by Larry Witham
Piero may have been encouraged to write the work by one of Montefeltro’s humanist advisers, Ottaviano Ubaldini, a connoisseur of Flemish art, astrologer, and regent when Montefeltro’s son, Guidobaldo, inherited his father’s rule in 1482. In the end, Piero dedicated Five Regular Solids to Guidobaldo, stating that it might go well on the Urbino library shelf alongside On Perspective. Piero described his offering as a kind of contrast in social rank; his work was like the “uncultivated things and fruits gathered on a humble table” now being put on the duke’s “most opulent luxurious table.” Yet by accepting the work, he says further, the House of Montefeltro will ensure his memorial, “a monument and reminder of me.” All humility aside, Piero then declares his work’s true novelty. Euclid and other geometers had dealt with this topic, but now Piero had handled it “newly expressed in arithmetical terms.”17 His little book would indeed pioneer making geometry mathematical, not just descriptive and verbal.
On the theory that Piero had already set up a scriptorium to make copies of his earlier On Perspective, he may have used the same setting to write Five Regular Solids, or he may have worked on it between Sansepolcro and his residency in Urbino, which had a library of ancient works. As with his previous writings, Piero first composed in the Tuscan vernacular. Then he had it translated into Latin (to which, in at least four Latin copies, he added the meticulous drawings of geometrical shapes).18 The Latin introduction to Five Regular Solids seems to have borrowed a good deal of rhetorical style from Vitruvius, the ancient Roman writer on architecture and art. In this reaching back to antiquity, however, Piero’s Five Regular Solids was clearly most interested in the Greek legacy of Archimedes, who had begun the high art of analyzing three-dimensional shapes.
Five Regular Solids is precise enough in its citation of Archimedes that Piero must have had a copy of the works of Archimedes with him, very probably as a personal possession obtained from his relative Francesco da Borgo in Rome, but alternatively as a copy in the ducal library of Urbino. The library also had a rare copy of Euclid’s Optics (which Piero cites only in Five Regular Solids).19
Whereas Euclid had worked primarily on plane geometry, Archimedes probed spheres and cylinders more extensively. It is said that he was most proud of discovering that the volume of a sphere is two thirds of the volume of a cylinder in which it fits exactly.20 In Five Regular Solids, Piero reveals his Archimedean fascination with fitting shapes one within another. Breaking new ground, he did this using mathematical calculations, not just visual demonstrations or verbal explications. His Abacus Treatise may have skipped over the “golden ratio” topic, but Five Regular Solids uses this golden proportion—1 to 1.6—to calculate and render polyhedra.
Some of his discussion also points to a kind of Platonist viewpoint on mathematics and reality, since, as Plato said, Ideas and essences are transcendent, and even attempts at achieving the highest forms of intelligible knowledge by precise numbers can be elusive. Or as Piero made the case in Five Regular Solids, “Truly demonstrated knowledge has not yet been discovered concerning the circumference of the circle. But since those who excel in geometry have determined it approximately, let us follow their statements concerning it. For they pose that the circumference is less than 31⁄7 the diameter and more than 31⁄8,” though to be practical in the real, sensible world, Piero thereafter uses 31⁄7 as his approximation in his exercises.21 (The actual numerical value, like the golden ratio [phi in Greek], is an irrational and therefore infinite number, namely 3.14159 … [known also as pi]).
Piero begins Five Regular Solids by working through the five basic Platonic shapes. He goes in sequence, according to the number of their sides. After this, he shows a new method to transform these Platonic shapes into increasingly complex polyhedra, the so-called Archimedean polyhedra. He then moves into demonstrations of fitting complex polyhedra precisely inside spheres or other complex solids.
