Daring Diagonal Virtual Museum

Writings on Diagonality

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Edgerton to LevinsonPerspective & Diagonality2004

Edgerton to Levinson
Perspective & Diagonality


Samuel Y. Edgerton, Jr. was a professor of art history at Williams College and was a member of the Institute for Advanced Study in Princeton, NJ. His major written work is The Renaissance Rediscovery of Linear Perspective, 1975. Between 1955 and 1957, he taught art history at Perkiomen Preparatory School in Pennsburg, PA. One of his students was Joel Levinson, founder of the Daring Diagonal Virtual Museum. Levinson reconnected with his teacher around 2015 and they exchanged emails concerning Levinson’s research into Diagonality. The following email is from Edgerton to Levinson on the convergence of their investigations.

Dear Joel: My apologies (for the usual reasons) for the late reply to your most interesting letter. My wife, Dottie, and I loved surfing your web-site and all those beautiful buildings you’ve created! I can’t wait to see the new academic center you’re going to design for dear old Perkiomen.

I was also quite fascinated to read about your theory of “diagonality,” and to inform you how remarkably it links to what has been much of my own life work during the past fifty years. I don’t know if you have ever come into contact with my several works – two books and a dozen or so articles – on the history and cultural implications of linear perspective, from its inception as a device for Italian Renaissance painters to its application as a tool of modern science – herewith is a copy of my CV which if you can wade through the pomposity will at least show you where I’ve been and where I’m going during all those decades since we last were associated.

Anyway, why Renaissance perspective may be important to your theory is that it was all about PERPENDICULARITY!! The reasons had to do with ancient Greek (Euclidian), Arab, and medieval optical theory, based on the notion of the geometric “visual pyramid” and the primacy of the *axis perpendicularis,” the “visual ray” that supposedly traveled from the center of (making right angles with) the object seen and the center of the eye, thus “certifying” the object most distinctly as “seen” in the cognitive center of the brain.

Not only was such perpendicular vision understood as the most certain, but it had a moral dimension as well. Medieval Christian theologians were convinced that God’s divine grace could also only enter the human soul at right angles (the eye was their model for this), but if the soul were stained with sin (like a cataracted eye), those righteous perpendicular rays of God’s grace could not enter and would be reflected or refracted away. Also, they believed (as we still do today) that you recognize an honest man because he looks you “straight in the eye”; that is, so “truth” can pass perpendicularly between you and him. However, if you encounter someone who gives you a sideways (diagonal?) look, you should judge that his “truth” is being reflected from you and that he should not be trusted!

All this is to suggest that your “diagonality” may well be a cultural reaction to medieval/Renaissance “perpendicularity” and all of its theological constringencies [sic]. “Diagonality” seems indeed to reflect a kind of evolutionary “in yo’face” attitude regarding the moral restraints of Western Civilization itself; sort of like an architectural variant of wearing one’s baseball cap backward. What do you think?

Before you tackle any of my writings, please read David Lindberg’s THEORIES
OF VISION FROM AL-KINDI TO KEPLER, U of Chicago Press, 1976, then my

Actually and coincidentally, I’m at work at this very moment on a new book
about this very matter!



Pentagons in Medieval Sources and Architecture

  • Krisztina Fehér,
  • Brigitta Szilágyi,
  • Attila Bölcskei &
  • Balázs Halmos

Even though the problem of the regular pentagon construction had been solved in Euclid’s Elements, it seems to have been neglected in the practical work of architects in the Middle Ages, even though it was known among masters of the liberal arts. Searching for new construction methods consumed enormous intellectual power and produced various methods of varying accuracy. The reason for their appearance could be either a quest for a simple method convenient for practical use or the characteristics of certain tasks in architecture. This is one of the most beautiful examples from cultural history where creative geometrical thinking appeared in the works of architects rather than mathematicians.

It can be seen that pentagons were not uncommon in medieval design. Half decagonal apses, pentagonal traceries, and even design details were related to geometric methods based on the construction of a regular pentagon. It is plausible that this figure, besides the obvious square and triangle, was also used in plan design of monumental constructions such as the cathedrals of Reims and Saint Quentin.

R. Buckminster FullerCover of TIME Magazine1963

R. Buckminster Fuller
Cover of TIME Magazine


R. Buckminster Fuller (1895-1983) was an American architect, systems theorist, author, designer, inventor, and futurist. When and where he grew up are clues to the man he became and to the unorthodox ideas he developed throughout his long and productive life.

