Clues in Geometry
Clues in Geometry
Clues in Geometry
Clues in Geometry
Clues in Geometry

Clues in Geometry

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Geometry is both rational and expressive, as much a means of contemplation as of calculation and construction.

To The Inhabitants of SPACE IN GENERAL
This Work is Dedicated
By a Humble Native of Flatland
In the Hope that
Even as he was Initiated into the Mysteries
Of THREE Dimensions
Having been previously conversant
So the Citizens of that Celestial Region
May aspire yet higher and higher
To the Secrets of FOUR FIVE OR EVEN SIX Dimensions
Thereby contributing
To the Enlargement of THE IMAGINATION

Edwin Abbott’s Flatland may be one of the most unclassifiable works of literature ever published. While it is acknowledged to be a classic of early science fiction, a work of Victorian social satire, and a religious allegory, it also presents, through its introduction to higher dimensions, an important contribution to the development of an area of mathematics that was eventually merged into non-Euclidean geometry. Flatland is an unusually effective work that spans disciplines and challenges divisional categories. Since its publication in 1884, the book’s popularity has continued today as its readers have embraced it as science fiction, popular science, and metaphysics. Working from the groundwork of philosophical issues raised by Plato’s Republic, Flatland merges social satire and geometry to produce a novel situated in two-dimensional space, a believable world populated by memorable inhabitants whose geometric shapes designate their positions in a complex social structure, one that bears some resemblance to the Victorian class structure.

When you read this book, keep two things in mind. First, it was written back in 1880, when relativity had not yet been invented, when quantum theory was not yet discovered, when only a handful of mathematicians had the courage (yet) to challenge Euclid and imagine curved space geometries and geometries with infinite dimensionality. As such, it is an absolutely brilliant work of speculative mathematics deftly hidden in a peculiar but strangely amusing social satire.

Second, its point, even about itself, is still as apropos today as it was then. We still do not really know what the true dimensionality of the Universe is. It seems somehow unlikely that it is just "four", even in terms of spacetime dimensions. String theory talks seriously about thousands of dimensions. Quantum theory implements very seriously infinite numbers of dimensions. And yet we are still stuck in our 3 space dimensions mentally, hardly able to visualize the 4 in which we live "properly" unless we study theoretical physics for a decade or three, and utterly unable to mentally imagine those four embedded in a veritable Hilbert's Grand Hotel of dimensions.

Ultimately, this is a book about keeping an open mind. A really open mind -- avoiding the trap of scientific materialism and the trap of theistic idealism and the trap of any other favorite -ism you might come up with. Our entire visible space-time continuum could be nothing more than a single thin page in an infinitely thick book of similar pages, that book one of an infinite number of similar books on an infinite shelf, that shelf but one such shelf in an infinite bookcase of shelves, that bookcase but one in an infinite library of bookcases, that library but one... but by now you get the idea.

We have a hard time opening our minds up to the enormous range of possibilities, preferring to live our lives mentally trapped in a single tiny period on just one of those pages, in pointland. We may be quite unable to actually perceive the space in which our tiny point is embedded, but our minds are capable of conceiving it, and Abbot's lovely parable is a mind-expanding work to those who choose to read it that way.

Rose is a rose is a rose is a rose
In 1817 Goethe wrote his book Zur Morphologie. This book was the start of a new science called Morphology, the Science of the Shapes. In his book Goethe describes the so called Uhrplant, the Primal Plant, which is based on the shape of the Leaf. Goethe believed that every Plant was a Leaf within a Leaf within a Leaf.

At the time of Goethe the concept of the fractal was not known. It was developed in 1975 by Benois Mandelbrot (“The Fractal Geometry of Nature”). A fractal is a self-similar structure. It’s shape repeats itself on every level of expansion.  Some fractals are scale-invariant.

Before Benoit Mandelbrot’s fractal mathematics and Gertrude Stein’s roses, Johann Wolfgang von Goethe wrote about a primal plant, “Urpflanze,” which was constructed as a leaf within a leaf within a leaf. I wonder if his Platonic vision for this plant, from which all other plants supposedly derived, was an early imagining of fractal mathematics and response to fractal forms in the natural world (coast lines, human migration patterns, Romanesco broccoli).

