Archimedes hesitates, transfixed by the rhombicuboctahedron hovering on the edge of the page, like a strange species never before encountered in nature. His body, rendered in layered chiaroscuro print, appears in a pose of almost vigorous, if haltingly arrested, contemplation. In Ugo da Carpi’s (1480–1532) dramatic rendering after Raphael, human and geometrical body are staged as a study in contrasts. At once rational, ordered, and finite, the rhombicuboctahedron is also out of this world. It stands both for itself and for the idea of itself, so much so that it appears to flicker in and out of focus, as if in excess of the real. Swathed in a cascade of wrinkled and tangled cloth, the philosopher is captured on the cusp of apprehending the crux of knowledge made palpable as a discrete object. He grips a tabula rasa in his left hand, grappling with the decision of whether to divert his gaze in order to ensnare it. Perhaps he wonders if, in so doing, he would run the risk of the polyhedron, and all it represented, vanishing beyond the reaches of his imagination.

Polyhedra are a spectral yet constant presence in the history of Western culture. Emblems of essence, they encompassed two major groups in the early modern period — the Platonic or regular solids (corpora regulata) and the Archimedean or semiregular solids (corpora irregulata): the latter forms derived by slicing up, truncating, or embellishing the former. There have only ever existed five completely regular and symmetrical polyhedra. No more, no less.¹ To artists, mathematicians, and philosophers at the turn of the sixteenth century, these regular solids — the pyramid (four triangular faces), cube (six square faces), octahedron (eight triangular faces), dodecahedron (twelve pentagonal faces), and icosahedron (twenty triangular faces) — radiated a tantalizing promise of divine symmetry, order, and perfection that spanned from the building blocks of matter to the proportions of the human body and the structure of the universe. The maturation of perspectival geometry had only made the representation of the solids a more pressing issue that verged on the realm of the existential. The capacity to produce disegni (drawings/designs), as theorized by Giorgio Vasari (1511–1574) and others, resulted from years of artistic training required to fully transmit onto the page “an apparent expression and articulation of the conceit that one has in the mind”.² Likewise, to successfully represent polyhedra in three dimensions was to boldly venture beyond the gauze separating the world of appearances from the world of causes and to return with an artifact that could be tangibly appreciated.

While the history of art and architecture has traditionally couched the Renaissance in terms of achievement and genius, the history of science and mathematics has conjured an ulterior reality of clashing ideals and concepts, a period swamped in irregularity, only beginning to grapple with the measurement and definition of newly discovered lands, trading in rumors of monstrous foreign creatures, consumed by the verification of data sets and the purging of translation errors from newly printed classical texts. The story of the early struggle to visualize the solids collapses these historical vistas and disciplinary registers. In a matter of decades at the advent of the sixteenth century, a form of Platonism — clothed in Christian, humanist garb — collided head on with perspective in all its novel complexity and contradiction. The force of this meeting would dislodge the solids from the text of Elements, Euclid’s epoch-defining treatise on geometry, and reconstitute them as the quintessence of the third dimension in its pure and unadulterated form.

The most popular, medieval version of Euclid’s text belonged to the mathematician Campanus of Novara (ca. 1220–1296), who had reworked earlier manuscripts of Elements, some of which had been translated into Latin from Arabic in the twelfth century. Written and compiled around 1250, Campanus’ Elements was reissued in 1482 by Erhard Ratdolt (1442–1528), a publisher from Augsburg working in Venice, and was notable for being the first printed book of mathematics to lay out explanatory diagrams to accompany Euclid’s technical terminology. Like the early publishers of Vitruvius’ De architectura, which had been rediscovered in the fifteenth century as a manuscript without illustrations, Ratdolt surely would have recognized the marketing opportunity that adding images to a classical text could provide. But unlike Vitruvius, whose explicit references to architectural descriptions and drawings left ample room for artistic intervention, the Euclidean paradigm was premised upon the painstakingly detailed instructions necessary to precisely construct geometrical figures.

