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.

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
“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

“FD IV, 5.17”Mark A. Reynolds2017

“FD IV, 5.17”
Mark A. Reynolds

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).”

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