Mathematical Art

Diagnosis Cancer, Titia van Beugen, Dutch mathematical artist.
She died of breast cancer in 2010.

“Mathematics, rightly viewed, possess not only truth,
but supreme beauty--a beauty cold and austere, like
that of sculpture.”  --Bertrand Russell
                                         British author, mathematician, & philosopher (1872-1970)

Very often artists, like the majority of other human dwellers upon this earth, are adverse to mathematics. Art and math would seem to be polar opposites--output from opposite hemispheres of the cerebral cortex. Few artists might argue with that, though, today, quite a number of mathematicians would. As a young man, I was endlessly fascinated by architecture (still am) but I was never very good at higher mathematics. I hated algebra, was barely on speaking terms with geometry, and terrified by calculus and trigonometry. Slide rules (remember slide rules?) always seemed like more trouble than they were worth, and I was born too soon to benefit academically from pocket calculators. Had all this not been the case, I might have become an architect rather than an art instructor/painter/writer.



The Meeting of Solomon and Sheba,
1450-52, Lorenzo Ghiberti, east door of the
Florence Baptistery--perfect perspective.

Actually the relationship between math and art is long and surprisingly intimate. By the 14th century there was a whole book on perspective by a mathematician named Alhazen, though it took mathematically inclined Florentine artists such as Lorenzo Ghiberti and his friend, Filippo Brunelleschi, to translate his by-the-book formulas into practical applications and teach them to all the mathematically adverse Renaissance artists to follow. Of course, architects such as Brunelleschi and Ghiberti, though artists, have, by necessity, embraced mathematics far more readily than did painters (for reasons outlined above). 

 And that's pretty much where things stood for five hundred years--mathematicians occasionally toying with art, but few artists, like myself, having more than a passing acquaintance with the underlying formulas governing the linear perspective they employed daily. Then came computers. Mathematicians, had, in fact, been graphing their obscure algebraic equations for centuries, but the calculations, not to mention the crudity of their tools, made such efforts way too time consuming to be more than a quickly passing fancy. Mathematicians had better things to do with their valuable time than draw pictures.

 

A Mandelbrot set, featuring
virtually infinite complexity.

With computers, first there was Mandelbrot--Benoit Mandelbrot (02-24-12), an IBM mathematician who, in 1979, with his pioneering work in fractal geometry, was among the first to grasp the fact that the ever-increasing speed of a digital processors erased the primary obstacle in the marriage of art and higher mathematics. But Mandelbrot was a mathematician first, and only an artist, of sorts, more or less by accident. But as computer technology eventually began to rely upon various graphic interfaces, the intricacies of Mandelbrot receded into the realm of fascinating oddity to be replaced by the practical mathematical and artistic demands of counting, collating, and coloring pixels for human consumption. 



Roots grown from multiple seeds using a constrained 3D DLA algorithm,
Paul Bourke, New Zealand mathematical artist



Frabjous, George Hart,
American mathematical sculptor

If you look at today's mathematical artists you find that, like Mandelbrot, they are virtually all mathematicians first, having developed a secondary interest in art. There's even a virtual math museum. Few, if any, of this new breed of artists even own a paintbrush, though some mathematical sculptors rely on traditional tools in giving substance to their computer-aided designs. Like all good artists, mathematical artists create work of great diversity visually, yet it is mostly of an abstract nature often utilizing a great degree of symmetry while having the "cold and austere" beauty Bertrand Russell found so enthralling.

The Apocalipse (Revelation), Anatoly Fromenko, Russian mathematical artist,
work not totally devoid of representational content.

  

 

Origami

Russian Peacock, 2012, Denierim (no scissors allowed)

This past spring, my granddaughter (along with my airman son and her mother) returned from a three-year stint in Japan. Being the exceedingly bright twelve-year-old she is, while in the land of the rising sun, she picked up far more than the rudiments of the ancient Japanese art of paper folding they call origami. (She's also into duct tape art, but we won't hold that against her). I've never been much of a sculptor and I'm doing good to fold napkins so they'll stand on their own, so she didn't even try to teach me even the simplest complexities of the art form. Suffice to say, her grandmother and I found her creations (and watching her create them) endlessly fascinating.