Piero did not give names to his shapes, but Johannes Kepler would take care of that (for Archimedes and Piero) in the future. One particular shape that Piero produced when he cut the corners off a three-dimensional triangle is what Kepler would designate as the truncated tetrahedron. In his earlier Abacus Treatise, Piero had demonstrated an Archimedean polyhedron that Kepler would call a cuboctahedron. In Five Regular Solids he demonstrates four additional “lost” Archimedean polyhedra, which Kepler nobly names the truncated cube, truncated octahedron, truncated icosahedron, and truncated dodecahedron. Using mathematics, Piero also found a new way to determine the volume and altitude of a tetrahedron, which is a solid with four triangular sides. Moving on, he shows the calculations necessary to inscribe an octahedron (eight-face) solid inside a dodecahedron (twelve-face) solid.
Finally, Piero brings this down to earth, possibly for use by a painter. The challenge here is to know the volume of irregular shapes, and none is more irregular than a human figure. On this problem, Piero turns to a human statue. His solution to finding its volume is comically practical (and no doubt borrowed from Archimedes): immerse the statue in a known quantity of water, and then calculate how much water is displaced. This is not analytical geometry. Yet Piero believes that every shape, no matter how irregular, has a precise measure that is sensible and can be expressed in numbers, at least to the highest possible human approximation. Accordingly, he presents a procedure for determining the surface and volume of a cross-vault, one of his most sophisticated results. Piero does this by relating a sphere, a pyramid, and ellipses to the curved vault section. In all of his geometrical procedures and solutions, Piero reveals his intuitive sense of how local space works, proved also by how he had already used many of these complex constructions in his paintings.
At this stage of his life, Piero was challenging the limits of human mortality, which included his brain’s mental capacities. He had labored on his Five Regular Solids in “my old age,” he recounts, “lest the mind should become torpid by inaction.”22 He was nevertheless filled with a remarkable “passion for geometry and Archimedes,” says Piero scholar James Banker, an interest that “eventually surpassed his commitment to painting.”23 This explains Piero’s next great project with geometry, which must have been equally taxing on his mind—the copying in Latin of the entire corpus of Archimedes, a compendium of seven Archimedean tracts.
The facts surrounding this project by Piero remain elusive, but they have been reconstructed well enough. By this time, Piero must have not only mastered Greek geometry, but he could conceptualize geometry in the Latin language as well. This allowed him to produce a 150-page copy of the Archimedean works that also featured 225 of his own geometrical illustrations. To carry this off, Piero may have used the little scriptorium he had presumably put in place for his previous copying exploits. The final evidence of this episode is a manuscript that now exists in a Florentine archive, and while it has no title or author’s name, it has not only the marks of Piero’s mind, but his own distinctive handwriting and quality of illustrations—all of it part of a modern-day detective story to be told in the late twentieth century (see Chapter 11).24
Piero’s fascination with complex Archimedean solid objects was at the high end of the general Renaissance interest in spatial reality. The concept of infinite Space itself, except when theologians spoke of infinity, was actually not a Renaissance concern, probably because the problem had no practical application and, again, theology had claimed that territory. In the tradition of the ancients, Renaissance artisans were attentive only to objects, their spatial relationships to each other, and their visual and numerical proportions.
Back in ancient Greece, for example, Plato had said that space was simply “where” a creator made things. Aristotle said about the same. Space was what an object occupied, what it “extended” itself into. This did not change much under Christian thought, except to conceive of the biblical God as perhaps the receptacle of all spatial reality and, to further mystify space, to conceive that God is everywhere at once and c
an incarnate himself also as a human being.
Following Plato, Aristotle, and Euclid, Renaissance perspectivists focused on the space of each object, finding its coordinates and then extending its surfaces accordingly. Nobody was yet talking about an infinite physical space into which visual rays disappeared. For all practical purposes, for drawing a picture, the physical horizon was the limit. Alberti’s On Painting was characteristic: “A painted thing can never appear truthful where there is not a definite distance for seeing it.”25 More than a century later, the art chronicler Giorgio Vasari offered a similar appraisal of space—what he called the “perspectives” of objects—as the local extension of one thing or another.26 Space was about extension, foreshortening, or diminution of objects: horses, colonnades, chairs, groups of figures or a figure alone. With the Scientific Revolution, the nature of space itself finally became a central conundrum. It was a chief philosophical puzzle for modern minds such as Descartes, Newton, and Kant, for non-Euclidian geometry, and finally for Einstein.
