**The Third Claim.**

To place dynamic symmetry on a secure foundation appeal is made to nature — the human skeleton, the maple leaf, the sunflower are called upon to give it character. Let us examine the witnesses. In several articles in The Diagonal (pp. 5, 27, 48, 71, 96, 118) measurements of two human skeletons are given, which, it is maintained, demonstrate the human skeleton to have dynamic symmetry of the root-five variety. As a matter of fact, the measurements neither do nor can show anything of the kind. The

distinction between commensurable or rational quantities and incommensurable or irrational is one of the most fundamental and important that mathematical theory recognizes, but the distinction belongs to theory not to practice (H. E. Hawkes, Advanced Algebra, 1905, p. 53). The most accurate of measuring operations is powerless to distinguish the one kind of quantity from the other. Hence, when we are told (D. p. 8) certain ratios, found by dividing measured dimensions of a skeleton, are “never ending fractions” what is one to infer? Simply that this is an assumption, a postulate which rests on no demonstration. When later these ratios are identified with certain from the root-five system, an assumption is again made, namely that the ratios from the skeleton exactly equal those from the root-five system. We have no means of knowing whether they do or not. We conclude then that the articles above mentioned do not prove the skeleton to be of root-five proportions, they assume it.

It may be asked does not the same argument apply to the measurement of a Greek vase ; is it not assumed to be root-two or root-five, as the case may be, without proof? This is indeed the fact, but with this difference — there is a degree of reasonableness in the supposition as applied to vases which is quite lacking in the case of any natural form. The vase was made by a Greek and much is known of their manner of living and thinking. They developed geometry and doubtless applied it in many ways to architecture and design. Whether they used dynamic symmetry is open to question, but it is at least possible. That root-five has any significance for the human skeleton is mere guesswork. Greek geometry has many possibilities to offer in the way of schemes of analysis, as we have seen, but for natural objects its restrictions are no longer pertinent. What nature’s designer may have used is unknown — the possibilities are endless.

Again, as we have shown, probably any object can be analyzed by the root-five system, to a fair degree of accuracy. Hence, to say that a vase and skeleton have some qualities of form in common, because they both belong to the root-five system, is like finding a relation between the architecture of the Woolworth Building and the New York Post Office by virtue of their both being laid out with a foot rule. Our author could hardly be defended by saying that what he really determines is that certain specific proportions are found in both the skeleton and Greek vases. In the first place, the way in which the proportions occur in the two would not in the least lead one to suspect any connection between them, especially as it is the living form and not its framework with which, as a rule, one becomes familiar. Even were there any aesthetic value in the pro- portions of the skeleton, one would not expect it to pass to the vase under the circumstances. Further, human skeletons vary greatly in their proportions and only rough averages are at all representative. These would be quite out of harmony with the exactness and incommensurability which distinguish dynamic symmetry.

Next comes the maple leaf whose form "strikingly resembles a regular pentagon. " To an examination of its trussing is credited the discovery of dynamic symmetry in nature (G. V. p. 30). Be that as it may, the rectangular subdivisions of the regular pentagon (G. V. Chapter Three) bear no resemblance to the serrated edge and internal structure of the leaf. It is interesting to note that the widths of the several rectangles obtained are determined not by the width of the pentagon but by that of its circumscribing circle. This results in a curious paradox. The root-five system is represented to be based on the maple leaf, that is, the regular pentagon, but when one calculates the rectangle enclosing the pentagon, and subdivisions of this determined by its vertices, they are found not to belong to any of the root systems, depending, as before remarked, on higher irrationalities. One cannot (exactly) analyze the regular pentagon by the root-five system. One wonders about the hundreds of other leaf forms of which nothing is said.