 

Wet-folded origami.

Origami is said to have originated in its earliest, simplest form around a thousand years ago, though it wasn't until the 17th century that the really good stuff like animals, flowers, and figures evolved. And to my surprise, there are actually a half-dozen distinctly different types of origami, including action origami, modular origami, wet-fold origami, Pureland origami, and Kirigami (which allows cutting of the paper). A few days ago I wrote on Mathematical art (07-17-13). It will undoubtedly come as no surprise to those adept at paper folding that the two are closely related. And, like mathematical art, the computer has intruded into the ancient art of origami design as well, eliminating much of the trial and error approach of the ancient innovators.

 

Origami for rich Republicans (impress your friends, use large denominations)

The starting point in origami is not the first fold, but the paper itself. Virtually any flat material capable of holding a crease may be used with varying degrees of success (money works well). However, traditional origami usually relies on a thin, tough, paper, cut to form a square, and usually colored on one or both sides. Size is irrelevant. Wet-fold origami requires a heavier paper with natural starches. This allows for curves, rather than the standard, angular folds. Often more than one sheet of paper, folded to interlock together, are used in the more extravagant constructions. Pure origami allows neither scissors nor glue, though some modern practitioners cheat a little on the glue restrictions.

It doesn't get much simpler than this.

For starters...

The ubiquitous crane (left, the bird, not the construction type) is often the first instructional piece. Directions (as seen above) almost always rely on drawings, and anyone with a modicum of eye-hand coordination usually achieves success. After that, it all comes down to patience, manual dexterity, and the ability to endure no small amount of frustration. Given enough paper and enough time, there is virtually no limit as to subject matter, even works inspired by M.C. Escher (a natural choice) and Vincent van Gogh (not so natural a choice). So far, I've not seen any origami portraits, but it might be fun to try. I'll mention it to my granddaughter the next time I see her.

Origami fashions, Bare Rose. They can only be worn once and are highly
restrictive of movement. (Heavy breathing causes tears and tears.)


 

 

Fractal Art

Photo by Don Archer

Benoit Mandelbrot, digital portrait, 2001

Imagine if you will, a whole, new type of art...an art that didn't exist, couldn't even be imagined, as recently as 35 years ago. It's an art based upon geometry; not traditional, Euclidean geometry, but a whole new type of geometry. It's an abstract art, also one that is largely if not entirely serendipitous. It's an art based upon numbers, real and imaginary, and it's an art that, until the advent of computers, couldn't even be produced on paper. And, while it's based upon a formula, it's anything but formulaic. The simple, yet elegant formula is Z=Z X Z+ C, with C being a constant added each time the multiplication of Z X Z takes place. The result is a series of points, that, when connected, create a graphic image of infinite complexity when enlarged. In nature, a snowflake is a crude example, as are mountains, clouds, aggregates and galaxy clusters. And even though the formula is simple, it was the incredible number of calculations needed to produce this new art form which made it unthinkable, indeed, unimaginable before computers came to be fast enough to perform them and create the images.

Though strictly mathematical in origin, Mandelbrot graphics can be exquisitely beautiful.

It's called Fractal art, and it's first practitioner was Benoit Mandelbrot (above), a Polish-born scientist of French descent who came to this country in the 1970s to work for IBM. It was there he developed both Fractal geometry as a new branch of mathematics, and also wrote some of the first computer graphics programs to print out the art his new, abstract form of geometry could create. Mandelbrot was born in 1924. He came from a highly educated Jewish family. While his father was a clothes merchant, his mother was a doctor and his two uncles were both mathematicians. They fled Poland in 1936 for France where Mandelbrot came of age during the strife and uncertainties of WW II. His education in mathematics, economics, engineering, and physiology was constantly interrupted and irregular. In fact, in many areas, he is largely self-taught. As a result, though primarily a mathematician, he came to have a much more abstract view of geometry than he might have had he attended regularly at a university.  He also came to have a much broader grasp of the other sciences and their interrelationship to geometry.

