Daring Diagonal Virtual Museum


Gallery 3.11

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1540-1545 AD

In the Indian fort of Kalinjar, in 1545, a mass of gunpowder exploded by accident. Sultan and Emperor Sher Shah Suri, a military genius and gifted administrator with shrewd political skills, lost his life. Architectural lecturer Kowshik Roy writes that the Sultan had gained control of the Mughals in 1549 and founded the Sur Empire. In his memory a red sandstone tomb that had been designed and built during the emperor’s lifetime by architect Aliwal Khan in the town of Sasaram in the state of Bihar was dedicated after his death.

Sketch of Sher Shah Suri by
Afghan artist Abdul Ghafoor Breshna

According to Takeo Kamiya, creator of the website INDO-ISLAMIC ARCHITECTURE: A CONCISE HISTORY, the tomb “… is appreciated as the best work among the octagonal mausoleums due to good proportion[s] and [the] brilliant formation arranging two-tiered chhatris, which are analogs of the main dome.”

The square stylobate (plinth or base) sits like an island in the middle of, and only a few steps above, the surface of a rectangular, artificially created lake. A causeway (barely suggested in the illustration above) connects the tomb to the mainland. The mausoleum’s base is an octagon 135 feet in diameter. Each of its eight sides is 56 feet long. A 20-foot wide gallery surrounds the central domed space, which is 71 feet in diameter.

The tomb is typical in design for that era, but the way it sits with its axis rotated to the axis of the stylobate immediately catches the eye of all who are attracted by instances of diagonality. The relationship of the axis of the stylobate with its stone banks and stepped moorings to the slightly rotated, much more massive upper podium on which the tomb sits was certainly not done for architectural reasons nor to achieve a deliberate expression of diagonality. Yet, according to Kamiya, it “generated a fantastic dynamism in its form, which is as if floating on the lake.”

Image: Wikimedia Commons

Takeo Kamiya asks whether this misalignment of axes was an error. It was … and it was not. It was an error in the sense that when the lower platform, the stylobate, was built, its positioning was off alignment relative to the four points of the compass. Then, when this error was discovered, someone decided not to fudge the difference by continuing the misalignment. Instead, the decision was made to build the tomb and its massive platform correctly oriented relative to the compass points.

Credit: Author’s manipulation of a Google Earth image.
The axis of the temple is aligned with the north compass point,
but the lake, the causeway, and the stylobate
are misaligned.

Would all architects have made that decision? Many would have lost lots of sleep while wondering whether the correction would be worse than concealing the error through second-phase misalignment. However, the orientation was corrected, resulting in the curious diagonal shift and the long thin triangles a few steps up from the water. These strange triangular fragments of paving must be obvious to visitors to the shrine. The drawing of the tomb in plan and the Google photo clearly show the correction, causing those who do not know the history of the tomb to wonder about this odd mystery of Indian architecture.

Image: Outlook India

When a visitor crosses the causeway, it is apparent that something is out of alignment because the path along the causeway results in not seeing the tomb square on. The center line of the causeway clearly does not align with the center arch of the entrance façade of the tomb. This off-axis situation is something like the axial rotation at Luxor Temple in Egypt, but the rotation at Luxor was for a different reason.
(See the Luxor story in Gallery 2.1)

Photo: Wikimedia Commons


Architecture 3.1
Renaissance and Baroque 2.5
Geometry 3.11


1240 AD

It’s a toss-up as to which is more fascinating: the building or the colorful guy who created it? Is it the 85-feet high, two-story 13th century building with its pure, simple, but not-from-all-perspectives obvious octagon-based geometry? Or is it the man who built the Castel del Monte: Holy Roman Emperor Frederick the Great (1194-1250) of the Hohenstaufen dynasty. He was at different times the King of Sicily (crowned at the age of four); and, the King of Germany, receiving the imperial crown in Charlemagne’s 8th century octagonal basilica in Aachen, Germany; and the King of Burgundy; and…if he wasn’t busy enough with all those titles and responsibilities, the King of Jerusalem.

Let’s begin with the building, then present a condensed portrait of Fredrick the Great, then return to the building, whose apparent simplicity belies endless geometric and historical complexity. The structure, alternatively referred to as a fortress, castle, and hunting lodge, is now a UNESCO World Heritage Site. The simple masonry mass with its eight octagonal stair towers is a fusion of Gothic, Romanesque, and Norman architectural styles with undercurrents of richly geometric Islamic architecture. Its geometric roots may even extend back to the octagonal Greek, Tower of the Winds, built in relatively nearby Athens in 42 BC and also to the Dome of the Rock in Jerusalem built in the 7th century AD.