Fuller was born and grew up in Milton, Massachusetts, an affluent suburb south of Boston. He spent many summers sailing off Maine’s coast. Back then, ships—even large ships—had masts, sails, and rigging. The wood compression system (masts and yards) was held in place by ropes, cables, and chains. To the eyes of a sailor, these were just parts of a ship that helped to catch the wind and achieve propulsion. To a deep thinker like Fuller, these systems of struts and guys held hidden meaning and promising potential. That may not have dawned on Fuller at the time. Perhaps it surfaced later, as sources of inspiration for creative people often do, but the ships’ parts and the forces acting on them were in his bloodstream. It would only be a matter of time before he could draw upon these inspirational structural elements and turn the principles into inventive new use.

Beginning around the middle of the 19th century, hundreds of octagon-shaped houses were built in the United States. They were part of the Octagon Fad sparked by the writings of Orson Fowler. Fuller would likely have seen some in Maine. One can’t help wondering what impact they may have had on him, given that his mind was so attuned to geometry. One may also wonder if Fuller, when he was developing his geodesic domes, knew about Alexander Graham Bell’s kites. Fuller said he didn’t, but Fuller also failed to credit the sculptor Kenneth Snelson for the invention of what Fuller came to call the tensegrity principle, central to his engineering ideas.

Perhaps we can take Fuller at his word when he said he was astonished to learn about the tetrahedral cell kites that Bell had been developing on Beinn Breah, Nova Scotia. This could be so because we are all bombarded with news stories that we can so easily forget soon after reading them. Fuller did have the opportunity to go through Bell’s notes at the National Geographic Society in Washington. He learned that Bell had been experimenting with kites with the goal of making stronger airplane wings. Bell explored what Fuller referred to as omni-triangulation. Fuller called it the octahedron-tetrahedron truss, and said that he had been immersed in his own investigations when someone asked, “Didn’t you know Alexander Graham Bell did it?”

When Dorothy Harley Eber interviewed Buckminster Fuller on the telephone in 1978, he explained how he saw the connections between tetrahedrons, geodesic domes, and the ways atoms are packed. “I didn’t learn about Bell until after the geodesic dome, and the geodesic dome comes quite a long time after what I call the synergetic mathematics — the way the spheres of unit radius close-pack. You just take two spheres and they just touch one another 00 that’s all. You nest a third one down between the two and you get a triangle. Then you nest another on top and you get a tetrahedron. If now you take two triangular sets of three unit-radius spheres and nest one on top of the other, you will make an octahedron unit. If you finally make two layers of spheres in closest packing, the spheres are the vertexes of the octahedron-tetrahedron truss. Many such closest-packed, unit-radius sphere layers, nested upon one another, produce the vertexes of what is known in physics as the ‘isotropic vector matrix.’ I’ve discovered it is the way atoms are packing. So it seems to be fundamental to nature.”

The way Fuller came to discover this triangulated method of joining structural members echoes Frank Lloyd Wright’s discovery of geometry using the Froebel toys. Fuller was very likely exposed to these toys in his “Froebelian Kindergarten” (Froebel had coined the word Kindergarten). “My first objective structural experimenting occurred in my pre-eyeglass, blurred-vision, 1899 kindergarten,” wrote Fuller. “The teacher gave us equilength toothpicks and semidried peas. She told us to make structures — houses. All the other children, none of whom had eye trouble, put together rectilinear box houses. The peas were strong enough to act as angle-holding gussets. Not having visualized the rectilinearity about me, I used only my tactile sense. My finger muscles found that only the triangle had a natural shape-holding capability. I therefore felt my way into producing an octahedron-tetrahedron truss assembly. I, of course, knew naught of such names.

“I can remember the teacher, Miss Williams, asking other teachers to come and look at the strange structure I had produced. Fifty years later I heard from that teacher, who clearly remembered that strange event.”



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.

The Age of Progress 2.7

Lewis Foreman Day’sGeometry Books1886-1909

Lewis Foreman Day’s
Geometry Books

Lewis F. Day (1845-1910) was a designer, writer, and critic who published several books that informed readers of patterns with which they might not otherwise have been familiar. These patterns covered a wide range of styles but noteworthy are the geometric patterns that involved diagonal relationships. This is critical because when Day’s books came on the market, it was when diagonality as a design motif in the modern sense of the word was just about to emerge.