Infinity, a swept fractal based upon the
Manowar set, a more recent application

If fractal geometry images came tightly bound with the development of computers, fractal art came bound with the Internet. A critical element in the definition of art involves it having an audience. It should come as no surprise then that the first fractal artists were some of the first computer "geeks" of the early 1980s. And the first art exhibitions came with one of the first broad, Internet communities in the early 1990s--CompuServe. But during these early years, the art they created was largely just a novelty traded back and forth among its creators. Then in 1994, a New York City high school English teacher named Don Archer, who also moonlighted as a massage therapist, co-founded the Museum of Computer Art (MoCA), not to be confused with the Massachusetts Museum of Contemporary Art (Mass MoCA).

Yes, even Mandelbrot tattoos

Even though this Cornell graduate has been creating and selling fractal art for several years now, perhaps Don Archer's greatest contribution has been in presenting, promoting, and preserving it (and other computer-generated images) through his Internet museum. Although in many ways it operates like any other museum, choosing its artist carefully, presenting them professionally, it has no brick and mortar address. Like Amazon or Ebay, it's only address is a URL, www.MuseumOfComputerArt.com. 

CGI

Glasses, 2006, Gilles Tran

I have deliberately titled this item with an abbreviation--CGI. Whether you're an artist or not, if you don't know what that means, in today's world, it's high time you learned. CGI stands for Computer Generated Imagery. Its a broad designation, basically including any graphic image generated from a digital source. The image above is NOT a photo. It's CGI. At its most highly refined, realistic best, in today's world of imaging, CGI not just rivals photographic imagery, in many ways, it surpasses it. If one were to look at a mixture of images, both CGI and photographic, the most telling difference would be that the CGI images might appear too perfect. The other major difference, even allowing for present day, state-of-the art photo editing software, would be that CGI allows artists to easily depict scenes impossible (or economically unfeasible) using any other means.

Gilles Tran

CGI is too broad to be a topic here. There are just too many manifestations, from fractal geometry to CAD (computer assisted design), to computer animation. I've already covered a couple of these areas, Fractal Art, Mathematical Art, and Digital Art in the past. Thus, to simplify, I'm going to highlight only one artist and only his CGI art. His name is Gilles Tran. He lives and works in Paris, and though his work is very much at a professional level, he still classifies himself as an amateur. Actually, his "day job" is that of an agronomist. The image at the top is by Gilles Tran. The image of him at right is not. He's a modest man. The photo did not come from his website but from the one where he works. It's the only photo on this page.

Gilles Tran uses POV-Ray software. That stands for Persistence of Vision Raytracer. This software allows the creation of graphic images based upon a text description. No, you can't just type in "draw a box."  As with all things digital, it's a little more complicated than that, though not much. However, there is a learning curve. Actually, the text looks more like the HTML code used to render this page.



<<<===That will get you this.

Here's a sample:

#declare the_angle = 0;

#while (the_angle < 360)
        box {   <-0.5, -0.5, -0.5>
                <0.5, 0.5, 0.5>
                texture { pigment { color Red }
                          finish  { specular 0.6 }
                          normal  { agate 0.25 scale 1/2 } }
                rotate the_angle }
        #declare the_angle = the_angle + 45;
#end

I pride myself in having a "way with words," but I would be speechless in describing the above figure in plain text. CGI has come a long way fast. Gilles Tran created Brittany Night (below, right) when he began experimenting with CGI in 1993. That was pretty much cutting edge, state-of-the art CGI back then. His Glasses (top) was done just thirteen years later in 2006. In addition to POV-Ray, Tran now also uses even more sophisticated software, Cinema 4D, FinalRender and Poser (among others). The two Gilles Tran images below offer a more concise, side-by-side comparison of CGI art then and now.




