Geometry
Gallery 3.11
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Crazy Quilts
and Kaleidoscopes
– – –
“Kaleidoscope” 1839
Probably made by Martha Ellicott Tyson
Maryland Historical Society
In 1816, Sir David Brewster (1781-1868), a Scottish clergyman-turned-scientist, applied for a patent on a glass arrangement in a metal tube—what today we call a kaleidoscope. Although something went wrong with his legal registration, by 1830 hundreds of thousands of kaleidoscopes were fabricated and sold, creating what Cindy Brick calls “a mania … for their colorful, fractured interiors [which] influenced styles in dishes, stained glass, and other expressions of decoration.”
A Bush Kaleidoscope
Charles Busch
The influential effect of kaleidoscopes cannot be overestimated. In 1819 it cost a penny to peek into a large kaleidoscope, many of which were set up on street corners throughout London. It has been reported that in Britain alone, two-hundred thousand were bought in the first three months of its manufacture. Those coming from wealth bought luxury versions of the device. One magazine of the time claimed that kids kept walking into walls, so enthralled were they by the intricate, radiating, churning images created by Brewster’s invention. A correspondent for the Quarterly Journal of Science and the Arts, published by the Royal Institution wrote, “A universal mania for the instruments seized all classes … and every person not only felt but expressed the feeling, that a new pleasure had been added to their existence.”
Kaleidoscopes enchanted viewers throughout Europe and in America during the rest of the nineteenth century, and indeed into modern times. Viewers were mesmerized by the tumbling fractured images they saw when they peered into and turned the magical tube.
Two years after Brewster invented the kaleidoscope, an English weekly began to be printed (1818 through 1831), starting originally as a folio of one page, then two years later turning into an eight-page folio. An article in Wikipedia reports “It consisted of slight original and selected articles in literature, science and art, and aimed at that happy combination of instruction and amusement which has since been more elaborately developed in still cheaper serials.”
In 1847, Charles Busch (1825-1900) emigrated from Prussia to the United States. He had a successful rope-making business and used microscopes to examine the fibers of his ropes. As a microscopist, he served for a short time as an assistant to Professor Louis Agassiz, which then led him to study telescopes and kaleidoscopes. Eventually, in 1870. he gave up rope-making and turned to manufacturing parlor kaleidoscopes, making them by the thousands. Reportedly, their quality was superb. When he moved to Claremont, New Hampshire in 1875, he set up a little room in his house and manufactured 5,000 more units working with his daughter.
In a March 19, 2019 posting on ETSY, Abigail Cain quotes Helen Groth, a University of New South Wales professor whose work focuses on Victorian literature and visual culture: “Unlike the specialization of science now, in the 19th century, there was this idea of instructive entertainment, of making scientific ideas and inventions accessible. That’s where the kaleidoscope comes from. … One of the things that’s really striking about the history of the kaleidoscope is that it’s the only device of its kind that’s still in constant production.”
Although only one crazy quilt bears the name “Kaleidoscope,” it can be reasonably presumed that this novel fractured and fragmented radial rendering of the gadget influenced many quilt designers from the moment it came on the market.
Although the Victorian Crazy Quilt fad is said to have begun in 1880, the use of irregular fragments started years earlier, the patchwork quilt called “Kaleidoscope” having been made in 1839, a date that is sewn into the fragile quilt on the backing fabric (some experts raise questions about the exact year). The quilt is made of forty-two patchwork squares, the pattern of which is almost impossible to discern because of the seeming randomness of the arrangement. Cindy Brick, the author of Crazy Quilts, states that the name “Kaleidoscope” was given the quilt by the family that made it.
Rebecca Palmer. Crazy Quilt, 1884 Brooklyn Museum
It is tempting to speculate that the Tyson family had acquired an actual kaleidoscope and that the crazy mesmerizing jumble of colorful fragments had indeed influenced the then-daring layout of this quilt.
Attribution under investigation
Attribution under investigation.
As the example below illustrates, not all quilts that have a diagonal pattern or rotated squares fall into the category of Crazy Quilts. Although appealing on any bed and made with extreme care, this pattern lacks the random asymmetry that characterizes those that rightly can be called Crazy Quilts.