Although no certain linkage is evident yet in my research, and while the diagonal motif was not the only kind of patterns Day illustrated, one can imagine that Day’s images of diagonal patterns did influence the world of design , especially the work of Frank Lloyd Wright.

Day’s titles include:

  • The Anatomy of Pattern
  • Ornament and Its Application 1904
  • Nature and Ornament
  • Pattern Design
  • Lettering In Ornament
  • Alphabets
  • Some Principles of Every-Day Art
  • Alphabets Old and New
  • Moot Points: Friendly Disputes Upon Art and Industry with Walter Crane
  • A Book About Stained Glass


To best understand the roots of Diagonality, it is important to distinguish between Diagonality’s antecedents and precursors. To understand antecedents, consider this hypothetical example: imagine that architects in ancient Egypt had reason to create a triangular room amid a floor plan consisting of otherwise rectangular rooms. The triangular room should be regarded as an antecedent if, over time, it is later rediscovered by architects, for instance in the 20th century, and is reinterpreted and put to use.

If, however, the original triangular room in ancient Egypt immediately starts a trend with architects in that culture to incorporate triangular rooms in other Egyptian buildings, then the first triangular room would be described as the precursor of those that followed because it is ‘built upon’ by subsequent practitioners.

In summary, if replications of a new approach to design occurs soon after the original, then the original should be regarded as the precursor, not the antecedent.

“Glass Architecture”Paul Scheerbart1914

“Glass Architecture”
Paul Scheerbart

Architect Bruno Taut was deeply influenced by fantasist German writer Paul Scheerbart (1863-1915), particularly his 1914 novel whose protagonist Edgar Krug was the “glass architect.” Taut dedicated his Glashaus to Scheerbart, whom he called his “Glass Papa.” This novel reveals Scheerbart’s commitment to the use of glass in modern architecture and is an outgrowth of his first novel, Das Paradies, in which he advocated a transformative new architecture of glass.

Scheerbart’s non-fiction treatise titled Glass Architecture (Glassarchitektur), which he dedicated to Taut, spoke of glass vehicles and a new civilization based on crystal cities and floating continents of chromatic glass. In 1913 he tried to organize a “Society for Glass Architecture.” After the Glashaus, Taut went on to even dreamier visions about glass in architecture in his 1917 book, Alpine Architecture, in which he writes about “glowing crystal houses and floating, ever-changing glass ornaments.”

What makes glass a key element in the Phenomenon of Diagonality is its connection to crystallography and crystal shapes, there being an inherent potential for sharp angularity. The idea of glass shards, sharp edges, acute angles, and asymmetrical angularity. These qualities taken together position glass as a building material at the very heart of the Phenomenon of Diagonality. Similar to the emergence of iron and steel in the 19th and 20th centuries, sheet glass enabled architects to give form to their angular crystalline visions, which were so central to the cubistic visions that arose in the first decades of the 20th century.

Erik Morse, in a 2015 article for the Paris Review wrote, “Scheerbart, an eccentric, Danzig-born poet and architectural theorist, is best remembered through obscure citations from Walter Benjamin, Walter Gropius, and Bruno Taut. But in the spirited era of Berlin’s café culture, he was a popular serialist, publisher, and proto-surrealist. From the late 1880s to his premature death in 1915, he wrote prolifically on science, urban planning and design, space travel, and gender politics, often in the course of a single text.” Morse goes on to say, “Like his French contemporaries Camille Flammarion, Villiers de L’Isle-Adam, Raymond Roussel, and Alfred Jarry, Scheerbart’s prophetic oeuvre oscillated between themes of technology and aesthetics in a genre known in the Francophone world as fantastique.

“Translations of Scheerbart texts have trickled into the English-speaking realm; Glass! Love!! Perpetual Motion!!!: A Paul Scheerbart Reader, edited by Josiah McElheny and Christine Burgin, is the first attempt at an English-language collection. Assembled from his fiction and critical works, drawings and photographs, and secondary texts from friends and acolytes, the book’s publication hopes to inspire what McElheny calls a new generation of “Scheerbartians.””

Full Article Here

The City CrownBruno Taut1919

The City Crown
Bruno Taut

Crystallography is the science of determining the geometric arrangement of atoms in natural crystal structures. This science is largely rooted in critical studies performed during the 19th century. One such investigator was the German mineralogist and physics instructor Christian Samuel Weiss (1780-1856), whose concepts regarding such notions as crystallographic axes and crystal systems created “…a formal edifice in which much of nineteenth-century crystallography found a compatible home and a place to grow.”

