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Dynamic Symmetry: A Mathematical Structure in New School History

While Orozco’s murals now speak of the past, students at Parsons today continue to learn about the Modular sequence in architecture and other courses.

In his autobiography, José Clemente Orozco described his murals at the New School for Social Research as an opportunity to investigate the “geometric-aesthetic principles of the investigator Jay Hambidge.” Hambidge, an aspiring writer, was the inventor and proselytizer of a newly-popular compositional theory, Dynamic Symmetry. Orozco learned of Hambidge’s ideas through his widow, Mary Hambidge. Like Orozco, Mary Hambidge was a member of the Delphic Circle, the salon led by Eva Sikleois and Alma Reed (who would become Orozco’s champion and dealer). The Delphic Circle’s influence on Orozco is well known; its championing of peace, international cooperation and the independence movements in India and elsewhere resonated with Orozco’s sympathies in the recent Mexican Revolution. These themes would inform his New School murals. Less remembered is the influence of the Delphic Circle on the mural’s composition through Dynamic Symmetry. Although Hambidge himself was never directly associated with The New School, his life and work link Parsons and the New School for Social Research through different moments in time.

The term Dynamic Symmetry has become obscure, but when Reed arranged for Orozco to paint the New School’s murals it was all the rage. Originally a journalist, Hambidge moved to New York City to work for theNew York Herald. He took night classes in the eighteen-nineties at the Art Students League, studying with William Merritt Chase in the period Chase was founding the Chase School of Art (later Parsons). Fascinated by the geometry of classical art and architecture, Hambidge embarked on a meticulous study of Greek vases, employing the assistance of curators at the Metropolitan Museum of Art and Boston Museum of Art. His measurements and subsequent analysis led to the 1920 publication of Dynamic Symmetry: The Greek Vase. In this book, Hambidge claimed to have rediscovered a hitherto lost set of geometric principles underlying Greek art based on a number of specific proportional schemes, many of them related to the golden ratio, . Application of these proportional schemes to the form and decorative organization of Greek vases gave rise to a quality Hambidge termed “Dynamic Symmetry.”

Hambidge contrasted Dynamic Symmetry with Static Symmetry, the “orderly arrangement of units of form about a center or plain.” In comparison, Dynamic Symmetry was “suggestive of life and movement… Its great value to design lies in its power of transition or movement from one form to another.” To Hambidge, Dynamic Symmetry was both a more aesthetically pleasing and more evolutionarily advanced system of geometric organization. Its basis in biological forms drew on work by D’Arcy Wentworth Thompson and others who found commonalities among leaf patterns and the growth of sunflower heads, even while the principle’s elevation of the geometric-abstract tradition reflected the Greco-centric narrative of mathematics history.

Hambidge’s work garnered immediate interest, both positive and negative. Strongly worded critiques were published in the American Journal of Archaeology and the Art Bulletin, challenging the precision of Hambidge’s measurements and his selective use of data, as well as the lack of historical evidence to suggest classical artists consciously adapted these principles. The mathematician Albert Bennett was equally skeptical.

Dynamic Symmetry nevertheless grew quickly in popularity. Hambidge lectured extensively in Europe and in the United States. He published a short-lived periodical, the Diagonal, in which he outlined the geometric principles of dynamic symmetry to students and artists interested in applying these methods to their own work.

Dynamic Symmetry also made its way into art schools, and in particular to the New York School of Fine and Applied Art (later Parsons). It first appeared in the 1920 summer catalogue and then through the nineteen-twenties, forming an integral part of the Life Drawing curriculum. The 1925 catalogue announced the inauguration of special Evening Courses in “Dynamic Symmetry with its application to figure drawing and composition,” designed for architects, painters, sculptors, professional commercial artists, and teachers. Saturday classes were added to meet “the growing interest in ‘Dynamic Symmetry’ as a fundamental principle both in nature and art and the constantly increasing number of teachers and artists who are returning to a belief in principle, or plan, instead of relying on the infallibility of ‘feeling.’” Dynamic Symmetry was later added to the curriculum of the Teacher Training Program and the programs in Clothes Design and Graphic Art.

What is Dynamic Symmetry? It is easiest described in terms of the basic “root” rectangles — rectangles with length-to-width ratios equal to  . In the Diagonal, Hambidge describes how these root rectangles can easily be constructed using traditional ruler-and-compass techniques, as shown in Figure 1.

At the heart of Dynamic Symmetry is what Hambidge terms the “reciprocal rectangle.” Figure 2(a) below shows a  ABCD and diagonal AC. Line BE is drawn so as to be perpendicular to AC. Adding line EF results in the reciprocal rectangle EFBC.


