Memorial remembers women at war
The article below appeared in the Sacramento State University newspaper, The Hornet.
CSUS Art Department Chair Kurt von Meier and a team of graduate students are participating in a national competition this spring to design a center commemorating a long-neglected and misinterpreted aspect of American history: the role of women in the military.
Most junior high school history books contain accounts and pictures of Molly Pitcher, in long flowing skirts, taking water to the American soldiers at the battle of Monmouth.
However, historians writing the guidelines for the memorial design competition have a different view.
Molly Pitcher, like many other women who have fought for this country throughout its history, wore trousers and a man's shirt and carried a flintlock rifle as often as a water jug.
The textbook view of women as auxiliaries, performing exclusively "womanly" tasks hie sewing uniforms and nursing the wounded, reflects our culture's schizophrenic attitude toward women in the military. Their participation has been encouraged — even demanded— in time of war, then minimized or ignored once the dust settled.
The Women in Military Service for America Memorial, to be erected on the grounds of the Arlington National Cemetery, seeks to address this emotional and social dichotomy.
When von Meier read about the competition to select a design for the memorial he was stuck by its
potential as a directed study project for his post-baccalaureate students.
"You would have to deal with all the symbolic and historical aspects, design considerations and political considerations," von Meier said. "It is a fascinating project to explore as a theoretical concept for students doing advanced study."
The site of the proposed memorial is immediately behind the gates to Arlington on a direct sight line between the Lincoln, Kennedy, and Robert E Lee Memorials. The design competition rules require a stricture that will be consonant with the solemnity of the present site and not interfere with significant vistas, particularly that from the Lincoln to the Lee Memorials.
The regiments are meant to guide competitors, but they present significant problems from the standpoint of the designer, according to von Meier.
"Do you put up a big bronze statue? Not anymore, you don't," he said. "A conventional rectangular building is also out You can't have anything with names inscribed because we don't even know the names of many of the-women who served. So it's something that's probably abstract, pure. A mathematical figure came to mind."
Von Meier recalled a model he had once constructed as a solution to a problem in finite geometry — a figure with "15 points and 15 lines, with three points on every line and three lines through every point." The design which met the criteria was something between a pyramid and a partially collapsed box.
"When I saw the announcement (of the design competition), I thought this shape might be the symbolic expression for women in the military," von Meier said. "The straight, rigid lines suggest the masculine nature of the institution. But it generates an elegant double curve which is an archetypal female symbol. The male and female principles are intrinsic and complementary to each other, part of an expressive wholeness."
As von Meier and his students explored their design, they found that it exhibits a number of symbolic properties which make it appropriate for the proposed memorial.
"It's symmetrical in three parts from one perspective and in two parts from another," von Meier pointed out. "In Chinese numerology the numbers two and three are assigned male and female characteristics. Whether or not you want to attribute any significance to the symbolism of numbers is another matter, but if you do, there it is. It's deep and beautiful and clean."
The figure also recalls a pyramid, which since Egyptian times, possibly even earlier, has been associated with memorial functions and the mysteries of life and death, von Meier said.
There is astronomical symbolism associated with the art department's design as well. With the structure oriented so that its spine points due north, the principle angle describes the arc of a bright star in the constellation Orion. "That star is called Bellatrix, the female warrior," von Meier said. "That's how the ancients memorialized their heroes; they translated them to the stars. There's one star in the sky devoted to female warriors, and this points to it."
Von Meier is delighted with the discoveries he and his students are making as they study their memorial proposal. Even if their submission is not selected during the judging this June for inclusion in the second of the competition's two phases, he said, the work they have done and the things they have learned have enlivened their study of art and its relation to history, mathematics and other disciplines. "What else should you be doing at a university if not integrating all this stuff which remains abstract nonsense in books on shelves," he said. "It's our proper focus as an art department."
Griff Field, Staff Writer
Sacramento State University - The Hornet
April 4, 1989
The article below appeared in the Sacramento Bee, April 9, 1989
Kurt conducted rigorous research and complied pages and pages of notes. Below is one small segment of those notes, material pertaining to the star "Bellatrix". As you can see, his notes include some in Greek, just one of many languages he had learned.