Lauren Block, quilter
THE ANATOMY OF PATTERN
LEWIS FOREMAN DAY
1887
In 1887, Lewis Foreman Day (1845-1910)—designer, writer, and critic—published The Anatomy of Pattern. The book was revised several times with some images removed and new ones added. It was one of several of his books that informed designers and students of patterns with which they were likely not familiar. Day’s designs and publications were part of the Aesthetic Movement in Britain unfolding at the very end of the 19th century and the beginning of the 20th century.
Day dissected decorative patterns to reveal their underlying geometric structure with the goal of bettering the decoration. However, there was a growing aesthetic thirst for the beauty of the underlying structure. The result was a rejection of decoration in the modern world and a movement toward structure-revealed and the exploration in art and architecture of geometric shapes and forms for their own sake. Day’s dissections were a far cry from the floral damask patterns (plate 41) that Day had been thinking his dissections (plate 8) would help to improve.
Lewis Day illustrated the lattice and the diamond with zig-zags and cross-lines. In discussing the triangle, Day illustrated the star, the hexagon, the lozenge shape, and the equilateral triangle. In a section dealing with the hexagon, he illustrated the honeycomb and diapers (a repeating geometric or floral pattern) based on “the compound of the hexagon.” The octagon is presented along with Arab lattice patterns. (plate 11)
These patterns covered a wide range of styles; noteworthy are the geometric patterns that involved diagonal relationships. This is critical because when Day’s books came on the market, the Phenomenon of Diagonality was just about to emerge. Although there is no definite evidence of linkage, one can imagine that Day’s images of diagonal patterns influenced the broad world of design, especially the work of the influential American architect Frank Lloyd Wright, who was always open to outside influences (although he rarely acknowledged them).
What Day chose to dismiss is what 20th-century artists and architects chose to embrace. Day writes, “An actually hap-hazard (sic) or eccentric scheme of composition, such as a Japanese will sometimes effect, is hardly in contradiction to what I have laid down. When a Japanese artist cuts a panel quaintly in two, after the manner of Plate 34, and treats each part of it as seems good to his queer mind, he is only doing what the Pompeian decorator did when he cut off a portion of his wall and painted it as a dado; though he does it more energetically, not to say spasmodically, and with less appreciation of proportion.
“So, again, when the Japanese strews buds and blossoms about a top box and breaks up the ground between with conventional, though very accidental, lines of crackle, as on Plate 35, or when he crams all manner of geometric diapers into a panel, as on Plate 21, he is merely doing in a more geometric manner what the European artist does, with greater regard for symmetry, when he disposes of his sprigs or what not on a geometric basis. If only he arise at balance, which he almost invariably does (so little is his instinct in this respect likely to err), there is no occasion to cry out against him. We, on our part, are perhaps too much disposed to design as though there were no possible distinction between symmetry and balance…”
Artists and architects in the modern era would delight in the chaotic collision of patterns in Plate 21. It is rich in diagonality and asymmetry, hallmarks of the Phenomenon of Diagonality. Consider also how the world renown Spanish architect of the 20th century Antonio Gaudi used the crackle pattern illustrated in Plate 35 in his tiled surfaces at Park Güell in Barcelona, Spain and how Day’s “queer” and daring Japanese diagonal in Plate 34 became an emblem of Diagonality in the modern era.
Galleries:
The Age of Progress 2.7
Pyramids Through the Ages
Egypt and Elsewhere
2,700 BC – Today
Ever since the Egyptians built their pyramids starting around 2,700 BCE, pyramids have been a subject of considerable mystery, fascination, and marvel. The Kush, in Sudan (south of Egypt), followed the Egyptian models 2000 years later, and pyramid-like structures were built in Mesoamerica roughly 1,000 years after those in Sudan. Following the construction of these ancient antecedents, a few lone pyramids were built in various parts of the world during the ensuing centuries. Then, beginning in the late 19th century, a revitalized interest in pyramids surfaced, particularly after the solid masonry approach to their construction was abandoned in favor of constructing hollow pyramids. New materials like steel, concrete, and, more recently, plastic, paved the way for hollow, habitable pyramids, which are illustrated in the galleries dedicated to developments in the 20th century. The introduction of abstract and geometric art furthered the appeal of the pyramidal form.
In the 20th century, architects explored the construction potential of large pyramids whose enclosing walls were thick enough to accommodate dwelling units or hotel rooms. Today, in the 21st century, there is renewed interest in the imagined health and environmental benefits of pyramids, an interest that surfaced during the “pyramid power” craze of the 1970s, which attracted a horde of enthusiastic followers.