Envisioned as the sort of diagrams recognizable in any modern mathematical textbook, Ratdolt’s portrayal of the solids ran aground on the inhospitable shores of the third dimension. Three factors had converged to frustrate the solids’ effective realization: the difficulty of comprehending the final books of Euclid, the mechanical production of the woodblock prints based on a nascent technology, and the conceptual difficulty of understanding what the geometrically composed third dimension would look like on the page.

The use of perspective to depict polyhedra had been experimented with before, most notably by Paolo Uccello (1397–1475) in his stellated dodecahedron on the marble floor of the basilica of San Marco in Venice.³ But the first printed text to theorize their construction and manipulation in the science of perspective was Pacioli’s influential edition of De divina proportione (1509). While the images would yield to future refinement, the book’s prints ignited a broad interest in how geometry was visualized. Yet De divina proportione also reflected the perspectival glitches the solids proffered.⁴ While the two earlier manuscript versions dazzle with vivid color drawings by Leonardo da Vinci or his circle, the woodblock-printed solids in the published edition were roughly and unconvincingly rendered with clumsy crosshatching and awkward shading. The dashed lines representing indirect light on the leftmost face of the Octaedron planum solidum, to take a representative specimen, do not read against the uneven distribution of shading across the solid's two darker faces.⁵ And the dark shading patterns, meant to articulate the edges of the solid, efface the very three-dimensionality the artist was trying to achieve. Similarly confused is the Octaedron elevatum solidum, where the orienting of contour lines conflicts with the object’s edges. The creator of these inchoate images, operating with a limited repertoire of stock textures (dashed, solid, and blank), had evidently deployed his patterns with only the most basic knowledge of how to bring each solid to life, unable to capture the way shadows cast their presence upon a subject. In the interval between geometry and its image, the caprices of pictorial description were already divulging their indiscriminate impact on the transmission of mathematical knowledge.

The desire to depict substance, and substantiveness, was born out of the unique priorities of perspectival representation in the Italian Renaissance. Whereas Euclid’s instructions for three-dimensional construction required deciphering several pages of highly technical Latin, within the visual realm — a realm newly essential to the pursuit of geometrical knowledge — significant contributions could be made by artists and geometers. On the heels of the success of De divina proportione, physical models of Platonic solids were adopted as commonplace objects in the Renaissance studiolo, particularly for instruction in geometrically modulated perspective. While few physical models from the period remain, their casual ubiquity is preserved through passing mention in texts and in various visual depictions of Renaissance scholars and artists at work.

Vittore Carpaccio’s (1460–1520) sketch of a scholar displays several polyhedral objects hanging from strings above a workspace, possibly astrolabes, armillary spheres, or indeed models of the Platonic solids. The scholar holds a book with his left hand and a compass in his right, presumably gazing through the open window upon the celestial scene he is measuring. Newe geometrische vnd perspektiuische Inuentiones (1610) by Johannes Faulhaber (1580–1635), a mathematician from Ulm and associate of René Descartes, includes a similar image in which the Platonic solids are clearly depicted as tangible objects hanging from hooks above the door of a workshop where a man is pictured setting out a perspectival drawing of a cube.⁶ Faulhaber situates the solids as integral components of a network of measurement devices that includes rulers, astrolabes, and compasses, suspending them adjacent to a Düreresque apparatus, drafting surface, and reference book opened to a page with a perspectival pyramid on one side and a generic architectural scene on the other.