It is doubtful that the great American architect Frank Lloyd Wright (1867-1959) knew of Weiss’s contributions, but the master was influenced by the teachings of Friedrich Froebel, who was not only a crystallographer in his early years but worked directly with Weiss between 1812 and 1816. It is therefore reasonable to assert that there is a direct link between 19th-century scientific investigations into the emerging world of crystallography and the world of design at the end of the 19th century and into the 20th century.

This influence can also be seen in the mindset of a German architect in the first half of the 20th century, when Wright’s engagement with Diagonality was in full swing. Bruno Taut (1863-1915) built the 1914 landmark expressionist Glass Pavilion, or Glashaus, for the Cologne Deutscher Werkbund Exhibition. (The Expressionism art movement was fashionable in Germany at this time.) The Glashaus, destroyed at the end of the Exhibition, was intended to “demonstrate the potential of different types of glass for architecture,” and to indicate how glass could “orchestrate human emotions.” The dome consisted of diamond-shaped panels held in place by a diagonal web of thin concrete frames, known as the Keppler system. It was intended to express the complex geometry of nature, a keen area of scientific investigation, as indicated above, throughout the previous century.

Taut was deeply influenced by the fantasist German writer Paul Scheerbart (1863-1915), particularly his 1914 novel, whose protagonist Edgar Krug was the “glass architect.” Taut dedicated his Glashaus to Scheerbart, whom he called his “Glass Papa.” The novel reveals Scheerbart’s commitment to the use of glass in modern architecture and is an outgrowth of his first novel Das Paradies in which he advocated a transformative new architecture of glass. After the Glashaus, Taut went on to even dreamier visions about glass in architecture in his 1917 book Alpine Architecture, in which he writes about “glowing crystal houses and floating, ever-changing glass ornaments.”

Frank Lloyd WrightFountainhead of DiagonalityStart of his Pivotal Decades 1910

Frank Lloyd Wright
Fountainhead of Diagonality
Start of his Pivotal Decades 1910

FRANK LLOYD WRIGHT – Fountainhead of Diagonality

There is no other architect in the world who has had a more profound and lasting impact on the emergence of Diagonality in the 20th century than the American Frank Lloyd Wright. From the start of his career to its end, Diagonality was a driving and sustained force. There are various factors that influenced Wright’s engagement and experimentation with the Diagonal motif—including Froebel Toys, Victorian architecture, Japanese Art, the Octagon Fad, and Cubism. The earliest and most impactful influence was a gift from his mother, the Froebel Toys.

In the summer of 1876, Frank Lloyd Wright’s mother traveled to Philadelphia—along with ten million other visitors from the US and abroad—to celebrate the 100th birthday of the signing of the Declaration of Independence at the Centennial Exhibition in Fairmount Park. Wright’s mother, Anna, a progressive schoolteacher, was certain her 9-year-old son, Frank, would become an architect. In the Friedrich Froebel Kindergarten exhibit, in the Education annex of the Woman’s Pavilion, she saw a Froebel Kindergarten chest of toys (1838-1840) that included wooden blocks in varied geometric shapes and 12-inch square sheets of German paper intended for paper folding. The latter was an activity upon which Froebel, a German crystallographer-turned-educator, placed great educational value. Upon returning home, Mrs. Wright purchased a Froebel chest of toys. Wright later wrote of their influence in his development as an architect. “For several years I sat at the little Kindergarten table-top…and played…with the cube, the sphere and the triangle—these smooth wooden maple blocks…All are in my fingers to this day.” Reportedly they influenced his design (and the interior decorations) for the Imperial Hotel in Tokyo and many other projects before and after. The hotel was one of the earliest examples of Mayan Revival, which took design cues from the architecture and iconography of pre-Columbian Mesoamerican cultures. Although the hotel complex is strictly orthogonal, Wright employed numerous diagonal decorative motifs throughout the wings of the building.

