figure composition. Fig. 15 shows the geometrical method for constructing a root-five shape in the center of a 1.382 rectangle. AB is a .382 figure and C and D are the centers of the two squares. EF is a root-five rectangle. Black-figured Amphora 06.1021.69 in the Metropolitan Museum, New York, Fig. 16, has a ratio, with the handles, of 1.3455 ^"^ without the handles, 1.382. The fraction .3455 is one-fourth of 1.382. The width of the lip is the end of a root-five rectangle of which the height of the vase is the side. The end of a root- five rectangle, of which the side is 1.382, is represented numerically by .618. The width of the foot is the end of a 2.472 rectangle described in the center of the 1.382 shape. This rectangle is composed of four whirling square rec- tangles; .618 multiplied by 4 equals 2.472. CG is one of these .618 rectangles. The compositional band at the base of the panelled picture, GH, is midway between the top and bottom of the vase. The line EF is one-fourth the total height. The angle pitch of the lip is determined by a line drawn to the center of the foot, or the diagonal of a root-twenty rectangle. Fig. 16. Black-figured Amphora 06.1021.69, Metropolitan Museum, New York. (Measured and drawn by the Museum Staff.) DYNAMIC SYMMETRY 99 d.£;_._ Cl -■■■.JXU-- !) ■■■si Fig. 17. Psykter in the Boston Museum. (Measured, drawn and analyzed by L. D. Caskey.) A psykter in the Boston Museum, Fig. 17, has a i .2764 shape. (See kalpis in this chapter.) The fraction .2764 is the reciprocal of 3.618. In Dr. Caskey's analysis AB and CD are whirling square rectangles. AE is also one and CF is the 3.618 and AF a square. r-l- V-..yL_ _::!_] Fig. 18. Black-figured Kalpis 06.1021.69 in the Metropolitan Museum, New York. (Measured and drawn by the Museum Staff.) lOO DYNAMIC SYMMETRY The ratio of 1.0225 appears in an early black-figured kalpis, 06.1021.69 in the New York Museum, Fig. 18. This shape is composed of .6 18 plus .4045, the latter fraction being the reciprocal of 2.472 or .618 multiplied by four. The rectangle contains a whirling square rectangle plus four such shapes standing on top of it. The width of the bowl, however, with the height of the vase is a 1.309 rec- tangle, i. e., a square AB plus two whirling square rectangles, AC. It will be noticed that the side of the square AB coincides with the neck and bowl junc- ture. T, QN and L are points which fix compositional divisions in the painting. SR, RC are two whirling square rectangles. The diagonal SR cuts GF produced at T. The diagonals of the two whirling square rectangles AC proportion the lip and neck at F and G. The whirling square rectangle IJ fixes foot propor- tions at K. The line NO relates the foot to the painted band under the pic- ture. AD is a square. r Fig. 19. Kalpis in the Metropolitan Museum, New York. (Drawn and measured by the Museum Staff.) The red-figured kalpis, 06. 1 02 1 . 1 90, Fig. 1 9, Metropolitan Museum, New York City, has a major shape of an exact square. The width of the bowl divided into a side of the major form produces, exactly, a 1.309 rectangle. The simple geomet- rical constructions incident to the comprehension of a 1.309 figure in the cen- A RED-FIGURED KALPIS IN THE METROPOLITAN MUSEUM, NEW YORK A handsome design within a square DYNAMIC SYMMETRY loi ter of a square and the resultant combinations of form are shown in the small diagrams. It is significant that the angle pitch of the lip is the diagonal of a .309 rectangle, /. e-., ML, Fig. 19, is a .309 rectangle. The width of the lip as shown by NL is one-half the width of the bowl. The width of the neck at its narrowest point is equal to the width of the juncture of the foot with the body. r~t 7^. .... ijT. ,4 £ I s s v^ vs '■ "" ■ s y-j ., ! 