Lyon Capitale, (magazine cover),
2009, Gilles Tran

Brittany Night, 1993, Gilles Tran,
Created using a PC 386 with 
MS-DOS, 4 Mb of RAM,
and POV-Ray 1.0

The Whitney Biennial

2012 Whitney Biennial, sculpture and paintings by Vincent Fecteau and Andrew Masullo.

We artists all enter juried art shows from time to time. It's fun.  It's like legalized gambling for artists. Ya pays ya money and ya takes ya chances. Worse, there is an inverse relationship between your chances of even getting in, and the prestige of the show. The greater the prestige, the more artists enter, the higher the entry fee, and the greater the unlikelihood you're art will be selected. Of course, just getting in doesn't guarantee any more than some bragging rights, perhaps a few column inches in your local paper, and the hassle of shipping or delivering your work. Naturally, the big payoff comes with a possible prize or maybe a sale. It's a little better odds than the lottery, but not by much.

Hearsay of the Soul, 2012,  Werner Herzog

The really prestigious shows in this country you can count on the fingers of one hand...two if you want to be generous. There's the Carnegie, and the Whitney...and a couple more on the West coast, one or two in the midlands, one I think in Seattle, and another one in Texas, but otherwise, they're all pretty much back room poker games for the locals.  This past year it was the Whitney Biennial which occupied center stage, from March 1, 2012 through May 27.  This year's show had four co-curators, writers and museum directors from various parts of the country. They met and compromised, and the result has been declared everything from mildly interesting to downright bland by writers and art people who get paid big bucks to decide such things. Jerry Saltz of the New Yorker  called it: "...a quiet, incomplete manifesto."  Not exactly a rave review.

Despite all the multi-media
installations, there were
still a few "traditional"
paintings such as Tom Thayer's
This Life is Nothing More Than
Waiting for the Sky to Open, 2011.

This is not good.  In previous years the show has been called “grim,” “flimsy,” and “pious.” Thus there were no TV cameras shooting lines wrapped around the block, no vandalism, no art thefts...in short, nothing to write home about. The show was mostly politically correct, ecumenical, independent, eclectic, geographically diverse, with bows to all media, and endorses an almost mathematical sexual diversity. Actually, there were more women artists in the show this past year than there were men. It blithely does what Biennials are suppose to do, report on the art scene from across the country. The difficulty is, there's not much to report. There were a few hot artist like Matthew Barney, Glenn Ligon, Janine Antoni, Charles Ray, Robert Gober, Charles Atlas, Mike Kelley, and Andrea Fraser, a few veterans like Werner Herzog and Mike Kelley, but most were relatively unknown, including a few who should remain that way. Almost like a convention, there was a delegation from Texas, one from California, and another from the East coast, with the rest of the 51 lucky ducks having been elected "at large." And for this they charge fourteen to eighteen bucks, unless you're under 18, then it's free. I might cross the street, but I wouldn't make a special trip to the "big apple" just to see it. I'll wait for this year's Carnegie International in October. Besides, for me at least, Pittsburgh is closer.
 

Working the No Work (detail), 2011, Georgia Sagri--more process than product.

 

Tessellations

Fall Leaves, by Jill Ethridge

Many, many years ago, when I first became a high school art instructor, I was somewhat startled to find that our high school math department also taught art. They were having students create tessellations. I'm ashamed to say it now, but at the time, I had never even heard of such things, much less how to create one. Yet students were coming to me for help in making theirs. I wasn't much help. I kept my mouth shut and observed (a tactic I'd advise for anyone knowing little or nothing about any subject). What I saw was really quite amazing. Students were coming up with incredibly complex designs (similar to the one above), some just short of mind-boggling. They were well-instructed. I don't recall for sure, but I think these were probably plane geometry students, which is not exactly higher mathematics but neither is it a playground for imbeciles. Mathematics being a rigidly left-brain subject, it would be fair to say this was an example of left-brain art...an art form ruled by rules.