According to some accounts, the Castel was built as a two-story hunting lodge enjoying a commanding presence atop a low rise. Other accounts describe it as a fort; perhaps it was both. As a fort, it is a departure from the rectangular forms of the Roman castra (a military encampment) and is unlike the central European fortresses of that time, which were irregular complexes sprawled in an irregular landscape. It is said that the Castel del Monte owes its geometry to more geometrically ordered Islamic prototypes, which in turn owe much to Pythagorean principles that focused on the diagonals of squares.


View looking up from courtyard
Photo: Wikipedia

The Castel del Monte, ‘crown of Apulia’, consists of a central two-story, hollow octagonal tower with eight subsidiary stair towers at the exterior corners. The central tower has a band of rooms surrounding an open-air, interior octagonal court that was likely filled with plants and a fountain similar to many impluvium-like courts built since the Roman era. The ring of rooms are separated, one from the other, by walls whose position in the floor plan aligns with imagined radials originating from the implied center of the court. These radials connect the vertices of the interior and exterior faces of the tower.

Although the Castel del Monte reflects the characteristics of Hohenstaufen architecture at the end of Frederick’s reign as emperor, it has been said that this building is unique, an apotheosis of what Frederick had been trying to achieve. In his book devoted to the Castel del Monte, author Heinz Götze writes,“…there is no fortress and no castle belonging to the period of the Middle Ages in Europe that could be regarded as a forerunner of this building concept… Nothing about it is a groping attempt, nothing is preliminary; rather, every aspect appears in itself to be consistent and consummate. Where did this collection of form originate?” Götze goes on to state that “forms and design principles of Hohenstaufen buildings have their roots in the Byzantine, but especially in the Arab-Islamic world and its mathematical-geometric tradition.”

Frederick II’s rebellious character, his interest in the sciences, his conflicts with the popes astonish even today’s historians. At the age of 3, he was anointed king of Sicily, then spent his youth in Palermo, where he came into contact with many cultures and scholars. When he took over government affairs at the age of 14, he was fluent in Italian, German, Latin and Arabic. Then he traveled to Germany to be crowned king. He owed his enthronement as emperor of the Holy Roman Empire to his extraordinary negotiating skills with princes and popes. After becoming emperor, he returned to southern Italy, where he loved the landscape of Apulia. Frederick distinguished himself from other rulers of his time by inviting numerous scientists and artists to his court, where Jewish, Christian and oriental influences intermingled. He became a mastermind of enlightenment and tolerance and was benevolent even towards the archenemy Islam. With Christian popes, however, he did not avoid conflict. Viewing himself as the representative of God on earth, he refused any form of subordination. This led to his excommunication several times. Nevertheless, he joined the Fifth Crusade, which succeeded in regaining Jerusalem for Christianity. But this victory was not with military force. Instead, he used his knowledge of the Islamic world and negotiated a ten-year peace treaty with the sultans.

According to an Islamic chronicler, Frederick was far from handsome; he had red hair, was beardless, and shortsighted. Calling himself ‘Lord of the World’, he was either praised as a wonder or reviled as anti-Christ, that is to say, one who, in Christian belief, represents on earth the powers of evil by opposing Christ, glorifying himself and causing many to leave the faith. In business, he was aggressive; his trading extended into Spain, Egypt and Morocco. He was also a gifted artist and a poet, supporting activities in science and philosophy with interests in medicine, mathematics, astrology and astronomy. Although Götze writes that Frederick “lived entirely according to Christian tradition…Frederick remained a continually questioning and doubting mind who did not accept the Church’s dictates about what to believe and how to think. An unquenchable thirst for knowledge drove him and inspired the most astonishing experiments, which reflect none of the prejudices of medieval thought…His bitter struggle against the Church’s claim to secular power nevertheless contributed considerably to the dissolution of the medieval world and thus to its emergence into modern times … The sense of reality became keener, and the beginnings of a new art and a new perception of nature became visible: beginnings of the world in which we live today.” After his death from dysentery, the Emperor was dressed in the robes of state then placed over Arabic garments with Kufic characters.