Piero’s approach to space might be taken in two different lights, one of them more mischievous than the other. As a master of mathematics, Piero seemingly had a playful knack of producing effects akin to optical illusions in his paintings. Modern-day interpreters would look back and call these “spatial games” and “optical tricks,” something that the mathematically minded Flemish master Jan van Eyck was also quite capable of achieving in his works.27 Where Piero toyed with such spatial illusions, they generally related to seeing a subject in the painting as both far and near, or perhaps to seeing two kinds of perspective in one painting (as in the Resurrection of Christ). As three of the most obvious examples, the Arezzo fresco of Constantine sleeping in his tent offers a sense that he is both inside, and outside, the enclosure, depending on the viewer’s far or near study of the scene; similarly, the two angels in the Madonna del Parto seem both inside and outside a royal tent featuring Mary; and perhaps most strangely, the gigantic foreground figures in Piero’s Montefeltro Altarpiece look as if poised both deep inside the church, but also right at the edge of the visual field.28 Although Piero was dedicated to the precision of geometrical perspective, he was also apparently aware of how perceptions can invariably distort it, and he used this to his advantage as a painter. As he said in On Perspective, “The intellect cannot distinguish that part nearest to the eye from that farthest from it without the help of lines.”29 With a few tweaks of those lines in a painting, Piero could cleverly keep the discriminating eye from distinguishing far and near too easily.
Piero’s second approach to space drew upon a theme coming down from antiquity, and that was the belief in harmonious proportions, often attributed to divine origins, as had been asserted by Pythagoras and Plato. Since ancient times, for example, proportions and predictable ratios had been found in harmonious musical scales. To cite just one more example, this proportional harmony was extended to ideal measurements of the human body, and then from the body to architecture. There was always a word of dissent against ideal proportions, a skeptical rejoinder that all people really had their own opinions on a beautiful shape. And yet the mathematics and visual power of certain proportions continued to support a belief in their divine nature: the ideal proportion showed a linkage between God, nature, and the human mind.30 It was also a metaphysical link between mathematics and works of art, and this is where Piero inserted himself.
Alberti had attributed the perception of beauty—the unity, variety, and balance of elements in a picture—to an ability in the innate human mind, a capacity that simply went along with how God had made the universe. Piero was the same. Like most natural philosophers of the Renaissance, both Alberti and Piero found geometry to be especially at the foundations of Beauty, revealing—in both mental and visual form—ideals, shapes, proportions, and laws not found so precisely in other kinds of visual experiences.
Of course, Plato had told them so, and even if that Platonist lesson originated in metaphysics, modern science could not escape some of Platonism’s central observations about human psychology. One of these is that the mind, or brain, seeks essences. A type of essence, moreover, is a kind of proportionality to things in the world. Furthermore, some of these proportions seem to produce mental pleasure, while others do not. In the modern view, the brain has evolved to grasp certain ratios, sizes, and shapes as normal, crucial to survival, and even enjoyably attractive.31 The simplest example is the brain’s ability to judge sizes, and thus distances. Symmetry is another powerful example of proportion. When perceived “out there,” it signals to the brain the presence of health, vitality, and order. Symmetry tells of a living thing, threatening or benign. When symmetry is broken in the visual world, the mind sees a deviation and it snaps to attention, just in case something unusual is happening.32
As much as brain science may want to finally retire the ancient Platonist doctrine of essences, this reality seemingly won’t go away. From a neuroscience point of view, the brain, whether viewing a painting or moving through the world, tries mightily to sort through the flux of reality and find what is essential and constant. If a painting excites the brain, it could well be because the brain has evolved to find in that imagery a confirmation of valuable and lasting knowledge. The brain’s ability to recognize an important essence is based on memory. In Platonist doctrine, memory plays exactly the same role, albeit with metaphysical origins: Plato proposed that each person’s soul has a divine memory, and that this is refreshed by learning true knowledge in the world. Similarly, to make use of its everyday perceptions, the brain must compare them with what is remembered.