 

Semi-regular floor tiling--hexagons, squares, and equilateral
triangles. Notice that all outside dimensions are equal.
(Ignore the cracks in the tiles.)

For those who are as uninformed as I was on the subject, a tessellation is a design created on a flat surface using tiling of one or more geometric shapes (called tiles), with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries. A common checkerboard would probably be considered the simplest example of a tessellation. The shape (perhaps literally a tile) would be the square (probably, but not necessarily) of two contrasting color values. This would be considered a periodic tiling since it has a repeating pattern. Such patterns can be broken down into two types: regular tilings utilizing a regular polygonal shape, each identical (the checkerboard); and Semi-regular tilings, regular tiles of more than one shape and with every corner identically arranged (above).

 

Bird Tessellation, ca. 1938, M.C. Escher

Undoubtedly the most famous artist to utilize tessellations was M.C. Escher, famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects (above). If the right contrasting colors are chosen for the tiles of the various shapes, amazing patterns are formed, and these can be used to decorate physical surfaces. Tessellations tiles need not have straight edges. Each tile may contain non-tessellating decorative elements as well. Notice the faint vertical and diagonal guidelines used to align the tiles.

Cone mosaic pattern columns, ca. 3300-3000 BC.

Tessellations are not the artistic stepchild of modern-day mathematics. They go back some five-thousand years to the Sumerian culture (above) of around 3300-3000 BC (located in modern-day Iraq). Since that time, they've been an element in virtually every civilization having developed an advanced decorative culture. In 1619 the German mathematician, Johannes Kepler, made one of the earliest documented study of tessellations. He wrote about regular and semi-regular tessellations in his Harmonices Mundi. He was the first to explore and explain the hexagonal structures of honeycombs and snowflakes. About two-hundred years later, in 1891, the Russian scientist, Yevgraf Fyodorov, in studying the arrangement of atoms in the crystalline solids, proved that every periodic tiling features one of seventeen different groups of isometries (reflections, rotations, and translations). Fyodorov was the first to engage in a mathematical study of tessellations.

An Alhambra tessellation as drawn by M.C. Escher, 1936.

Being the farthest thing from a mathematician or scientist, from this point on tessellation go way over my head. Suffice to say, many other types of tessellation are possible under different sets of rules. For example, there are eight types of semi-regular tessellations, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner. Irregular tessellations can also be made from shapes such as pentagons and polyominoes (a plane geometric figure formed by joining one or more equal squares edge to edge, as in the electronic game of Tetris). Besides floor tiling, tessellations are also sometimes used in wallpaper design. In fact, tilings with translational symmetry in two independent directions are categorized in what's they call wallpaper groups. Unless you're a mathematician, on a par with Yevgraf Fyodorov, you don't want to know the details, other than to recall his magic number of seventeen such groups. It has been claimed that all seventeen are represented in the Alhambra palace in Granada, Spain (above), though there is some dispute in this regard.

A Republican tessellation

If you're a RINO
(Republican in name only)
you might like this tessellation more.

 

Nathan Walsh

Evening in Paris, 2014, Nathan Walsh

One of the things I'm quite conscious of in writing about art is that I don't play favorites. That is, I try not to emphasize only artist whose work I really like as opposed to those whose style and content, while often quite fascinating, interesting, and adept, don't garner the respect and admiration I have for painters such as the British artist Nathan Walsh. I love to see hyper-realism and I'm endlessly fascinated by the urban landscape, reflections, movement, and atmosphere. Moreover, Walsh's urban landscapes are huge. One might be tempted to call them "life-size," though given the subject matter, that might be stretching the term a bit.

 

Walsh's exactitude is incredible.