1 s ^s Fig. 20. Fig. I shows the geometrical method of constructing a 1.309 shape in the center of a square. AB is a whirling square rectangle comprehended in a square. The diagonals of two squares, CD and DE, cut the side of the whirling square shape AB at F and G. Fig. 2. EC is a 1.309 rectangle. AB is the diagonal to two squares. DF is a square and DE two whirling square rectangles. The point G fixes the two com- posing elements of the 1.309 rectangle. Fig. 3. A 1.309 rectangle is divided into two parts. Each part is composed of a square plus a square and two root-five shapes. Fig. 4. A whirling square rectangle applied to a 1.309 rectangle leaves a square plus a root-five shape. Fig. 5. The construction for the lip angle of the kalpis, AB and DC are .309 shapes. The remaining area in the center of the square is a .382 shape. Fig. 6 is a .691 shape applied to a square. The .309 remainder is divided into two shapes, one being .191 and the other .118. Analysis of design for symmetry is slow and often difficult. Especially is this 102 DYNAMIC SYMMETRY true of Greek designs. The first step is -the approximate determination of the containing rectangle. This is done arithmetically from direct measurement. The rectangle thus obtained may, frequently, be verified arithmetically by measurement of details. If a root-two rectangle be obtained, for example, /. ^., a rectangle whose ratio is some recognizable one connected with the root- two series, and the width of the foot, hp or neck either divided into, added to or subtracted from this ratio, or divided into the width or height of the whole, produces other ratios recognizable as belonging to the root-two series, a theme in root-two is almost sure to be found. Usually root-two and root-three are easier to recognize than themes in the compound forms. This is due to the fact that root-two and root-three do not modulate or unite with other shapes. Com- paratively, the synthetic use of symmetry is simple; the artist, however, must understand basic principles and be familiar with simple geometrical construc- tion or the use of a scale. The scale to use is one with units divided into tenths because the ratios may be read off directly as numbers. The technique of area or figure dissection is based upon the diagonal not only to the major shape but to its composing elements. The relation of the foot and lip of a stamnos of Stamnos 06.1 021. 176, Metropolitan Museum, New York. (Measured and drawn by the Museum Staff.) DYNAMIC SYMMETRY 103 this chapter, Fig. 21, shows clearly the employment of two sub-diagonals. A well-trained designer who understands his symmetry will work rapidly, use a simple machinery and his key-plan will be unintelligible to any inferior in sym- metry knowledge to himself. In most cases his working plan will not show more than a few diagonals. Dynamic symmetry produces in a design the correlation of part to whole observable in either animal or vegetable growth. It is a satis- fying harmony of functioning parts which suggests a thing ahve or a thing which has the possibility of life. Design without an understood symmetry is the negation of this. Stamnos 06.1021. 176, Metropolitan Museum, New York, Fig. 21, is a simple square and the elements of the vase are proportioned by the dynamic sub- division of the containing shape. AB is a rectangle of the whirling squares. AC is a diagonal to one-half this shape. It cuts the diagonal of the whole at D, which point determines the width of the foot. This foot width is equal to one-third of a side of the en- compassing square. AP is a .382 rectangle and AF is the diagonal of half this shape and it intersects the diagonal of the whirling square rectangle AE at G. It Fig. 22. Kalpis 06.1021. 192, Metropolitan Museum, New York. (Measured and drawn by the Museum Staff.) 104 DYNAMIC SYMMETRY also cuts the diagonal of the square HI at J. The line GM cuts the diagonal of the whirling square rectangle KL to determine the line NO, which fixes the width of the bowl. The hne NO meets the diagonal of the whirling square rec- tangle AB at O and the diagonal of the .382 shape at N. This shows that the lip and the foot of the vessel are proportioned respectively in terms of the two main divisions of the overall shape, i. e., .618 and .382, and that both foot 'and lip are directly proportioned to the width of the bowl. The theme is an arrangement in diagonals of the two main subdivisions of the containing square and diagonals of half these shapes. Kalpis 06.1021. 192 in the New York Museum, Fig. 22, is contained in a square. A small error is shown at the points where the handles do not quite touch the. sides of this square. The width of the bowl and the height define a 1.2764 rectangle. The fraction .2764 is the reciprocal of 3.618, /. e., two squares plus a whirling square rectangle. The area of the lip and neck is com- posed of these two squares, while AC and DE, added, form the whirling square rectangle. AB is a square. The width of the foot is the side of the 2.618 shape FG. FH is a square, andHKis 1.618. FI is a whirling square rectangle. The area of the foot elevation is composed of two whirling square rectangles plus a square or the ratio 4.236. ^ p^ o >H ^ w '^ D w ^^ CO W) (1) t3 tin S ■to ,« 4-> :^ 5^ (1) s 1 O oq a" CIh ■43 S •** .S 1h W -) J5 ^ -« is ^ 0) ^ (ij r/1 ^' :-< O 03 J3 4-1 ^ S c ^ t/j -^ "^ 2- < p^ p o H-H ta !