Visual depictions of fractals have no beginnings or endings in time, no inside or outside in space, and self-similarity and repetition occur at all discernible scales. To my eye, these aspects of visual fractals are pleasing. I am also dissatisfied by the undeviating periodicity of fractals. Despite an idyllic peak-n-valley and never-ending nowhere-everywhere, and unlike some poems and subatomic particles, in fractal video art that mesmerizingly moves the eyespot through fractal terrain, I can tell where I the eyespot am, where I the eyespot am going, and where I the eyespot has been. Fractals, while on the surface psychedelic, do not quantum jump. There is no dark matter. Fractals are like lite matter. Everything is seen.

Mandelbrot first used the term, “fractal,” in 1975 to consider how natural forms are aligned with theoretical and topological fractional dimensions where substructures go on infinitely and replicate themselves at all distinct scales. In Alice Fulton’s 2005 essay, “Fractal Poetics: Adaptation and Complexity,” originally published in the UK’s Interdisciplinary Science Review and which drew from an earlier essay of hers, “Of formal, free, and fractal verse: singing the body eclectic,” published by Poetry East in 1986, Fulton responds to formalists who argue that only metered forms in poetry have structure by saying that free verse has a structure much like fractals in nature have structure, where there is a “dynamic, turbulent form between perfect chaos and perfect order.”

Fulton also discusses the uncertainty principle in quantum mechanics in relation to the feminist physicist Karen Barad and her interrogation of the constructed boundaries between the subject and object to reject transcendental, universal theories, which can be applied to poetry and science, in favor of “contextual and embodied approaches.” Fulton claims that Barad’s critique of scientific authority represents a direct challenge to patriarchy. Fulton says:

Heisenberg’s uncertainty principle challenged the Cartesian split between agent and object by suggesting that the observer does not have total control of matter: the world bites back. ‘Neither does the object have total agency, whispering its secrets, mostly through the language of mathematics, into the ear of the attentive scientist’, Barad writes. ‘Knowledge is not so innocent’. Thus ‘nature is neither a blank slate for the free play of social inscriptions, nor some immediately present, transparently given “thingness”.’ Nature is slippery: a neither/nor. Light cannot be both particle and wave. Yet it is. The two categories dismantle one another, ‘exposing the limitations of the classical framework … Science is not the product of interaction between two well-differentiated entities: nature and culture.’ Rather, ‘it flies in the face of any matter-meaning dichotomy’. As Barad sees it, subjects and objects both have agency without having the ‘utopian symmetrical wholesome dialogue, outside of human representation’ posited by objectivist accounts. She proposes ‘not some holistic approach in which subject and object reunite … but a theory which insists on the importance of constructed boundaries and also the necessity of interrogating and refiguring them.’ Her theory of agential realism calls for ‘knowledges that reject transcendental, universal, unifying master theories in favor of understandings that are embodied and contextual’.

The work of feminist scientists and philosophers (such as Sandra Harding, Donna Haraway and Karen Barad) critiques the authority of science, which—like every powerful belief system—needs such self-scrutiny lest it become smug and claustrophobic. Of course, the categories unsettled have enormous real world consequences. The old association of women with nature and men with culture undermined women as artists, thinkers, and human beings. Binary constructions of reality, unchecked by s[k]epticism, have a pervasive, destructive magnitude. When one considers their effects, it’s evident that the questions posed are ones literature needs to address until the world is just—which is to say, forevermore.

If we find that audiences remain uncomfortable with abstraction, uncertainty, multiple positionality, and lack of closure—one hundred years after the discoveries of the theory of relativity and Heisenberg’s uncertainty principle—clearly the artist’s job at the beginning of a new century remains unfinished.

This point by Child and Waldrop can be used to argue for the continued relevance of abstraction, uncertainty, multiple positionality, and lack of closure in literature and art within this contemporary moment where breakthroughs in science of the last and current century are too often ignored by writers and artists, and where conceptual approaches to art and literature are becoming more common. While audiences and readers are more accepting of abstraction, uncertainty, multiple positionality, and lack of closure now than in the past, there still seems to be a lack of awareness as to how these modes are in conversation with breakthroughs in physics. Should more people seek to investigate these breakthroughs in physics, consensus reality will become more complex as will our poetry.