Manfred Mohr
Artist
1969-Present
In the mid-1960s, after discovering information aesthetics as taught by Professor Max Bense, Manfred Mohr (b. 1938) turned from abstract expressionism to computer-generated algorithmic geometry. Born in Pforzheim, Germany, he lived also in Barcelona and Paris before taking up permanent residence in New York. Almost every one of his works in his newly adopted computer-generated geometric mode of expression reveals a pivotal embrace of Diagonality. This engagement with Diagonality, however, did not result from an understanding of the historic roots of the diagonal motif nor of the cultural Phenomenon of Diagonality that swept the world in the 20th century. Like almost all modern artists working in the angular genre, it was a “blind” engagement with Diagonality. However, this blind engagement in no way diminishes the significant artistic merits of Manfred Mohr’s works. In fact, it elevates his body of work because his use of the diagonal springs from subconscious artistic sources.
From 1957 to 1961, Mohr was a jazz musician (tenor sax and oboe). In 1960 he turned briefly to action paintings until he came under the influence of Bense’s information aesthetics in 1961. In 1962, living in Barcelona, his palette turned exclusively to black and white. Between 1964 and 1967, he studied at the Ecole des Beaux-Arts in Paris. Experiments in geometric themes led to hard-edge painting.
It was in 1969 that Mohr started drawing with a computer. In 1972 he began producing sequential computer drawings and then computer-generated animations. In 1977 he introduced “Diagonal-Paths” into his work. Diagonal-paths continued as a major motif in his work even though he continued to explore multi-dimensional mathematical forms of expression. Mohr states in the exhibition catalogue “Manfred Mohr CUBIC LIMIT II, Generative Drawings” Galerie Pierre Weiller, Paris 1977, “While always maintaining the rigid structure of the cube, I destroyed the three-dimensional illusion as well as the symmetries of the cube, drawn in two dimensions, thus creating generators of two dimensional ‘êtres graphiques’.”
George Hart
Sculptor & Mathematician
2000 – 2020
George Hart’s The Triangles Which Aren’t There, refers to an interesting visual effect. It is very difficult to perceive the underlying geometric structure from a photograph, but in person, one at first sees it as a set of bars which form twenty interlocked triangles. This interplay is at the heart of Hart’s work as a sculptor of geometric forms. He uses a variety of media including wood, paper, metal, plastic, and assorted household objects to investigate geometry that has been given physical form.
Hart describes his works as classical forms pushed in new directions. He says, “The integral wholeness of each self-contained sculpture presents a crystalline purity, a conundrum of complexity, and a stark simplicity.” These are qualities one associates with modern-era Diagonality. The artist has been interested in and has a section of his extensive website devoted to this topic polyhedra and art. However, he explains, that his geometric constructions blend math and art more evenly than his purely mathematical treatises. A visit to George Hart’s website will impress with the sheer depth of his engagement with this interdisciplinary potpourri of topics. Hart also shares links to many other geometric sculptors that reveal the degree to which the diagonal motif has permeated an art form once focused primarily on the smooth contours of the human figure.
Hart is a research professor now retired from the engineering school at Stony Brook University. His works are installed around the world, particularly in university settings. He was a co-founder of the Museum of Mathematics in New York city. His book Zome Geometry, co-authored with Henri Piccotto is available through the DDVM bookstore.
“After Black Forest”
Robinson Fredenthal
c. 1984
Simple Beginnings is a central theme in the work of architect-turned-sculptor Robinson (Robin) Fredenthal(1939-2009). In his youth, Robin excelled in lacrosse, football, and basketball, but around 1960, when he was in his early 20s and starting his architectural studies at Penn, he discovered that he couldn’t move as fast as he once did. Nor could he continue to sit at a drafting board or maintain control over pencils, triangles, and T-squares. By 1964 his studies at Penn were interrupted by the illness that would later be diagnosed as Parkinson’s disease. But his love for design, geometry, and sculpture endured and filled him with a passion to give form to the geometric shapes that continued to fire his untiring imagination. He also saw for the first time the work of sculptor Tony Smith, on exhibit at Penn’s Institute of Contemporary Art in 1966-67. That artist’s steel forms, brimming with angularity, had a profound impact on the development of Robin’s art.