In Piero’s day, such mental powers typically were attributed to innate spiritual origins, not neurons and nodules in the brain. That conundrum aside, Piero seemed to paint an idealized world in which the brain and memory readily find essences, characterized by his use of geometry, symmetry, and unusually effective proportions. Amid the flux of the world, Piero’s images prod the brain to remember essences that are pleasing. The ultimate question remains: Is such a mental act of pleasurable remembering purely biological, or does it have a transcendent nature as well?
On such existential matters, Piero would have turned to his religious beliefs, not his geometry. A painter of theological topics, Piero must have been aware of the role religion played in his era. After so much turmoil in the papacy over previous centuries, the laity had taken religion into their own hands. Popular preaching was at its all-time height. The religious ferment balanced off the new emphasis of the Renaissance humanists, which was to reject older Christian beliefs about the “renunciation” of the world and to embrace it in all its glory. On this they quoted the Roman orator Cicero: “The whole glory of man lies in activity.”33 The new ideal of the Renaissance man, following the Roman code of virtù, was to combine classical learning, good style in manner and dress, cleverness in speaking and writing, military courage, and a general boldness.
As a result of the classical revival, Greek science influenced Renaissance religion as well, producing a re-Hellenization of Christian thought. This orientation allowed Christianity to adapt to the new sciences and achieve a new tolerance in beliefs, at least until the Reformation and Counter-Reformation made religion a matter of doctrinal combat once again. The Middle Ages had already produced a method of avoiding conflict between theological dogma and philosophical speculation, and this was the doctrine of “double truth.”34 Oddly, it was men of faith, often working in the Church, who manifested this duality by presenting the sharpest arguments against many Church doctrines, all the while being upholders of the faith.
What the Renaissance added to this discourse by way of reconciliation of theology and philosophy (and science) was a more coherent and systematic restatement of Platonism. The Platonist doctrines provided an avenue to achieve a tolerant ambiguity between conflicting claims. Classic Platonism, as read in the dialogues of Plato, put limits on the human knowledge that was achieved through the physical sense
s (the sensible). It was also careful, and even skeptical, about obtaining the true knowledge of transcendent essences (the intelligible). Nonetheless, Platonist thought and its Renaissance subculture still promoted secular learning and spiritual self-cultivation, and indeed what one late Renaissance Platonist called “the dignity of man.”35 The Platonist outlook motivated rational inquiry, mathematics, experiment, and the acceptance of faith and mystery.
Where Piero stood on all of this can be inferred only from his environment and his acute interest in Greek science and its interplay with Christian thought. His paintings also tell that tale, and perhaps his two final surviving works provide a summary of how Piero existed within the philosophical tautness of his age. One painting is a fresco of the mythological figure Hercules. The other is a private, quirky, and joyous panel painting of a nativity scene, to be known as Adoration of the Child (and as the Nativity for short). They might be taken to represent a kind of double truth for Piero, or a Platonist reconciliation of two opposing viewpoints.
The Hercules has survived as a fresco fragment, suggesting that it had been part of a series of figures—perhaps both pagan and Christian—that Piero had painted as a mural in the ancestral family house in Sansepolcro, much as the princes had had done in their houses. The Nativity, by contrast, may have been his final religious painting. For burial, his family had space for tombs at the abbey, but not a chapel. Hence, Piero probably painted the Nativity for his home as well, sharing its ambience with Hercules and perhaps other famous characters that Piero had painted in fresco around the house.36
Like other Roman myths, the story of Hercules had made a comeback during the Renaissance. Wide use of the Golden Legend had proved the popularity of heroic figures, sometimes rulers but mostly saints. They were visual topics for mosaics, stained glass, and paintings. Speaking to Renaissance sentiments, this genre evolved into a great-men theme. It had been encouraged in the humanist triumph-and-glory literature. Not a few courts around Italy had begun to commission cycles of portraits of storied personages of Greek and Roman myth. One example, having survived intact, is the Montefeltro studiolo in Urbino, where the upper walls are embellished with portrait paintings of great figures in literature.