'Fractal Poetics': A rose is a leaf is a rose is a leaf AMY CATANZANO

Anne Tyng 
In 1969, Anne Tyng claimed her stake in the radical potential of classical geometry with the publication of Geometric Extensions of Consciousness: a word and image essay of research into geometry, and its overlap with investigations into the human psyche. The text was a portion of her Graham Foundation supported research in 1965– Anatomy of Form: The Divine Proportion in the Platonic Solids.  In her research she developed a theory of hierarchies of symmetry—symmetries within symmetries—and a search for architectural insight and revelation in the consistency and beauty of all underlying form. She weaves between Jungian cycles, city squares, and the cosmos. Throughout, geometry is both rational and expressive, as much a means of contemplation as of calculation and construction.

Zodiac editor Maria Bottero had assembled an extraordinary group of international architects around the themes of geometric studies and the influence of natural systems on design– such a visionary approach to geometry seems encapsulated within the compelling image on the cover of Zodiac 19. Two equilateral triangles, drawn in two-dimensional space, sit side by side with the caption: “1+1=2”. Another diagram, apparently composed from the same 6 rods that make up the two triangles, floats above; one set of rods is smooth and the other hatched. Brought together in three dimensions, they form a tetrahedron. A quizzical notation, “1+1=4,” provokes the viewer to look beyond this mathematically incongruous formula. While this formula seems nonsensical, by taking one’s time, one can see that we are dealing with the power of geometry. The arithmetic becomes exponential, and logarithmic relationships speak to the higher levels of organization and complexity in nature. We see also how this power may be harnessed to influence man-made systems.

Anne Tyng’s essay provides an important node of connection by linking mind and matter through the study of three-dimensional geometric form. She studies the five Platonic Solids (the tetrahedron, cube, octahedron, dodecahedron and icosahedron) and the relationships among them. In doing so, she revisits the wisdom of the ancients. The Platonic Solids are the only equiangular and equilateral polyhedra and were described by Plato in his Timaeus as the smallest units and the origin of all nature; their combinations could account for the unique complexity of all manifestations of matter. 

Tyng proposed an idea of dynamic symmetry, through which she would relate the “biological roots of man-made forms.” Drawing on the ideas of Henri Focillon, who related plastic form to the workings of the mind, Tyng‟s idea of dynamic symmetry posits that the workings of the mind can inspire the designer to move past the static conditions of form. Tyng wrote emphatically, “The chief characteristic of the mind is to be ceaselessly describing itself. The mind is a design that is in a ceaseless flux, of ceaseless weaving and then unweaving and its activity, in this sense, is an artistic activity.”

Following this inspiration, Tyng proposes geometries that open up and follow natural laws of growth. She writes, “I have found a geometric progression from simplicity to complexity of symmetric forms linked by asymmetric process.” The significance of this is that while the Platonic Solids are by their nature symmetrical and static, she realizes that the relationships among them are by no means static. She demonstrates, through a series of diagrams, the movement in two dimensions of the innately static equilateral triangle with the addition of an asymmetric geometric figure. This figure, called a gnomon, is described by the pioneering mathematical biologist D‟Arcy Wentworth Thompson as “any figure which, being added to any figure whatsoever, leaves the resultant figure similar to the original.” Indeed, Tyng’s investigation can be seen to pick up and continue from Thompson‟s widely influential 1917 book On Growth and Form, where he points out that in “any triangle…one part is always a gnomon to the other part.” And so we see in her drawings, she begins with an equilateral triangle and adds a gnomon to produce a new right angle triangle; then by doubling this, she then creates a similar figure to the original, a larger equilateral triangle. Thus the gnomon is the crux of Tyng’s ingenious method of transitioning from a symmetrical figure to an asymmetrical configuration in the production of a proportionally larger geometric figure. Maria Bottero describes Tyng’s process of form-finding as such: “Geometry, through oscillations from the symmetrical to the asymmetrical offers…the key to the processes and the phases of becoming, both organic and of consciousness.”

Tyng’s work provides a critical link between the static nature of platonic geometry and the open growth evidenced in Kiesler’s endless spiraling. What is crucial for Tyng is that in this spiral (think of the helical spiral of the Nautilus shell), appear clues to natural growth. In the spiral Tyng sees that growth occurs proportionally: “the shell, like the creature within it, grows in size but does not change its shape; and the existence of this constant relativity of growth, or constant similarity of form, is of the essence, and may be made the basis of the definition of the equiangular spiral.”