Robin was also influenced by the work of his Penn architectural professors Louis Kahn, Robert LeRicolais, Romaldo Giurgola, and Anne Tyng, who were themselves exploring a vocabulary of geometric forms that Joel Levinson groups under the conceptual umbrella of Diagonality. Philadelphia at the time was a hotbed of experimentation in this rebellious geometric motif. Students and professors alike were twisting conventional box-like forms into sharp-angled designs that challenged the dominance of the age-old, reliable right angle. Expressions of Diagonality can now be found in every city and museum around the world.
One of the solid shapes upon which Diagonality was based back in the 1960s is the regular tetrahedron. Hard to visualize for some people, this four-sided pyramid is composed of equilateral triangles on all four sides, including its base. This form, the simplest of the five Platonic solids, resonated deeply with Robin, who said to a colleague, “I feel like geometry is God-given.” When Robin was asked what he would do or say if aliens visited him from outer space, he said he’d hand them a model of a tetrahedron. A fitting answer from an artist who understood the elemental nature of the diagonal and its pervasive artistic expression in Diagonality.
“FD IV, 5.17”
Mark A. Reynolds
2017
Mark A. Reynolds is devoted to developing geometry as an art form. Not surprisingly, many of his hand-drawn works and paintings involve diagonals and diagonal relationships. His works have been produced during the 21st century, so in that regard, his output and explorations are an “offspring” of the Phenomenon of Diagonality that occurred in the 20th century.
True to a core feature of Diagonality in its purest form of expression, Reynolds’ drawings and paintings are often not symmetrical; he revels in asymmetrical relationships. This gives his work a dynamism and motion that is truly modern. And yet he works with geometric relations that go back to ancient Greece and even further back to ancient Egypt.
As Reynolds writes, “Some of the artwork is based on discoveries and inventions that I have made through the daily practice of drawing and experimentation. The work develops as much from an artistic and creative process as from any pre-planned calculations, although the perimeter ratio is always predetermined in order to define the specific geometric system I will be working with. It is through an organic process of overlays, tracings, revisions, exploration, and experimentation with geometric systems – specifically, certain ratios and proportioning systems found in rectangles, squares, and triangles – that I have been able to produce the drawings and paintings presented here.”
The octagon has been central to the unfolding of Diagonality through the ages. Based on what has survived through history, the earliest use of the octagon in architecture is the Tower of the Winds still standing in Athens, Greece. It was a dominant geometric architectural motif during the Romanesque period and Gothic era. It was heavily used through the Renaissance, which is the period that Mark Reynolds turns to, particularly the drawings and designs of Leonardo da Vinci.
On his website (markareynolds.com), in the article, The Octagon in Leonardo’s Drawings, Reynolds reveals the depth of his research as he writes, “The construction demonstrates that Leonardo’s explorations were far more than rudimentary. A drawing for the plan of the city of Imola, in 1502 (Windsor, RL 12284) shows a plan view of the city drawn in a circle divided into eight parts (with four subdivisions of each of the eight sections). Several drawings of octagon-based fortifications done in 1504 can be found in Cod. Atl. folio 48, v-a. Cod. Atl. f 286 r-a, of technological studies and wooden architecture (an “anatomy theatre”?), shows a circle divided into eight parts, each containing its own circle. There is also a famous sheet of sketches for the Last Supper and geometrical drawings in the Royal Library (Windsor RL, 12542).”
Intarsia, Italy
Various intarsiatori
15th century
During the Renaissance, the creation of panelized scenes using inlaid wood, a craft called intarsia, was the geometrical art par excellence. This melding of the art of perspective with interest in geometry resulted in an art form that arguably did much to increase awareness and appreciation of angularity and a “fractured” image as design motifs worthy of further exploration.
For people living in the latter half of the 15th-century and the first quarter of the 16th-century, intarsia functioned like today’s movie theaters and TV; it was a form of entertainment. Perspective and complex geometric forms were dramatically and precisely created by intarsiatori, the masters of perspective. Just in Florence alone in 1478, 48 workshops were churning out these meticulously crafted “paintings,” not on canvas but in wood. One might say it was a craze. Street scenes and architectural complexes, both real and imagined, were precisely rendered using as many as 1,000 pieces of ebony, cypress, boxwood, walnut, and fruitwoods selected for just the right color and gray value to create the illusion of depth and verisimilitude. As Dr. Judith Tormey has written, “The sudden flourishing and the subsequent fortunes of intarsia coincided with the Renaissance effort to give art a mathematical basis … intarsia dramatically exemplifies the fusion of art, mathematics and philosophy during the Renaissance.”