Finally, for those of you interested in the nitty-gritty, here's Kurt's "not for publication" analysis of the proposal.
WIMS -- WOMEN IN MILITARY SERVICE MEMORIAL
The principal feature of this proposal is a commemorative work in the form of a monumental sculptural figure. This construction represents a mathematical figure from the field of projective geometry: it is composed of fifteen principal structural elements. Each of these is a rigid beam (?) made from a section of (cast? rolled?) aluminum (or stainless steel?), and measuring very nearly fifty feet in length. These structural elements may be seen as corresponding to lines in the abstract geometrical figure. They are arranged so that on each of the fifteen lines there are three and only three points of intersection with other elements or lines. Precisely fifteen geometrical points are generated, each of which becomes the locus for joining three lines in the mathematical figure, or three structural members in the monumental piece of sculpture. That is to say, at each of these fifteen points (or joints) there come together--by springing from, or by transecting --three and only three lines (or structural member); and on any given line or structural member there are three and only three points.
Thus, the figure may be described technically as composed of fifteen points and fifteen lines, with three points on each line and three lines at each point. In the conventions of mathematical notation it may be represented: 15(3), 15(3). Since this is known as a self-dual configuration, it may be indicated by the shortened form: 15(3).
[H.S.M. Coxeter, "Self-dual Configurations and Regular Graphs," Twelve Geometric Essays, Carbondale and Edwardsville, Southern Illinois University Press (1958), p. 107.]
Such theoretical configurations were invented and studied in the late nineteenth century by the great Italian geometer, Corrado Segre; and in mathematical literature, such a configuration as the particular 15(3) example here may be called a "Segre's Figure." But abstract mathematicians had neither need nor very much interest in actually constructing either this or similar figures in the ordinary three- dimensonal space of the world we commonly experience as real.
In the early 1960s, I served my first academic appointment as Senior Lecturer in the History and Theory of Fine Arts at the University of Auckland in New Zealand. That institution boasted a gracious faculty club in typical British university tradition, where weekly luncheons occasionally provided delightful diversion. On such an occasion, one of my older and far more distinguished colleagues, then Professor Emeritus H. G. Forder of mathematics introduced a graphic (two-dimensional) representation of "a Segre's figure," the very 15(3) example which was to inspire the essential form of the commemorative sculpture.
Professor Forder's illustration may have been from either volumes II or IV of H. F. Baker's Principles of Geometry, for in both did Baker use the configuration 15(3) in four dimensions as a frontispiece. Coxeter states in his 1950 article, originally published by the American Mathematical Society in Bulletin 56, "a 15(3) in two or three dimensions can be derived by projection." [Coxeter, Essays, p. 135].
Citing Coxeter, Forder proclaimed his confidence that a three- dimensional example of the figure could be constructed, although he confided that he had never seen such a figure—doubting, moreover, that anyone else ever had either--and that he could not imagine precisely what it might look like. While contemplating this problem later at home, I joined fifteen lengths of dowel with some rubber bands until the resulting construction satisfied the necessary conditions of the mathematical figure. By then attempting to regularize this configuration, making each of the dowels approximately the same length, a beautiful three-dimensional sculpture began to take shape. The skeletal elements together implied a solid form that would be enclosed by three surfaces, each of which bounded by a skewed quadrilateral. Most simply, the gross exterior could be imagined as two tetrahedra joined base to base; but that figure would need only nine lines for perfect definition, defined by five points in space: two of the points at the apices of a vertical axis, and the three other points in the equatorial plane arrayed as apices of an equilateral triangle.
But the fully constructed figure of fifteen lines suggests a more subtle and complex three-dimensional form: enclosed by three symmetrical surfaces: each of which is doubly curved, saddle-shaped, or more technically, a hyperbolic paraboloid. One of the interesting features of such a form is that its three surfaces--each with their elegant double curves--can be generated by adjacent straight lines, or for practical purposes constructed from straight pieces of lumber placed side by side, but with each timber "twisted" slightly.