Chapel of Charlemagne
Aachen, Germany
Photo: Joel Levinson

Plan at entrance level, Chapel of Charlemagne

Cross section, Chapel of Charlemagne

As regards his design for the Castel del Monte, it is worth noting that in 1214 Frederick was elected to be the German king and the following year received the imperial crown in the Chapel of Charlemagne in Aachen, Germany. “His succession to the heritage of Charlemagne and of his grandfather Frederick Barbarossa made an indelible impression on him and remained of enduring importance for him from that time forward,” writes Götze. In the chapel, annular galleries are separated from the octagonal chapel by piers at the corners of the octagon with infill screens composed of columns, railings and arches, a scheme somewhat similar to the more solid facades of the interior court at the Castel. Where the octagonal layout of the chapel in Aachen was modeled on octagonal San Vitale chapel in Ravenna, Italy, it is not too much of a stretch to imagine that the chapel in Aachen was the model for the Castel del Monte.

San Vitale chapel in Ravenna, Italy
Photo: Joel Levinson

Frederick the Great had frequent contact with Leonardo Fibonacci, “the most eminent European mathematician of his time. One of his most noteworthy accomplishments was to impart the achievements of Arabic mathematics to the Western world. According to author Götze, he used the ‘Fibonacci sequence’ to create “a mathematical expression for the harmonic ratio of the golden section, already known to the Greeks.”

Frederick II built more than 200 buildings including citadels, fortresses and palaces.
In some small way his Castel del Monte represents part of the slow shift from the medieval world to the modern world and to the emergence of diagonality.


Architecture 3.1
Middle Ages 2.3
Geometry 3.11

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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.”

SforzinaFilarete 1464


A dramatic cultural shift occurs as the medieval concept of geography is taken over by the Renaissance perspective. Medieval maps of the world were fictions and did not relate to the world as it was. Instead, a Christian fiction of the world was fabricated with the holy city of Jerusalem placed in the center. In other maps, Christ’s body became the cartographic shape into which geographic facts were squeezed. This corruption of reality was dramatically overthrown during the Renaissance. Geometrically correct representations were the overriding objective. Alternative ‘projections’ were tried and the most enduring occurred in 1569, with the development of Mercator’s map of the world. However, the view of the cosmos remained rooted in another fiction, which was that the earth was the center of the cosmos. This belief continued until the time of Copernicus in 1543. According to Christian Norberg-Schulz, “With these two approaches the Renaissance image of the world unified empirical, homogenous space and ideal, centralized space.”

These images are synthesized in the notion of the ideal city. “Whereas the medieval city was ideal inasmuch as it was a living concretization of the Civitas Dei, the Renaissance city aims at ideal form. Scientific studies were made of town design, although it was not until 1593 that an ideal city was built, Palma Nova, by Savorgnan and Scamozzi…numerous architects published treatises where the problems of the city are analyzed with the aim of working out ideal plans. The first plan for an ideal city, Filarete’s Sforzinda (1464), is based on a circle with a star inscribed…”

Sforzinda is an eight-pointed star inscribed within a circle with another smaller independent circle concentrically placed within. An orthogonal arrangement of buildings in the center of the town is connected to one of the points (perhaps a gate in the circular fortification wall) by a singular radial element of unknown purpose. A nine-sided polygon (nonagon) is the shape of Palma Nova, a figure that is repeated in a series of concentric streets culminating in the town center. These streets are interrupted by radial streets, which lead to the arrow-shaped fortification projections. Both these plans, one a proposal, the other a still extant example east of Venice, are interesting departures from the organic plans of the middle ages and the orthogonal plans that existed from the dawn of civilization.

Alberti, in addition to being credited with the codification of the rules of mechanical perspective, is important also as regards Diagonality in terms of his theories on architecture and the design of ideal cities. Before he ever designed a building, Alberti wrote De Re Aedificatoria (between 1443 and 1452). This treatise was the first coherent presentation of Italian Renaissance architectural theory and its significance as to its impact on the course of architecture through subsequent centuries cannot be over-estimated.

Although Alberti appreciated and saw great beauty in small towns, particularly hill towns with fortifications, as Kostof points out (Kostof 1991) “he favored in principal the geometrically organized city form.” In another chapter Kostof continues (Kostof 1991), “In the next century Alberti and others would equate a city with a palace. “The principal ornament of a city,” in Alberti’s words, “is the orderly arrangement of streets, squares and buildings according to their dignity and function.” Beginning with Filarete’s Sforzinda, Renaissance architects projected ideal cities with fixed perimeters and fully composed parts. In them the use of circle-based geometry as a proportioning device in urban design is clear.” One must be careful not to draw the conclusion that circle-based geometry means a city plan composed of a circle and other circular components because the plans in fact reveal a significant degree of angularity as in the aforementioned plan for Sforzinda.