Lastly, there is the sunflower (D. pp. 2, 45). The lines separating its seeds are logarithmic spirals arranged in two sets. Those of one set are congruent curves winding to the right, those of the other are likewise congruent among themselves but wind to the left. Pine cones exhibit an analogous structure. The interest centers on the ratio of the number of curves in one set to that in the other. Each sunflower furnishes but one ratio but different flowers have different ones. These curiously, with but few exceptions, belong to an infinite series of fractions, namely %, %3, 1 %i? 2 %4 which have appeared in various mathematical investigations of the past (G. V. pp. 152-157). The successive fractions in the series have values which more and more nearly equal the ratio of the “golden section” or the ratio of the rectangle of the whirling squares of Mr. Hambidge. Thus is the connection with dynamic symmetry established. Without in any way wishing to belittle the scientific value and interest of these facts for botany, it is difficult to understand how any one could hope to have any emotional response, which the flower might produce in the beholder, carried over to the vase through any such long and intricate mathematical argument. The ratio for any one flower is not based on any conspicuous features of its form, nor does it depend on the nature of the curves separating the rows of seeds, only on their number. Further, the connection cannot be established by one flower, one must have a series of all sizes, and even then assume that were they to grow to unlimited size the ratios would follow the law of the above series. We doubt whether this or other varieties of phyllotaxis teach any lesson with regard to the ratios of value to art, but if they do, it must certainly be — ''Use rational ratios" — “Make static designs.”

However it may fare with the above details, the real essence, the great secret of dynamic symmetry has still to be considered; namely, the sides of the rectangles, the lines of the diagrams, though incommensurable are “commensurable in square,” also “'dynamic symmetry deals with commensurable areas” (D,». pp. 14, 48; G. V. Note III, p. 145, Note VI, p. 157). Our author explains that if squares be constructed on two adjacent sides of a rectangle and their areas are found to be commensurable, then the two sides of the given rectangle are “commensurable in square.” In algebraic terms this means simply that the ratio of the rectangle in question is the square root of an integer or fraction. Thus, in the root-five system, the sides of the root-five rectangle are commensurable in square, as are also the sides of five or six others that have been used in analyses, whose ratios are multiples of the square root of five. The property is not however true of the rectangle of the whirling squares, nor of nine-tenths of the rectangles used in the Hambidge diagrams.

If the statement “dynamic symmetry deals with commensurable areas” means “commensurable in square” there is nothing more to be said. If it means what it says, it is no more correct than the other. For example, if the height of Fig. 9 be unity, each S has area equal to one, each E equal to V5-^-4, and the area of the whole is (4+V5)-H2. No one of these areas is commensurable with any other. We remark in passing that all rational rectangles are both commensurable in square and in area. Even were the author’s statements not incorrect, no aesthetic qualities can rest on the distinction between commensurable and incommensurable. The most accurate measurement fails to separate the one from the other, what hope is there to do so by inspection? Further, in the case of vase designs, the areas in question are not present in the design itself, they pertain to the rectangular scaffolding by whose aid, it is assumed, the design was laid out.

**The Fourth Claim.**

It will not be necessary to discuss the relative merits of Greek and other art, the Gothic for example. Each quite accurately expresses the environment, life, thought, and aspirations of its creators, and though both are part of our artistic heritage, the art of this age, in so far as it is a spontaneous expression of present day conditions, cannot duplicate the past. In particular, we cannot hope to carry over into modern art any of the excellencies of the art of the Greeks by the employment of dynamic symmetry, even overlooking the absence of proof that it was ever used. Our previous argument has shown that probably any design admits of classification under all the several dynamic types and the static as well. A design having been analyzed in the Hambidge manner, does not on that account, possess any special excellence of form or of artistic qualities. The same is true of any modern design laid out by dynamic symmetry, it may prove to be quite commonplace or have exquisite beauty, just as might result from the employment of other methods.