Whether in its purely abstract mathematical form, or as a representation of that eternal form in materials of the temporal world, the 15(3) "Segre's Figure" displays two different orders of symmetry. Viewed as it were in elevation, the figure displays a two- part symmetry, as though reflected by the equatorial plane; and viewed along the vertical axis, the figure displays a three-part symmetry.
The numbers two and three are regarded as symbolizing the fundamental attributes of female and male--yin and yang--in the ancient Taoist and Confucian traditions of China. They are also deeply related to other symmetry functions of rare beauty and fascination suggested by the figure. Some of these, like "Sylvester's Duads and Synthemes," are directly related to properties of the number fifteen, as described by Coxeter in another essay.
Sylvester (1844, p.92)noticed that six digits 1,2,3,4,5,6 form fifteen pairs of duads such as 12 (the same as 21), which can be taken together in fifteen sets of three such as (12, 34, 56), called synthemes. (It is understood that (12, 34, 56) is the same as (12, 56, 34) or (34, 12, 56), etc.) Duads and synthemes are in (3,3) correspondence: each syntheme consists of three duads, and each duad belongs to three synthemes. The synthemes occur in six families of five, each exhausting all the fifteen duads.
["Twelve Points in PG(5,3) with 95040 Self-transformations," p. 151; reprinted from the Proceedings of the Royal Society, A, Vol. 247 (1958).]
These eternal qualities of formal, mathematical relationships, especially well-suited for a memorial function, are a part of the intrinsic embodiment and aesthetic expression of the commemorative work. In its form as memorial, the figure symbolizes the service and sacrifice of women throughout the entire lifetime of our nation.
In particular, this architecturally-scaled figure is intended to symbolize two fundamental, complementary aspects of women's service and sacrifice in the armed forces of the United States of America. The first of these is a male or yang quality -- interconnected rigid metal structural elements symbolic of the preeminently masculine nature of military activity. The second is a female or yin quality-- the subtle double-curves symbolizing the compassion and caring provided by women as nurses and staff supporting men on the front lines, who on many historical occasions even disguised themselves to join men in actual combat.
The commemorative work will be situated in the area called the "Memorial Court" by the original architectural firm of McKim, Mead and White. This is a grass-covered half circle, surrounded by the 1930s Beaux-arts symmetry of the Memorial Entrance, surrounded by the pylons and fences of the gateway complex. Above which rises a slope toward the west with the Kennedy Memorial. Atop this hill and culminating the sightline is the neo-classical mansion, dedicated as a Memorial to Robert E. Lee. Thus, the monumental sculpture will form a dramatic focus for the new Memorial, located directly on the sight-line from the Lincoln Memorial, across the Potomac River and up Memorial Drive to the main entrance of Arlington National Cemetery. The service buildings will be located to the south-east and north-east of the present Memorial Gate largely below-grade (and connected by a subterranean walkway that passes under Memorial Drive ). Their form is intended to mediate aesthetically between the geometric expression of the commemorative structure and the traditional architectural statement of both the recently erected Visitors Center and the Memorial Entrance.
One may imagine the structural elements in the figure as bones, and the connecting points as joints.
The figure stands upright resting on two bases, in any of six symmetrical positions. Professor Forder had originally introduced the problem of constructing a Segre's Figure during a conversation on the manifold and mystical properties of the number six. See Coxeter's remarkable preamble to Chapter 6, in Essays, p. 107.
Brief summary of associated ideas:
Euhemerus, ca. 300 B.C., Greek philosopher credited with the theory attributing the origin of the gods--perhaps understood as the celestial constellations--to the deification of local heroes. Historical reference for connecting the idea of a memorial function with the stars. One of the principal vertical aspects of the figure, as viewed from inside, points toward the Great American nebula.
We know that the Great Pyramid at Giza was precisely oriented to the stars: the "descending passage" was directed toward an asterism which was at that time within a few degrees of the north celestial pole.