Triangular Lodge at RushtonSir Thomas Tresham1593-1597

Triangular Lodge at Rushton
Sir Thomas Tresham

Sir Thomas Tresham (1545-1605) was a well-educated gentleman architect and builder in Northamptonshire, England during the Elizabethan era. In 1593 he designed and built, over a period of four years, a remarkable triangular lodge (then known as Warreners Lodge) on the edge of his estate in Rushton, a town in Northamptonshire, to manifest his fervent belief in the Holy Trinity and, as some believe, as a play on an abbreviation of his last name (Tres). The footprint of the building is an equilateral triangle.

The Triangular Lodge (known also as Rushton Lodge) is what is known as a folly, a building primarily constructed for decoration but suggesting through its appearance some other purpose. A folly can call attention to itself through unusual details or form. This lodge satisfies those criteria on all counts but also rises above them because it is a very early precedent for a subsequent stream of triangular buildings that followed in the next few centuries—and then exploded in frequency in the modern era.

The three-part manifestation of the Trinity is designed into every detail of the Lodge, even into the very unusual triangular floor plan of the building itself. Its three walls are each 33 feet long (note 33) and each has three windows, triangular in shape and surmounted by three gargoyles. The Tresham family had an emblem or crest in the form of a trefoil. The geometry of the trefoil is used repeatedly in the shape of the windows on what in the U.S. is called the second floor. The “basement” windows are much smaller but they, too, are trefoil in shape and have a triangle in their center. The shape of the first-floor windows is a rotated square. The triangular corners of the building on all three levels are separated by walls from the central space, which is hexagonal. One corner contains a spiral staircase, the other two are small rooms with currently unknown purpose. They have delightful windows that make them more appealing than one would imagine for a tight triangular space.

The famous architectural historian Nikolaus Pevsner considered the building so architecturally significant that he used a photograph of it for the front cover of the first edition (1961) of his The Buildings of England: Northamptonshire. His selection of this unusual triangular building for the cover of a treatise dealing with a country whose architecture was for centuries dominated by the right angle is perhaps best explained by the fact that in 1961, the diagonal motif was flourishing around the world. According to Joe Jarrett, a researcher in mathematics, Tresham owned a copy of Henry Billingsley’s 1570 edition of Euclid’s Elements. Jarrett writes, “That he chose to write such snatches of Stoic wisdom in a book of mathematics is surely no coincidence. As a devotee of architecture, Tresham attributed great significance to lines, angles, and numbers.”

Several other triangular buildings were designed and (most) constructed before the modern era. These include:

  • Caerlaverock Castle, southern coast of Scotland – 1270
  • Longford Castle, south of Salibury, Wiltshire, England – c. 1576
  • Wewelsburg Hill Castle, North Rhine-Westphalia, Germany – 1646
  • Andrea Pozzo’s proposed Triangular Church and Monastery -1700
  • Jan Blaise Santini-Aichel, St. Anna in Three Persons Chapel in Penenske Břežany near Prague -1705
  • J.M Prunner, Stadl Paura Church in Austria – 1714
  • Jan Blaise Santini-Aichel, Chapel in Ostrunzo (Czech Republic) – 1720
  • Jan Blaise Santini-Aichel and František Ferdinand Kinský, Karlova Koruna, Czech Republic – 1723
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
The ShardRenzo Piano2000 – 2013

The Shard
Renzo Piano
2000 – 2013

This 95-story tower in Southwark, London is the work of the brilliant Italian architect Renzo Piano. Over 1,000 feet high, it is the tallest building in the United Kingdom. Construction started in 2009 and it opened to the public in 2013.

The project goes back to 2000 when the developer had lunch in Berlin with the architect. Over lunch, Piano sketched a spire, and a deal was struck. The proposal to build the Shard was initially opposed by many conservative group in London, due in large part to its height and perhaps its unconventional tapering profile. Reportedly, Piano was inspired by London’s spires as depicted in an 18th century painting and other historic engravings as well as by the masts of sailing ships (Piano grew up in Genoa).