The significance of the reciprocal rectangle stems from the fact that it is similar to the original rectangle; see angles in Figure 2(a). Thus the ratios of corresponding sides are equal, leading to the relationship

The “dynamic” quality stems from applying these relationships in succession. Figure 3(a) (below) shows the “golden rectangle” ABCD with length AB=ϕ and AD=1. Its reciprocal, EFBC is also golden, and since AC is perpendicular to BE, it can be used to find the reciprocal of the reciprocal: the golden rectangle EHGC. Figure 3(b) shows the next several steps of this sequence in blue with the accompanying golden spiral in green. Hambidge called this pattern the spiral of the “whirling rectangles.”



The familiarity of this image to most modern readers is in no small part due to Hambidge. In the inaugural issue of the Diagonal, he laid out the case, still new at the time, for the ubiquity of the golden spiral and Fibonacci sequence — 1, 1, 2, 3, 5, 8, 13…, in which the successive ratios 2/1, 3/2, 5/3, 8/5 approximate ϕ — in the natural world. Succeeding issues of the Diagonal analyzed the properties of root rectangles and suggested how arrangements of reciprocal rectangles could be used to divide up the root rectangles in harmonious ways.

There are few records of how Dynamic Symmetry was actually applied in teaching exercises, although notebooks from this period show students exploring these ideas formally and in clothing design.

Dynamic symmetry soon appeared in high school curriculums and department store windows. Mathematics teacher Alma Ekholm describes the enthusiasm of her students in Girls Commercial High School in Brooklyn on learning that the “dragonfly and the growing iris” fit the root 3 rectangle and that the golden spiral models “the path of a rolling coin, the arrangement of the sun-flower seeds, water running out of a tub, the snail shell, the closed hand…” She narrates how one student arrived breathless in class to announce that Abrams and Strauss had new blouses on display with a poster reading, “Blouses featuring dynamic symmetry.” The 1956 Parsons Alumni Bulletin describes an event from the sixtieth anniversary week in which an alumna proudly demonstrated how to apply Dynamic Symmetry to floral arrangements.

Dynamic Symmetry was also taught at the New School for Social Research in the course Pictorial Analysis. Instructor Ralph Pearson devoted a chapter of his 1923 book, How to See Modern Pictures, to the aesthetic value of Dynamic Symmetry, speaking favorably of renaissance painters Tintoretto and Titian, who he claimed followed its guidelines, and bemoaning the “decline in vitality of design organization” in the seventeenth and eighteenth centuries.

What about Orozco? Aware of the artistic “furor” over Dynamic Symmetry, Orozco wrote in his memoir that he wanted to discover “how convincing and useful those principles were and what their possibility was.” Biographers speculate that Dynamic Symmetry provided Orozco a classical-infused and structured way of thinking that drew his practice away from the folk-inspired styles of his fellow Mexican muralists. Orozco’s sketches of the New School murals document his experiments with diagonals and proportions to organize the panels. Art historian Lawrence Hurlburt provides a detailed analysis of the left half of the Struggle in the Occident, a portion of which is reproduced in Figure 4 below.


Critical reaction to the New School murals was decidedly mixed. Laurence Schmeckebier declared them “worthy of being called the most remarkable examples of modern fresco work yet produced in the United States.” But many reviewers criticized the figures as stiff, and blamed an over-reliance on geometric principles! Alfred Neumeyer described the murals as “dry and didactic,” the use of Dynamic Symmetry resulting in a “decorative stylization of design in which the subject matter is forced in to a mechanical system of order.” Orozco himself seems aware of potential drawbacks. He writes “I abandoned the over-rigorous and scientific methods of Dynamic Symmetry but I kept what was fundamental and inevitable in it… I had the explanation of many former errors and I saw new roads opening up.”

Meanwhile, the brief fad that was Dynamic Symmetry began to wind down. It remained in the Parsons curriculum through the nineteen-thirties and forties, but largely disappeared from the New School’s. In the preface to his second edition of How to See Modern Pictures, Pearson includes a prolonged mea culpa for “overstressing” the role of Dynamic Symmetry.

One of the last hurrahs for Dynamic Symmetry at The New School occurred in a 1930 course “Introduction to the Arts.” Taught by artist Amédée Ozenfant, the description of the Week Four lessons on “Good and bad forms” reads:

Why? The need of symmetry, but of dynamic symmetry, time as a factor, and the special value of forms of growth. A study of reactions to an Egyptian stele, Michelangelo, Orozco, Rivera, a grain elevator, a poem, Stravinsky.

Ozenfant went on to become synonymous with the “Purist” movement in France, along with his friend and colleague Charles-Edouard Jeanneret, better known as Le Corbusier. The artistic connection is fitting; Le Corbusier’s influential Modular drawings are based on a series of ratios also related to the golden ratio which were surely influenced in part by the prominence given to this number during the heyday of Hambridge’s work. While Orozco’s murals now speak of the past, students at Parsons today continue to learn about the Modular sequence in architecture and other courses. The spirit of Dynamic Symmetry lives on at The New School.

Jennifer Wilson is Associate Professor of Mathematics at Lang College, The New School. When not working on the mathematics of fair division, she likes to think about the relationship between mathematics and art.

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