Overhead, the Pleiades would have been significantly aspected; and we know that the Great Pyramid was geodeisically oriented with extraordinary care. The location of all pyramid complexes on the west bank of the Nile supports their association with death, and hence with a memorial function. The general shape of our figure is accordingly intended to recall the pyramids, although the geometry is of course distinct. One of the most important ideas in relation to the pyramids--even though their function as actual tombs is quite problematical--concerns the very high degree of care and accuracy with which they were constructed. As Peter Tompkins says, despite their monumental size, they were built with the precision of an optician.
The memorial work whould be constructed with similarly exquisite attention to fine detail. As a piece of sculpture and as a work of architecture, the Memorial should be built with the care and craftsmanship of a jeweler, with the very finest of appropriate materials and the highest technology of which this nation is capable.
Precise stellar navigation forms an essential part of the military tradition. For example, practical navigation at sea still depends upon solar and stellar reckoning. Ordinary citizens understand these as obvious considerations for the Air Force and in our exploration of space. And so, while such attention to matters of precise angles within the figure and its accurate siting and orientation might appear tedious for another project, for the WIMS competition, they illustrate intrinsic design qualities that express an important theme in the history of the nation's military.
If a person stands under the "entrance X" facing north, the line of sight follows a sloping vertical member called the "tent pole" which terminates at the "peak." While standing at the same point and facing south, the line of sight focuses on the tip of the gable, where the two cornices join; this is the direction and sight line that points toward the star Bellatrix. Known to astronomers as Gamma Orionis, since it is the third brightest star in the constellation Orion, the name Bellatrix translates as "The Female Warrior."
There are several ways in which Antiquity represented this idea of the femal warrior. In Greek tradition there was the goddess Athene, as well as Artemis the Huntress, and the myth of the Amazons. One of Athene's principal attributes is her battle helmet, that sometimes takes a shape very much like that of our figure. Rembrandt's dramatic 1633 painting of a female in warrior's armor with a helmet that is similar, and obviously based upon his idea of the ancient archetype.
An appropriately cut diamond, or at least a rock crystal should be at set at the peak of the figure. Perhaps the proximity of Dulles International Airport will require the installation of a red flashing light. This raises the question of how high the figure can be, and thus how long each of the structural members must be--a critical factor for engineering considerations that, in turn, indicate materials decisions.
Throughout the figure there is a repeated angle that is just short of 82 degrees. Mathematically it is computed at 81.786789 degrees. One of the astounding facts pointed out by Professor Scott Farrand, Chairman of the Mathematics Department at CSUS, is that the cosine of this angle is exactly equal to 1/7 (one seventh), cos = .1428571428... which is a repeating decimal. While functions of the number seven and fractions with seven in the denominator are common enough in number lore--all the way back to ancient Egypt, in fact-- they are quite scarce in modern mathematics. Farrand says that when he showed this figure to several mathematical colleagues, their jaws dropped!
However, recent correspondence from Coxeter provides another, unrelated example of seven-foldedness.
If one considers the three angles just shy of 82 degrees that describe points of the "equilateral triangle" situated in the equatorial plane, in the figure they are bisected, each by a structural member that together form the "crossing." This bisecting thus creates two angles of approximately 41 degrees at each of three points. This angle is exactly that marking the midpoint of the primary rainbow, since the visible light extends from 40 to 42 degrees. This rainbow reference symbolizes, perhhaps in a slightly subversive way, the theme of peaceful consequence for justly conceived military activity. Should the structure be of aluminum, it could be anodized and given a rainbow effect as well.
At the "crossing point" in the center of the figure, we propose to set, facing outward (to the south), the recto of the Great Seal of the United States. This is an image of the Pyramid and the Eye of Providence, as shown to the left hand side on the back of the one dollar bill in the design based upon the 1935 series. This face of the Great Seal has never been struck officially, although is was mandated by an act of Congress in the 18th century. Iconographers refer to this as the "feminine" side, perhaps because of its associations with mystical symbolism derived from or related to Freemasonry. Certainly it has been suppressed, although the reasons for this are obscure.