^ U < J PQ CHAPTER NINE: SKYPHOI URING the entire classical period, Greek designers seem to have been searching for certain ideal shapes for certain purposes. The large drinking bowls, which we recognize under the general name of Skyphoi, in their general proportions, illustrate this. The overall shape scheme of these vases approximates a ratio of one and three-quarters. Modern designers would frankly accept this ratio and not trouble themselves about subtle refinements on either the plus or minus side of so obvious, and consequently commonplace, an area. The employment of ratios either a little less or a little more, than one to one and three- quarters, suggests conscious effort to get away from an ordinary rec- tangle. Again, the skyphoi shapes curiously parallel the Nolan amphorae forms, the difference of the outstanding or containing rectangle in most cases being simply that of position. The sides of the skyphoi rectangles rest horizon- tal, the sides of the amphorae shapes, perpendicular. Also, Greek classic artists wasted little design material. This is shown by their use of curves. Practically all convex curves of one design are repeated as concave curves in other creations. For example, the convex curve of the pelike is the concave curve of the pyxis. The convex curves of the lekythos are the concave curves of the calyx krater. Convex cups have their concave counterparts, a sort of reverse echo in forms which may be termed an inversion of a theme. Fig. I . Black-figured Skyphos in the Metropolitan Museum, New York. (Measured and drawn by the Museum Staff.) io6 DYNAMIC SYMMETRY The two skyphol, Figs, i and 4 of this chapter, should be compared. The ele- vation of each shows the same rectangle. One vase is in the Metropolitan Museum, New York, the other in the Museum of Fine Arts, Boston. The rec- tangle was a favorite as it appears repeatedly. ^,5. KV.^. I^.S m,/ ji /■3f>. Fig. 2. Fig. 3- The early black-figiired skyphos in the New York Museum, 06.1021.49, Fig. I, has an overall ratio of 1.854, three whirling square rectangles, .618 x 3, AC, CD and DB. The bowl ratio, however, is 1.382, GI. By construction, as shown by the line GH and the area HB, it will be noticed that the general scheme is that of two overlapping whirling square rectangles, IB and AG, the overlap being the 1.382 shape in the middle. Fig. 2 shows the three whirling square rectangles. Fig. 3 shows the overlapping whirling square, rectangles. A 1.382 shape divided by two equals two square and root-five areas, IK, KH, and IJ, JH in Fig. i are squares, MK, KL two root- Fig. 4. Adams Skyphos in the Boston Museum. (Measured, drawn and analyzed by L. D. Caskey.) DYNAMIC SYMMETRY 107 five areas. The width of the foot is determined by the point R and SL is a square. OP and OA are squares. Thus the vase without its foot would be a root-four area. HB is composed of two whirling square rectangles plus a square. A black-figured skyphos loaned to the Boston Museum by the late Henry Adams, Fig. 4, has the same shape as No. 06.1021.49 in the Metropolitan Museum, New York. The overall ratio of 1.854 (.618 multiplied by three) is divided in exactly the same manner as is the one from New York. The Adams vase has a slightly narrower foot as shown by the point A, the center of the small square BC. The bowl is 1.382 and the vase minus the foot equals two squares as shown by the line DE, the diagonal to a square. FE and GH are two whirling square rectangles overlapping to the extent of GI, the 1.382 shape. Dr. Caskey has suggested the sequence of subdivision in the three small diagrams, Figs. 5, 6 and 7. The picture on this vase shows clearly that the Greek artist at the time was incomparably better as a designer than as a figure draughtsman. The figures of the men riding the dolphins are crudely suggested, but the picture as a design composition is superb. ■A s / ■ vy' s S w w rs r.s' > '. : Fig- 5- Fig. 6. w •. w Fig. 7. A small black-glaze skyphos at Yale, Fig. 8, has an overall ratio of 2.1213 or three root-two rectangles, .7071 