The intarsia craze ended about 1550, but one wonders whether intarsia was an antecedent that directly influenced the art of the modern era, particularly Cubism and other expressions of Diagonality. “Some intarsia panels showed the Renaissance mastery of constructing polyhedrons. Piero della Francesca (1410-1492) wrote several illustrated documents showing how to construct these three-dimensional forms built up of polygonal facets. His works included Trattato d’abaco and Libellus de quinque corporibus. Illustrations include the icosahedron, the dodecahedron, and the cuboctahedron, drawings that remind one of the drawings by the 20th century architect/engineer R. Buckminster Fuller.
Origami
————
17th century to present
It is believed that origami, the practice of folding paper into figures, emerged in Japan, China, and Europe during the 17th century, seemingly as culturally unrelated traditions. In Japan ori means “folding” and kamimeans “paper.” The goal is to fold a flat square piece of paper into a sculpture or geometric structure without cutting or gluing.
However, in Japan since the Edo period (1603-1867), cutting the paper and using pieces of paper that are not square have been acceptable. In the early 1900s, Yoshizawa-Randlett began systematically diagramming the crease lines, inspiring a renaissance of the art form.
Medical stents, packaging, airbags in motor vehicles, and space-probe technology have all employed principles of origami, bringing the tradition into the modern world. Its significance regarding Diagonality, however, can be traced to the German educator Friedrich Froebel (1782-1852), a student of the pedagogue Heinrich Pestalozzi. Froebel created the concept of the kindergarten. The designs published in connection with his curriculum are attributed to the English tradition of folding napkins in exotic shapes for the dinner table.
David Lister has written extensively about Froebel, noting that Froebel did not do well in school, in part because his mother died shortly after he was born and his father basically abandoned him. Froebel had a dreamy nature and was fascinated by everything in the natural world. Lister writes, “He began to perceive that there was an underlying unity in all things…” Froebel began to study geometry and mathematics. He succeeded in convincing his father to send him to Jena University, but after three terms he was imprisoned for debt. He then took to an assortment of jobs to support himself. In Frankfurt am Main, he briefly studied architecture, then was swayed by Dr. Gruner, headmaster of the Frankfurt Model School, to give up architecture and teach at this school along the lines of Pestalozzi, whom Froebel later met. After some hardships, Froebel opened the “Universal German Educational Institute.” This project led to other educational ventures and he moved to Lucerne, Switzerland. Froebel started educating teachers on how to teach and, in 1837, in a century of progress in many fields, Froebel created his first Kindergarten. As Lister explains, Froebel “did not see the human mind as a receptacle to be filled with all manner of diverse and unconnected knowledge. Man learns only through self-activity, through his own seeking; the mind must be free to develop as a plant grows. Parrot repetition had no place in Froebel’s methods.”
Lister goes on: “Froebel devised a series of objects of play which proceeded from the simple to the more complex. He called these his “Gifts.” Lister concludes, “This, then, is the background for one of Froebel’s ‘occupations’: the occupation of paper folding. The gift to a child of one of the commonest substances of life to be folded, but not cut or torn. Used imaginatively it had great possibilities for both science and art and for the discovery of the principles of mathematics. But it was much more: by sending the child on a voyage of discovery it stimulated his own ability to create and to make things of beauty. In this way, paper folding awakened a child’s ability to share in the act of creation and led him to an appreciation of beauty as he manipulated the paper to form an endless array of graceful patterns. In the trivial, the child would find the profound.”
In 1876, Frank Lloyd Wright’s mother attended the Centennial Exposition in Philadelphia and visited the Kindergarten exhibit, where she saw a Froebel Kindergarten Chest of Toys. Mrs. Wright was certain her nine-year-old son, Frank, would one day become an architect, so she later purchased a Froebel Chest that included wooden blocks in varied geometric shapes, and a set of toothpicks and peas that invited children to assemble 3- dimensional structures. Decades later, Wright said that he had played with the cube, the sphere, and the triangle, stating “All are in my fingers to this day…” Throughout his career, Wright used various geometric motifs, but the diagonal, arguably, had the greatest impact on other architects during his lifetime and after. A building project symbolic of Wright’s lifelong engagement with the diagonal is Beth Shalom synagogue outside of Philadelphia, which is featured in this Daring Diagonal Virtual Museum.
Origami has moved into the scientific engineering realm. It has transformed how products and structures have been conceptualized and built. Significant efficiency results from folding material and creating creases and fragmented planes.
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