This is where we see Tyng’s significant and radical contribution to the study of more static, yet geometrically complex, close-packing than we see in the work of her contemporaries. In her drawing of the “Spiral Extension of the Helix using the Golden Rectangle plan of the ‘whirling square,’ Tyng uses Hambidge’s Divine Proportion progression to relate a series of cubes through polarity through rotation within a set of proportionally related squares. While this diagram shows only the spiral progression through cubes, she states that “any of the 5 solids may be substituted for the cube and indicated with a complexity of overlapping forms to articulate polarity within rotation within polarity—the geometric hierarchies of spiral form…”

She carries this engagement with movement to her concise and beautiful drawing of the “Helical Extension of Rotation” in a complex Platonic Solid, the dodecahedron. In this drawing, Tyng shows the effect of the Divine Proportion progression on the perpendicular extension of the form. She uses the proportions to lift the dodecahedron in spiraling turns and inscribes two lines along the edges of the form, which she likens to the double helix structure of the DNA molecule.

Throughout her texts, drawings, and projects, Anne Tyng translates the geometry present in ineffably beautiful and complex natural objects—such as the double spiraling seen in the head of a sunflower. She then transposes this knowledge to her design. Hambidge wrote, “Any dynamic subdivision of a dynamic rectangle is like a seed endowed with the external principle of growth. It possesses the property of expanding or dividing until it includes the entire dynamic system. Like an osier twig planted in congenial soil which soon develops into a beautiful tree, dynamic shapes have in them the life impulse which causes design to grow.” It is precisely this sense of the power of geometry as the invisible driving force in natural forms that Anne Tyng brings to her research, her astonishing drawings and notes, and her architectural form.

Gyorgy Kepes: bridging art & science
Hungarian designer & artist Gyorgy Kepes established the Center for Advanced Visual Studies at MIT in 1947. As an artist working among scientists in the post WWII, fast changing technological era, he was determined that artists and scientists must learn to collaborate in an advancing world.  He realized that understanding the natural world was a significant unifying factor that motivated both artists and scientists.

He began to organize exhibitions in which he presented hundreds of natural and biological images side by side, drawing from the science and art worlds.  Ultimately these exhibitions were published as a series of books, titled Vision + Value.  Chapters are organized by non-scientific titles like “Image, Form, Symbol”; “Thing Structure, Pattern, Process; and “Analogue, Metaphor.”  It is up to the viewer to continuously interpret and reinterpret the images, both scientific and artistic, by the simple act of viewing these images side by side.

Kepes “attempted to present in pictures the new visual world revealed by science and technology, things that were previously too big or too small, too opaque or too fast for the unaided eye to see.

In 1951 he organized an exhibition titled “The New Landscape” presenting affinities among visual arts and recent visualizations of research models. Macro, computer images, photographs of the moon surface were shown along visual artists’ production in an attempt to let the similarities emerge. The focus is on recurrences of patterns, forms, growth and logical systems, rhythm.

The obvious world that we know on gross levels of sight, sound taste and touch, can be connected with the subtle world revealed by our scientific instruments and devices. Seen together, aerial maps of river estuaries and road systems, feathers, fern leaves, branching blood vessels, nerve ganglia, electron micrographs of crystals and the tree-like patterns of electrical discharge-figures are connected, although they are vastly different in place, origin and scale … Their similarity of form is by no means accidental. As patterns of energy-gathering and energy-distribution, they are similar graphs generated by similar processes. - György Kepes

Claude Bragnon
Throughout his life, the architect and theosophist Claude Bragdon published several books about nature, architecture, yoga, consciousness and four-dimensional space. 

The architectural practice of Claude Bragdon (1866-1946) was one component of a larger synthesis of creative practices and spiritual convictions. Based in Rochester, New York, he became committed to the notion of organic architecture emerging from Chicago initiated by Louis Sullivan. By the mid-1930s, despite having earlier been recognised as a leading American modernist, his reliance on ornament was considered regressive, his work outdated and antithetical to the principles of the dominant International Style. Almost entirely ignored by twentieth century critics and historians, his architectural theories and practice became forgotten figures in the conventional historiography of modern architecture.

A Primer of Higher Space
Euclidean geometry had, since antiquity, formed the mathematical basis for describing objects in the world; this comprised a series of axioms which formed a logical geometric system. In the seventeenth century, the invention of analytic geometry transformed Euclidean principles into algebraic equations expressed as a co-ordinate system. The transformation of axiomatic logic into a symbolic and abstract system allowed mathematics to challenge the integrity of Euclid’s system. By the mid-nineteenth century, Euclid’s fifth postulate, concerned with parallelism, was interrogated by several mathematicians. Notions of n-dimensional space (or hyperspace) began to emerge which posited that geometries with four or more dimensions were possible.