People who enjoy the type of work Walsh does might often forget that before the first brushstroke, comes hours and hours of research, critical decisions as to scale, viewpoints, and perspective (especially with the eyelevel street scenes Walsh seems to favor. Then there's the drawing. In featuring another urban artist whose work is similar to that of Walsh's, I suggested that that the artist used a projection device to save time and effort. Within days I received an angry e-mail from the artist emphatically stating otherwise and demanding a correction. I can't remember whether I obliged him or simply took down the offensive posting. In any case, let me make it clear, here and now, Nathan Walsh does not use a projector. He's a master at freehand drawing and good old pencil and yardstick draftsmanship as seen in the gif photo of Evening in Paris (top) and the New York City street scene NYC6AM (above).

 

Though Walsh's immense urban landscapes sell in the low six-figure range (in dollars), the man earns evert dime of it.

Nathan Walsh was born in Lincoln, United Kingdom in 1972. He earned his BFA degree from Liverpool School of Arts and his MFA from University of Hull. He has been painting for approximately fifteen years. His works have been exhibited in Korea, London, Switzerland, United States, and North Wales. He is represented by the Bernarducci Meisel Gallery in New York. He currently lives and works in York, England. Walsh's giant pieces have reportedly been valued at as much as £80,000 ($125,000).

Rockefeller, 2013, Nathan Walsh

Before beginning work on his paintings, Walsh sometimes peruses more than 300 images and photographs of the location before committing to it. The overriding theme behind his works is an interplay between culture and location. He often draws inspiration from European and American art, such as Pointillism, Post-Impressionism, and (of course) Realism. From there he blocks areas with color and, over the subsequent months, layers are built up and sanded away. Though he seems to favor New York City, Walsh has also put together hyper-realistic images of cities such as Paris (bottom image) and Barcelona. For larger works, the entire process from start to finish can take up to a year, while some of his smaller pieces, though still grand in size, can take (only) about six months. Walsh's Rockefeller (above) is some 63 inches x 88 inches. Some of Walsh's paintings are over eleven feet wide.

New York Reflections, Nathan Walsh

Apple, 2011, Nathan Walsh

By using simple mathematical ratios Walsh begins by describe concrete forms within his picture plane. Over a period of time he draws and redraws buildings, manipulat-ing their height, width or nature in relation to other pictorial elements. By intro-ducing spatial recession to these elements the aim is to present a world the viewer can enter into and move around. Some of the more recent works deal with layers of information, such as the de-piction of reflective surfaces or the combination of inside and outside spaces as seen in his New York Reflections (above). Walsh explores the potential for re-resetting reality, sandwiching what's in front of, and behind, the viewer together much like his Apple (above, left). Duplicating the flatness of a photograph or a series of stitched together photographs is of no interest to Walsh. The reproduction in paint of these mechanical processes serves to negate the human experience of responding to the world.

Lake Street, 2018, Nathan Walsh

Of course the urban landscape is nothing new, nor is hyper-realism. However, Nathan Walsh belongs to a new generation of artists who are extending the boundaries of realist painting. His paintings demonstrate an ambitious effort in combining photographic source material with the traditional skills of the representational artist. This is not easy painting but the bracing clarity of his work and the satisfaction we can derive in spending time with it shows a significant achievement. Walsh has managed to combine architecture, painting and photography all into one. His works are not just highly detailed, but full of texture, and an exceptional sense of color. Walsh's Lake Street (above) showcases his adroit translation of atmospheric color to canvas.

Catching Fire, 2016, Nathan Walsh

Photorealism is a loaded and complex term coined by Louis Meisel in 1969. Walsh aims for his work to be as convincing as possible, but only on his own terms. Online or in print his work may seem photographic in appearance, but the actual nature of his paintings is very different. He finds his sketchbook to be of increasing importance even if just for notes on color or whatever he happens to be thinking about at the time. This immediate personal response to the environment plays an important role back in his studio where he is reliant mostly on the photographs he's taken. The end result is a heavily worked surface which aims to maximize the potential of the paint, and present an alternative reality to our own. Over the past three years Walsh's paintings have become more about describing particular weather conditions and atmosphere. He sees this as playing more of a role in future work where eventually the cityscape is just a stage to investigate the weather through paint.