x 3 =2.1213 (compare Kylix, Fig. 15, Chap- ter IV), AB, BC, CD are root-two rectangles. AE, BF are squares. These squares divide the area AB into three squares and three root-two rectangles. The gen- eral proportions are all obtained by this subdivision of the root-two shape AB. io8 DYNAMIC SYMMETRY Fig. 8. Black-glaze Sky phos at Yale. (Measured and curves traced by Prof. P. V. C. Baur.) Fig. 9. Skyphos 76.49, Boston. (Drawn, measured and analyzed by L. D. Caskey.) Red-figured Skyphos 76.49, Boston, Fig. 9, furnishes an overall rectangle with a ratio of 1.8944. This is a square plus two root-five rectangles. The eleva- tion of the bowl however is i .236, or two whirling square rectangles, and the logical subdivision of one of these determines the proportionate relation of the details of foot and decorative bands. The points C, D and E in the rectangle AB are clear. DYNAMIC SYMMETRY 109 Fig. 10. Boston Skyphos 13.186. Skyphos 13.186 in the Boston Museum, Fig. 10, has a bowl ratio of 1.309 and an overall area of 1.809. The whirling square rectangle AB is derived from the overall shape. The center of the square DE fixes the width of the bowl. The relation of the bowl to the meander band beneath the picture is shown by C and F. The points G H show that the meander band at the top of the picture is related to the foot. IA.F. A- oi«"7ft Fig. II. Boston Skyphos 01.8076. (Measured, drawn and analyzed by L. D. Caskey.) no DYNAMIC SYMMETRY Skyphos 01.8076 in the Boston Museum, Fig. 11, has a bowl ratio of 1.236 and an overall area of 1.764. This latter ratio frequently appears in Greek design. Fig. 12. Boston Skyphos 01.8032. Skyphos 01.8032 in the Boston Museum, Fig. 12, has a bowl ratio of 1.236, and, apparently, an overall area which is a root-three rectangle. This is the only case in over four hundred examples of Greek design where a root-three figure was apparently used in connection with a whirling square rectangle. Fig. 13. Yale Skyphos 398 Black-glaze Skyphos 398, at Yale, Fig. 13, has a bowl ratio of 1.236 and an overall area of 1.809. ^-^ is a whirling square rectangle from the 1.809 ^^e^- GB is the diagonal to a square and the point H shows that without its foot the vase is a root-four area. DYNAMIC SYMMETRY 11 1 Fig. 14. Skyphos 06.1079, Metropolitan Museum, New York. A skyphos from the New York Museum, Fig. 14, has a bowl ratio of 1.236 and an overall area of 1.809. ^^ i^ ^ whirling square shape from the bowl while EC is a similar figure from the 1.809 ratio. \ 1 '■ .•-' ' ' ^' '■ , I 1 / __-— - 1 X 1 / \ I s ■. i ' ' / \ 1 ^ ' 1 / \ "^ ■ ' -•' \ ^ ■ ' ' ' / \ • . ■*> i // \ ' ^ ••■ '■ \ ' 1 ' ! N-. 1 1 '/ ' \ ' ' V.I '1/ ■ \| ' ' Ul/a- !• r"\; ; ■/ ^ ' \ /' N. \ i 11 ; i) Fig. 15. Boston Skyphos 10.176. Skyphos 10.176 in the Boston Museum, Fig. 15, has a bowl ratio of 1.236, while over all it is 1.809. The picture composition is placed within the square AB. 112 DYNAMIC SYMMETRY -^^ Fig. 1 6. Yale Skyphos 397. Black-glaze Skyphos 397 in the Stoddard Collection at Yale, Fig. 16, has a 1.854 ratio. As AB is the diago;n-al to a square the area of this vase with- out the foot is equal to a root-four rectangle. The bowl, as shown by E, has a 1.236 ratio or two whirling square rectangles. G is the center of the square DC. (See Figs, i and 4, this chapter.) Fig. 17. Yale Skyphos 399. Yale black-glaze Skyphos 399, Fig. 17, has an overall ratio of 1.854 while the bowl is 1.236, and the vase without the foot is a root-four rectangle. DYNAMIC SYMMETRY 113 Fig. 18. Yale Skyphos. A black-glaze skyphos in the Stoddard Collection at Yale, 400, Fig. 18, has a ratio of 1.854 or three whirling square rectangles. The bowl ratio is 1.236 or two whirling square rectangles. AC, CG are two squares. The point H is the intersection of the diagonals of the square HJ and the two whirling square rectangles AI. CHAPTER TEN: KYLIKES ^ n ^HE adjustment of the handles on a kylix to maintain a pro- portional relationship with the bowl and minor elements of the design seems to have been a difficult technical problem to the Greek potter. The great width of the bowl compared to its height and the delicacy of both stem and bowl supplies an uncertain foundation for the attachment of the two, comparatively, heavy handles. When the kyhx was first submitted to analysis the varying height of the handles suggested that the pottery designers had frankly met the difficulty