By the 1880s n-dimensional geometry became popularized, resulting in the emergence of hyperspace philosophy. Four-dimensional geometry, whilst theoretically feasible can only be imagined and represented, never built or experienced, as human perception is constrained to recognising space only in three dimensions. This aspect of hyperspace was exploited within the writings of Charles Hinton (1853-1907) which transformed this mathematical theory into a philosophical framework. Hinton proposed that by learning to perceive, through the power of thought, spatial dimensions outside the constraints of one’s sense perceptions, would lead to understanding a higher reality outside of the limitations of his concept of a materialist three-dimensional world, a process he called ‘casting out the self’.

A Primer of Higher Space and Man the Square became popular additions to hyperspace philosophy and literature; Bragdon’s most significant contribution to these fields was due to his ability to clarify difficult geometric concepts via illustration. The strength of Bragdon’s graphic communication contributed to the significance of these publications serving as reference texts to understand the principles of four-dimensional geometry for the Avant Garde artists Marcel Duchamp and Kazimir Malevich.

Bragdon admired Louis Sullivan’s ornamental compositions based on vegetal motifs, but believed they relied too heavily on the subjective impulses of the individual. Believing that most Americans lived divorced from nature, his version of an authentic organic architecture was based not on the appearance of nature, but rather the geometric and mathematical logic underpinning the natural world.

Projective Ornament proposed a system of ornament comprised of motifs generated from two-dimensional axonometric representations of four-dimensional geometries. Bragdon proposed that learning to see in four dimensions would allow people to become free from the constraints of their own subjectivity and selfhood and realize that consciousness was not individual, but universal. Learning to see in four dimensions would thus lead to revealing the illusions of the three-dimensional world, including the falsehood of the self. The publication included diagrammatic explanations for generating decorative patterns based on four-dimensional geometry. These were illustrated in axonometric projection and based on three-dimensional solids and four-dimensional hypersolids as ‘folded down’ to form graphic patterns as well as two-dimensional axonometric representations of four- dimensional geometries. 

Marcus Fajl, The Visible Invisible: X-Rays and Claude Bragdon's Fourth Dimension

The Dot and the Line: A Romance in Lower Mathematics
Once upon a time there was a sensible straight line who was hopelessly in love with a beautiful dot. But the dot, though perfect in every way, only had eyes for a wild and unkempt squiggle. All of the line's romantic dreams were in vain, until he discovered...angles! Now, with newfound self-expression, he can be anything he wants to be--a square, a triangle, a parallelogram....And that's just the beginning! First published in 1963 by Norton Juster and made into an Academy Award-winning animated short film, here is a supremely witty love story with a twist that reveals profound truths about relationships--both human and mathematical.

The story details a straight blue line who is hopelessly in love with a red dot. The dot, finding the line to be stiff, dull, and conventional, turns her affections toward a wild and unkempt squiggle. Taking advantage of the line's stiffness, the squiggle rubs it in that he is a lot more fun for the dot. The depressed line's friends try to get him to settle down with a female line, but he refuses. He tries to dream of greatness until he finally understands what the squiggle means, and decides to be more unconventional. Willing to do whatever it takes to win the dot's affection, the line manages to bend himself and form angle after angle until he is nothing more than a mess of sides, bends and angles. After he straightens himself out, he settles down and focuses more responsibly on this new ability, creating shapes so complex that he has to label his sides and angles in order to keep his place. When competing again, the squiggle claims that the line still has nothing to show to the dot. The line proves his rival wrong and is able to show the dot what she is really worth to him. When she sees this, the dot is overwhelmed by the line's responsibility and unconventionality. She then faces the now nervous squiggle, whom she gives a chance to make his case to win her love. He makes an effort to reclaim the dot's heart by trying to copy what the line did, but to no avail. No matter how hard he tries to re-shape himself, the squiggle still remains the same tangled, chaotic mess of lines and curves. He tries to tell the dot a joke, but she has realized the flatness of it and he's forced to retreat. 

To give the squiggle an unkempt appearance, the animation drawings were inked on rice paper. The ink bled, creating a textured line that was then photocopied onto cel.