Carousel, 2015, Nathan Walsh

P.S. Walsh also paints Ballerinas (2016)

 

The School of Athens (in depth)

The School of Athens, 1508-11, Raphael

Raphaello de Sanzio,
Self-portrait, 1506

Just about everyone has heard of the Renaissance, the period in Italian art of some forty years from roughly 1480 to 1520. And anyone familiar with the arts is no doubt familiar with the half-dozen or so landmark painting masterpieces produced during this period (or shortly before or after it). They would include at least one each from what I've termed the "big three" of the Italian Renaissance, Leonardo, Michelangelo, and Raphael--artists so prominent their first names alone should suffice. Two of the three lived long, productive lives while the third, Raphael de Sanzio died young. Born in 1483, he died suddenly (on his birthday, no less) in 1520 at the age of thirty-seven, his lifetime perfectly coinciding with the Italian High Renaissance. We're all too familiar with Michelangelo's Sistine Ceiling and Leonar-do's Mona Lisa. But Raphael's comparable fresco, The School of Athens (top), located in the Vatican's Stanza della Segnatura is as underexposed as the other two are overexposed.

A "Who's Who" of Greek philosophy with likely names in black and Raphael's possible models in red.

The School of Athens shows Plato and Aristotle in conversation. Plato, on the left, upwards while Aristotle, on the right, points down. The here and elsewhere, heaven and earth are the subject of these discussions. The fresco, painted between 1508 and 1511 (dates vary) conveys an impressive synthesis of the world-view of the two great Greek philosophers that was formed in the course of the 15th-century and would have been completely inconceivable just a century earlier. This was the result of the rediscovery of Plato which took place in Florence thanks to the efforts of the Platonic Academy and the activities of Marsilio Ficino and his circle. Restored to his master's side, Aristotle, who had never suffered the same neglect, could now speak, and his words took on a new significance.



The elder Plato walks alongside Aristotle.
School of Athens (detail). Leonardo is said
to have served as the model for Plato.

Plato lived in Athens during the 5th-century BC. He was a disciple of Pythagoras' school of philosophy which interpreted the universe as a mathematical system. Plato believed that a link existed between mathematics and music, and understood the heavenly bodies as entities separated by rhythmic intervals similar to those found in music. The heavenly spheres followed the same principles of harmony as those applied in music--heavenly music (so to speak). According to Plato, the entire world of creation, which we perceive with our senses is merely the shadow of the real world--a world of godly causality--the world of music. Further, he believed that only those minds which have been trained in the contemplative use of reason could know the only true world, a world of pure harmony. If that sounds pretty "deep," it is, and Plato's teaching was as much a lost cause in Europe during the Middle Ages as the study and knowledge of the Greek language itself. Apart from individual quotes used by Latin authors, all that was known of Plato's work was the Latin translation of the treatise on mathematics, the Timaeus (an anachronistic bound edition which Raphael depicts Plato holding under his arm).

Zoroaster, Ptolemy, Raphael as Apelles, and Perugino or Timoteo Viti as Protogenes, are arrayed on the right as followers of Aristotle.

Plato was Aristotle's mentor, but he moved away from his teacher's ideas in that he believed it possible for man to understand the laws of the universe with his senses and study them with the help of logic. Aristotelian mind is not contemplative in itself. The main doctrine of the medieval church was based on established Aristotelian thinking, which influenced biblical interpretation and the understanding of the relationship between God and man. Logical mind games were something of an intellectual passion among the medieval schools of theology. Moreover, they were completely comparable to those we know today, which have led to the invention of the computer. The problem was, having been engulfed by logic, left a degree of uncertainty with respect to the body of Aristotle's teaching. In an attempt to explain the world, Aristotelian reason tended to lose itself in a roomful of mirrors.