of adjustment by making allowance for an error. This was found to be true because, while the handles were sometimes high and sometimes low, there was one feature of this arrangement which was practically stable. This was their width in relation to the bowl. The makers of the kylix, therefore, must have raised or lowered the handles, after they were attached and while the clay was still workable, so the width should remain true. Of course, the handles of the kylix may be ignored, as they may also be in the skyphoi, and the analysis confined to the bowl, foot and other details; but the Greek, apparently, did not ignore the handle adjustment in any type of pottery when they extended beyond the rectangle of the bowl, a fact clearly shown by the amphorae. In this vase class there are many examples with han- dles both inside and outside the bowl rectangle; when outside they are almost invariably finely worked and highly finished, when inside the reverse occurs. The Greek pottery collection in the Boston Museum of Fine Arts is unusually rich in kylikes and Dr. Caskey has given them careful attention, as the table in this chapter shows. This table contains seventeen examples of red-figured kylikes completely examined. The complete list comprises fifty-four examples. This table is interesting. First it shows that five out of the seventeen are themes in root-two while the other twelve are design arrangements in the com- pound figures derived from the proportions found in the dodecahedron or the icosahedron. The relation of the details, to the overall shape as shown in the classification is striking. Of the seventeen there are six where the width of the foot is equal to the height of the bowl, or one side of a square in the overall shape. The reader will recognize the tabled ratios as representing dynamic areas which have appeared frequently in the vases so far described. Ip every example the details, as sub-ratios, show a recognizable theme in terms of the overall shape. Of the root-two shapes there are three overall ratios of 3.4142, or two squares plus a root-two rectangle. CD o PQ c/f H < o > ^ o -E 2 o -s D W CO D C ni- 1—1 h-l w w Q W O I— I u < h-l pq bo d Ji ^ DYNAMIC SYMMETRY CASKET'S TABLE OF R. F. KYLIKES 11 Museum No. Overall Shape Bowl Foot Stem Base of Stem* Projection of Handles 89.272 3.000 2.382 -8944 -309 95-35 3.090 2.472 .8944 .236 -6552 -309 01.8074 3.090 2.236 1. 000 •309 .427 95-32 3-^3(> 2.618 1. 000 .691 -309 00.338 3-236 2.528 1. 000 -236 .764 ■354 01.8020 3-236 2.528 1. 146 -236 .691 ■354 01.8022 3-236 2.618 •927 -764 -309 10.195 3-236 2.618 1. 000 -236 .618 -309 89.270 3-382 2.618 1. 000 -309 -545 .382 1353-15 3-382 2.764 1. 000 .764 -309 01.8038 3-528 2.764 1.09 .764 .382 01.8089 3-854 2-854 .OOT-T 1-545 WO SE [APES 13-83 3.0606 2-3535 -9393 ■353S -3535 95-33 3.4142 2.4714 1.0672 .4714 .4714 98-933 2.7071 1. 000 00.345 3.4142 2.7071 1 .0606 .7071 •3535 13.82 3-4142 2.7071 .2929 -3535 The overall shape of the early black-figured kylix, 03.784 in the Boston Mu- seum, Fig. I, is represented by the ratio 2.854. The bowl ratio is a root-five rec- tangle. The width of the foot is a side of the square in a root-five shape. The diflf'erence between the square root of five, 2.236, and the ratio 2.854 is .618; consequently, the handles, as represented by AE and DF, are each equal to two whirling square rectangles. The bowl fills two whirling square rectangles as shown by AG, GD, and the area of which the foot is a side is composed also of two such shapes as shown by CG and GB. The scheme of the kylix, therefore, is a theme throughout in double whirling square rectangles. ' *Base of stem is the slightly raised ring on top of the foot. ii6 DYNAMIC SYMMETRY Fig. I. Black-figured Kylix 03.784 in the Boston Museum. (Measured, drawn and analyzed by L. D. Caskey.) Fig. 1. Boston Eye Kylix 13.83. (Measured, drawn and analyzed by L. D. Caskey.) A large Boston eye kylix. Fig. 2, is a theme in root-two. The overall area ratio is 3.0606. The bowl area is 2.3535. The two handle areas, added, represent .7071, the reciprocal of root-two, and therefore a root-two shape. Each handle area must then be composed of two root-two areas. The bowl area, 2.3535 ^^ com- posed of two squares plus .7071 divided by two, or two plus .3535. BE, FC are the squares and FG is the area composed of two root-two figures. The areas HI and JK are each a root-two