There is practically nothing in Dynamic Symmetry : The Greek Vase and The Diagonal relative to the procedure to be followed in creative design, but the deficiency has in part been made good by enquiry among designers using the method. In the first place it is evident that dynamic rectangles, say those of the root-five system, are quite as inert and dead as are door-nails or roofing slates. We ask them in vain whether the head-board of a bed should be higher than the foot or not; whether a pitcher should be twice as broad as high or the reverse. They tell nothing. We are informed they are not expected to. First, one must know the kind of article he is to design, the service for which it is intended, the period and style to which it belongs, its general size and shape, the material of which it is made and the technique to be employed. The design as thus blocked out still admits a limited measure of freedom in the selection of final dimensions and pro- portions — these dynamic symmetry is called upon to determine. A diagram made up of dynamic rectangles is devised to harmonize with the blocked out design in such a way that none of the variations of size and shape which it permits are overridden. From this diagram the final dimensions are determined. That there may be several such diagrams either of dynamic or static rectangles of several types seems not to have attracted attention. Nor does it seem to have occurred to the users of the method that the forms one may thus determine are so extremely numerous that they fail to be characterized by any remarkable properties, and might just as well be obtained by mere caprice or the throwing of dice. The selection of one figure from the hundreds geometry has to offer is strikingly like the last mentioned procedure.

It is difficult to understand how the followers of the method — now numerous and lacking neither faith nor enthusiasm — can credit a collection of simple rectangles with having occult power to decide the last subtle gradation of proportion necessary to the production of a masterpiece, be it a pottery vase, a silver bowl, a marble statue, or a figure composition (D. pp. 121, 133-138, 153, 155-161).

**Summary of Conclusions.**

The analyses of Mr. Hambidge do not in any way constitute a proof that vase designs and others not arranged about a center admit — as to form or aesthetic significance — of being classified into static and dynamic types. The proof could not be carried through without an almost end- less examination of thousands of constructions, and in the end there is little doubt that each and every design would

be found to belong to all classes at once. That the Greeks ever employed dynamic symmetry does not seem capable of proof by geometry. The making of further analyses of the kind already published will not help the situation. The claim that dynamic symmetry in any way expresses the essentials of plant and animal forms is without rational foundation. The statements that the diagrams of dynamic symmetry are “commensurable in square” and composed of commensurable areas, are for the most part incorrect. The attempt to base differences of artistic quality on the distinction between rational and irrational quantities, whether of length or area, is bound to fail — the eye is powerless to make the distinction.

The rectangles of dynamic symmetry are of themselves inert and lacking of any directive force. They stand ready, as do the rational rectangles, to be selected for such service as the intelligence of the designer may elect. As a method for modern designers, dynamic symmetry has nothing of value to offer, and by imposing false standards and needless restrictions can but hamper the freedom of creative inspiration.

Note: When the above paper was in process of publication my attention was called to one with identical title by Prof. Rhys Carpenter, which had but recently appeared in the American Journal of Archaeology, Vol. XXV, 1921, pp. 18-36. The two papers are, fortunately, not to any great extent duplicates, but rather each supplements the other, and their conclusions, in so far as they treat of the same phase of dynamic symmetry, are in substantial agreement. This is the more interesting since each was produced independently of the other, the one by an archaeologist with an interest for mathematics, the other by a mathematician attracted to art. Prof. Carpenter confines his attention to the archaeological questions involved while I have, in addition, discussed the reputed foundations of dynamic symmetry in nature, and its value to designers of the present day. Prof. Carpenter has done well to insist that note be taken of the “complete irrelevance of these (dynamic) rectangles to the actual areas of the vase, and especially to the contour curves which are so largely the animating life of the ancient vase” (p. 36), and by showing much simpler methods of design which may have been used. Ratios need not have been thought

of at all by the Greek potters, the dimensions of the several parts of a design would have sufficed.

New York, April 17, 1921 Edwin M. Blake

*My apologies for any grammatical/spelling errors found in this text. It was translated from the original print and I may have missed some of them. Just let me know if you spot any errors and I will correct them as quickly as I can.*