The dramatic departure of the building’s profile from the predominantly orthogonal facades of London up through the 20th century can be attributed to the acceptance of the Diagonal motif in all aspects of design as well as to the gradual embrace of non-orthogonal geometry by designers—architects in particular—during the 20th century. Had this shift in sentiment and practice not occurred, it is likely that this marvelous work of architecture would never have been given final approval by the Royal Parks Foundation and English Heritage.

The top of the Shard ends in something of a sky-reaching finial reminiscent of the Art Deco top of Bruce Goff’s tower for the Boston Avenue Methodist Church in Tulsa, Oklahoma, which Piano may have seen in his readings as a student of architecture. (Art historians believe that credit for that dramatic church structure should be shared with Adah Robinson, an art teacher who was later Goff’s client.)

Although there is no certain evidence that Piano was influenced by the Tulsa church, one can’t overlook the common feature it shares with the Shard. About the Tulsa church, Wikipedia has this to say: “At the top of the tower, as well as on many of the other high points and used much in the same manner that churches in the Middle Ages utilized crockets and finials, is a stylized sculpture that represents two hands raised upward in prayer. This motif of praying hands is one that is echoed throughout the building and is one of the areas of design that can be traced back to the early drawings by Robinson.” Some wit has said that the finial atop the Shard represents two hands, palm-to-palm, praying that the adventurous Shard would finally be built.

George HartSculptor & Mathematician2000 – 2020

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.

Read more here.

Poplar ForestThomas Jefferson1806

Poplar Forest
Thomas Jefferson

In 1823, 80-year-old Thomas Jefferson made his final visit to Poplar Forest in central Virginia. He had built the house as a secret escape from his then too-frequently-tourist-visited mansion at Monticello, ninety miles to the north. The three-day trip on horseback must have been a profoundly sad journey for Jefferson because this would be his last engagement with this little-known house that was so intimately tied to his abiding passion for the use of the octagon in architecture and furniture design. That eight-sided geometric figure had appeared in so many of the President’s designs that one could argue it contributed in no small way, decades later, to the emergence of the Era of Diagonality: first through what became known as Orson Fowler’s “Octagon Fad” in the middle of the nineteenth century, then, just before the start of the twentieth century, in the revolutionary work of Frank Lloyd Wright.

In 1800 (half a century before Fowler’s work), during his second term as President, Jefferson was still looking for every opportunity to employ the octagon, in which he saw such beauty and utility. It was then he designed an octagonal house for his daughter Maria at Pantops, in Albemarle County, Virginia. Construction was started—then, very sadly, halted when Maria died unexpectedly in 1804. Roughly two years later, Jefferson dusted off the plan and used the octagonal scheme, mostly unaltered, for the design of Poplar Forest. It was only five years earlier that he had completed the octagonal dome over the heart of his beloved architectural experiment at Monticello.

Poplar Forest is a six-room, two-story house whose basic footprint is an octagon. Although the body of the house, exclusive of porches and two enclosed exterior stair towers, is a pure octagon, no rooms within the house, curiously enough, are themselves regular octagons (regular meaning all eight sides are of equal length and the corner angles are all equal). On the main level, a perimeter ring of four elongated octagons frames a square, windowless, two-story, sky-lighted dining room—a spectacular and memorable architectural space. The house is an elegant and elemental mosaic of spaces that is truly expressive of Jefferson’s discriminating and creative mind. As historian Hugh Howard observes in Thomas Jefferson: Architect (2003), “The house at Poplar Forest has the purity of a geometry lesson.”

If no room in Poplar Forest is itself a regular octagon, one is apt to ask: Why did Jefferson fit all these non-octagonal and irregular-octagon spaces into an overall octagonal volume? Did Jefferson begin with that simple overarching octagonal form and then devise an elegant geometric solution to accommodate his required functional spaces? Or, did he begin doing space diagrams (what modern architects call bubble diagrams), which, as they evolved, naturally led him to the octagonal footprint? Or, did Jefferson steer the arrangement of rooms, subconsciously or consciously, in one evolutionary sketch after another, into the octagonal shape because that was a shape that was embedded in his geometric psyche? This is much the way non-orthogonal shapes are embedded in the psyches of architects during the Era of Diagonality.

Pyramids Through the AgesEgypt and Elsewhere2,700 BC – Today

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.

Read More 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).”

Intarsia, ItalyVarious intarsiatori15th century

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.

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