Cosimo de' Medici, 1545,
Agnolo Mariano

The revival of Platoism began its slow spread in the city of Florence when Manuel Chrysolaras from Greece was invited to give a series of lectures at the University of Florence sometime in the early years of the 15th-century. Chryso-laras' student circle included the young Cosimo de' Medici (right). He and others who were interested in the study of philosophy, gathered around Ambrogio Traversari in the monastery of Santa Maria degli Angeli. The young Cosimo was also a member of that group. Traversari, the general prior of the Camaldoese monks was one of the few men of his time who was fluent in both Latin and Greek. He set about trans-forming the Monastery of Camaldoli, high up in the Casentin mountains, into a workshop for the translation of classical authors. Cosimo withdrew from his phil-osophical studies at the age of forty following the death of his father, Giovanni de Medici in 1429. He was obliged to take over the family business. However, he continued to buy books and spend part of his vast fortune on the support of humanists and their work. One such project carried out with Cosimo's financial aid was a search by Poggio Bracciolino and Niccolo Niccoli of Europe's monastery libraries for the ancient classical texts, which had been preserved for centuries thanks to the efforts of the Benedictines. In 1437, Cosimo de' Medici was present at the Council of Ferrara which brought representatives of the two great Christian churches--Greek and Roman--together in a last-ditch attempt at reunification. There Cosimo met the Greek scholars from the Byzantine delegation and the Emperor of Constantinople, John VIII Palaiologos.



And on the left, the school of Socrates (in the tan robe, a follower of Plato), The School of Athens (detail),  Raphael.

When the town of Ferrara was no longer able to accommodate the Council, Cosimo offered to foot the cost for it to continue in Florence. This single, magnanimous, yet seemingly incidental gesture was enough to change the course of European intellectual history. The Greek scholars who moved to Florence with the Byzantine delegation were the main impetus for "the new Plato." There followed a series of memorable lectures by Georgis Gemisto Plethon at the University of Florence, which was attended by all the humanist scholars living in the city at the time. The importance of the lectures by Plethon, who was over eighty years old at the time, was connected with the fact that Plato's dialogs had already reached Italy a decade earlier thanks to the efforts of Giovanni Aurispa. Aurispa, a humanist, was a bibliophile antique dealer who was constantly on the road between Constantinople and Rome. He had managed to save a considerable number of classical works.

Raphael's School of Athens (right) as seen in the Stanza della Segnatura.
The Cardinal Virtues, also by Raphael, is on the left.

Raphael's rival, Michelangelo,
depicted as Heraclitus,
School of Athens (detail).



Cosimo de' Medici commissioned a young man named Ficino with the task of translating Plato and hence starting a Plato Academy. Ficino translated the Hymns of Orpheus and several other Greek works into Latin. In 1464, he be-gan translating Plato's dialogs. Cosimo was first able to read Plato's words from Ficino's translation while on his deathbed. The Platonic Academy (Euro-pe's first modern academy, was truly es-tablished through the efforts of the small group gathered around Cosimo's death-bed listening to Ficino's translations. It's hard to overemphasized the influence of the Plato in 15th-century Florence on the fine arts and their flowering during the golden age of the Tuscan city. The works of Sandro Botticelli such as his Adoration of the Magi (below), from 1475, are not generally intepreted as a rendering of the Platonic mythology in painting. However, the art of Domenico Ghirlandaio (bottom), who reached the summit of his artistic career in Florence, appears to be based on the philosophy of Plato. Yet the sublime tranquility of the figures rendered by both artists with their imperturbable calm mark them as "ideal" in the Platonic sense. A generation later, Raphael's The School of Athens was one of the direct results of the birth of Cosimo de' Medici's Plato Academy.

Adoration of the Magi, 1475, Sandro Botticelli

Zachariah in the Temple (detail), by Domenico
Ghirlandaio, depicts four humanist philosophers
under the patronage of the Medici.