rectangle and JF is the difference between .7071 and unity or .2929. DYNAMIC SYMMETRY 117 Fig. 3. Yale Kylix 167. A heavy red-figured kylix at Yale, Fig. 3, has an overall area ratio of 2.618. The bowl ratio is 1.927, the fraction being. .618 plus .309. The width of the foot is the end of an .809 shape. The major area is divided curiously. The total area of the handles gives a .691 shape,one-half of which is .3455. The area AO, there- fore, is a square and a root-five; AP is also such a figure, consequently it is the reciprocal of AO, and the diagonals to both shapes meet at right angles at Q. EF is composed of four root-five rectangles. FG equals two whirling square rectangles; AH and ID are square plus root-five shapes. The points J, K, L, M, N are clear. Fig. 4. Kylix 92.2654, Boston. (Measured, drawn and analyzed by L. D. Caskey.) ii8 DYNAMIC SYMMETRY Kylix 92.2654 at Boston, Fig. 4, has an overall ratio of 1.882, the bowl 1.382. This leaves for the handles .5 or two squares. When .5 is divided by two it will be noticed that the space on each end in excess of the bowl is composed of four squares. The 1.382 rectangle divided by two furnishes two .691 rectangles, each of which is composed of a square plus a root-five rec- tangle. The relation of the foot to the bowl is shown by the intersection of diago- nals to two squares and the two .691 forms. The area AB, which is determined by the line formed by the juncture of the lip with the bowl, supplies the ratio 1.7236, i. e., a square plus a 1.382 shape, .7236 being the reciprocal of 1.382. CB is this form and it is divided into two .691 shapes by the line DE. Fig. 5. New York Kylix by Nikosthenes. (Measured and drawn by the Museum Staff.) A large eye kylix in the New York Museum, 14.136, Fig. 5, signed by Ni- kosthenes, has an overall area of three squares. The bowl area however is 2.4472, i. e., two squares plus root-five. The width of the foot in relation to the height is .9472, which is root-five, .4472 plus .5 or two squares, or i .4472, a square plus root-five, minus .5 or two squares. The foot area AB is composed of two squares, and CD is one square. The areas EF, BG are each one and one-third. The areas EH and GI are each composed of two squares plus a whirling square rectangle. There is much evidence in this vase that the designer had been trained in static symmetry. The method of arranging the units of form have a distinct static flavor. A large red-figured kylix, 06.1021. 167 in the New York Museum, Fig. 6, supplies an overall ratio of three squares. The width of the bowl in relation to the height however is 2.4I42, i. e., a root-two rectangle plus a square. The two root-two rectangles AB,CD have ends equal in length to half the diagonal of one of the major squares. DYNAMIC SYMMETRY 119 Fig. 6. Kylix 06.1021. 167, Metropolitan Museum, New York. (Measured and drawn by the Museum Staff.) ^ i> J\^J\ \^ -^^^^-^^ • ^^--^^^"^^ ">p • '■. / i V •• .i; ■■-. \ Fig. 7. Boston Kylix 95.35. (Measured, drawn and analyzed by L. D. Caskey.) A large kylix, 95.35 in the Boston Museum, Fig. 7, has an overall area of 3.090 or five whirling square rectangles, .618x5 =3.090. The bowl area is four whirling square rectangles or 2.472. This latter fraction subtracted from 3.090 equals .618, therefore the handle areas are each composed of two whirling square rectangles. In the whirling square rectangle BC the line representing the width of the foot passes through the point D. Therefore the foot width is equal to the end of an area represented by two root-five rectangles. AB is one of these. The overall ratio of the black-figured kylix, 06.1097 in the Metropolitan Museum, New York, Fig. 8, is 2.472 or .618 multiplied by 4. The bowl ratio is 1.854 or .618 multiplied by 3. AB is the major square in the reciprocal BC of the whirling square rectangle BD. 120 DYNAMIC SYMMETRY Fig. 8. Black-figured Kylix 06.1097, Metropolitan Museum, New York. (Measured and drawn by the Museum Staff.) Ring-foot Kylix 01.8089, Museum of Fine Arts, Boston, Figs, ga and 9^. Overall ratio 3.854, bowl, 2.854. Three whirling square rectangle reciprocals, .618, multiplied by three, equal 1.854, a common shape in Greek design, espe- cially among the skyphoi. The ratios 3.854 and 2.854 are apparent. In one case it is 1.854 plus two squares, the other 1.854 plus one square. f 7!^. ■'■*'■ I • — -X^ ', .-•■'' '■■■x ] ^^ 1 ■-k A ■■ \