Froebel’s toys did not involve just blocks. They also included the weaving of straws and folding paper horizontally, vertically, and diagonally. Having been a crystallographer, Froebel had a special affinity for diagonal lines

Given the fact that the Froebel blocks profoundly influenced Wright (he was a lifelong sponge of external influences), one wonders if this modern master had also been influenced by the intricate and complex angular geometry of cut glass, so popular during his youth. Although no hard evidence yet supports this notion, Wright’s possible inspiration by the prismatic geometry of cut glass would not be surprising. If alive today, however, he’d probably reject the suggestion because it would reference too directly his roots in asymmetrical Victorian design, which he later felt the need to disparage, no doubt to inflate (unnecessarily) his own originality. As time passed, however, Wright became a fountainhead of diagonality in his stained-glass window designs, wall surface decoration, furniture, and the geometric organization of his floor plans, which are clearly crystalline in the geometric patterning and three-dimensional massing and spatial organization of the spaces he created. Froebel’s influence on Wright was rooted in part in lattice building exercises, which had a demonstrably powerful effect on Wright as evidenced in the floor plan below— one of many such plans that he executed throughout his career.

It is reasonable to assume that Wright and his mother would have read news reports about discoveries in the field of crystallography in the 19th century, given the fact that Froebel himself was a crystallographer and because Mrs. Wright became deeply involved with everything Froebel. One is therefore left to wonder whether Wright was influenced, consciously or unconsciously, by the prismatic cuts in the decorative glass of that era.

It is also reasonable to imagine that makers of cut glass were influenced by the stream of reports about discoveries in crystal structure that surfaced throughout the 19th century, beginning in 1826 with Moritz Frankenheim’s Crystallonomische Aufsätze in which he presents his discovery of thirty-two kinds of crystal shapes. Some of these geometries seem echoed in the patterns Wright saw in the designs of his mentor Louis Sullivan. With this background, it is understandable that Wright became a fountainhead of ideas and examples in the emergence of Diagonality throughout the 20th century.

Architectural historian Anthony Alofsin points to 1910 as a pivotal year in Frank Lloyd Wright’s design methodology. In the previous decade the square and rectangle had been the predominant geometric module in his architectural designs even though there were instances of some diagonality in his work. Rigid axiality and the orthogonal grid resulted in a predominance of bilaterally symmetrical floor plans. As Alofsin points out, even in projects where the overall plan was asymmetrical, “local symmetries” were to be found. But in 1910 the architect returned from his travels in Europe with a new vision.

Wright’s first experiments with asymmetrical diagonality involved non-orthogonal rotation and alterations to square elements in architectural ornament. This progression is much like the progression from diagonal patterning in mosaics that prefigured floor plans with incipient forms of diagonality nudging the layout out of the grip of the right angle. Although Wright had previously used trapezoids and angled elements in the composition of earlier ornament, the effect was mostly symmetrical. Alofsin suggests that Wright’s methodology of ornamental composition used grids, matrices, and geometric rules developed under the influence of Louis Sullivan. He lists the Midway Gardens as a beginning point in Wright’s experiments with asymmetrical diagonality in ornament. The repeating motif, which Wright called “Dancing Glass,” was a series of ornamental cast concrete panels. The motifs were composed of altered lattice grids within a square upon which a double square rectangle rotated by 30 degrees is overlayed. These rectangles are linked together with another rectangle rotated 60 degrees in the opposite direction forming a running border pattern. These, along with other angular sculptures and moldings, helped to produce, according to Alofsin, an “exotic effect.”

Wright’s experiments in asymmetrical diagonal ornament found “unparalleled achievement” in his design of the Imperial Hotel in Tokyo, mentioned above. Here, as with the Midway Gardens, Wright used a rotation system of rectangular elements as the prime generator of ornamental motifs. These ideas were expanded upon, using techniques of multiple frames and the layering of multiple diagonally rotated matrices, to create a sense of visual tension that Wright often balanced and contrasted with the inclusion of orthogonally organized, primary geometric forms.

From ornament, the next step in Wright’s evolution in diagonality was the utilization of this methodology in the composition of his building plans. His earlier plans, as illustrated in the design for the Nakoma Country Club, utilized square, rectangular, and regular polygonal forms radiating from a core structure along diagonal axes. The diagonal method of spatial organization allowed Wright to use non-rectilinear polygons (such as the triangle in the un-built San Marcos hotel resort and the hexagon in the “Honeycomb House”) as the primary spatial modules in the generation of his plans.

Wright’s embrace of the diagonal motif and his trailblazing experimentation was a challenging but potentially rewarding departure from the right angle architecture of the past and would seem to justify referring to the first quarter of the 20th century in this architecture-oriented virtual museum as the Age of Wright.

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