The presence of reference books in such images was common, but the engraving of the philosopher Diogenes by Giovanni Jacopo Caraglio (ca. 1500/1505–1565) after a drawing by Parmigianino (1503–1540) adds a further layer of scholarly consultation — and in doing so perhaps offers a critique. In the copperplate engraving, Diogenes points toward a reproduction of a dodecahedron as he also consults an open book whose pages are concealed from view. While the twelve-sided solid can be identified as being from Pacioli’s De divina proportione, the book on the stand is not so easy to pinpoint, though its size suggests it could be the recently published Elements, populated with Ratdolt’s diagrams. With his stick pointed toward Pacioli’s text, Caraglio’s Diogenes might inadvertently disclose the inadequacy of Ratdolt’s depictions: how the more fully realized perspectival images of De divina proportione were critical for grasping the reality of the solids.⁷

Wielded, brandished, held limply or with vigor, the trope of the pointed stick or compass introduced didacticism into portrayals of mathematical learning. Both literal and metaphorical, the stick hints at the divide between culture and nature, the need for guidance, discipline, and punishment. In the spatiality it engenders, the viewer is conscripted into a performance of knowledge, becoming witness to the act of transmission. Two masterful paintings of stick-bearing figures, further explicate how polyhedra were used as complementary participants in the acquisition of knowledge.

In de’ Barbari’s Portrait of Luca Pacioli, the mathematician, resplendent in a habit of the Franciscan order, stares intently at a suspended glass rhombicuboctahedron, half-filled with water to dramatize the volume of its container and de’ Barbari’s painterly skill in perspective. Each of Pacioli’s hands unseeingly traces the contours of the same information: a chalk drawing of a tetrahedron inscribed in a circle on his right and the corresponding text from Ratdolt’s Elements on his left. Pacioli is flanked by an aristocratic apprentice or student whose gaze is directed askance at the viewer, its vector pulling focus to a wooden model of a dodecahedron resting on Pacioli’s desk.

In a similar vein, Der Nürnberger Schreibmeister Johann Neudörffer mit einem Schüler (1561) by Nicolas de Neufchâtel (ca. 1524–after 1567) depicts the “writing master” Neudörffer training his attention upon the vertex of a skeletal, wooden dodecahedron. To his left, a student takes notes, presumably attempting to draw the dodecahedron. Behind them both, a wooden cube is suspended with its vertices pointing up. On close inspection of the painting, the cube appears to have been hung on a painted nail protruding from the rear wall of Neudörffer’s workshop. Neudörffer presumably will hang his new dodecahedron next to the cube once he finishes working on it.

Both stick-bearing scenes take place in the contiguous space of the studiolo or Werkstatt, artificially darkened by each painter to spotlight the actions of the main protagonists. Unlike Carpaccio or Faulhaber, or the many others who produced cluttered depictions of contemporary workshop activity, de’ Barbari and Neufchâtel make use of a stripped-down and fathomless backdrop to ensure that their paintings will be read as allegories of pedagogy and not merely as depictions of more everyday mathematical practice. All extraneities have been eliminated in order to emphasize the essential components of study and teaching. At once real and unreal, the painterly polyhedra occupy a hybrid reality halfway between mental constructs and objects made physical.

Models were useful for helping to visualize ancient Greek geometry and as stereometric drawing aids. But if polyhedra served the purpose of clarifying Euclid, they did so primarily through their presence in printed editions of his Elements and in Pacioli’s De divina proportione, often available for consultation in the study or workshop of an artist/mathematician. In turn, these printed images began to exert their own magnetic pull. As cases in point, Fra Giovanni da Verona’s intarsia in the monastery of Monte Olivetto Maggiore near Siena and in the church of Santa Maria in Organo, Verona, both completed around 1520, display a seventy-two-sided sphere, an icosahedron, a truncated icosahedron, two elevated icosidodecahedra, a cuboctahedron, and a cube with equilateral pyramids affixed to each face, all derived from De divina proportione.

The technical mastery on display in the works above obscure the more challenging impasses of polyhedral geometry in the first decades of the sixteenth century. Not everyone agreed that the effort of producing geometric corpora was worth the result. For Vasari, when the representation of geometry in perspective was the central purpose of an image and not subservient to the realization of a greater pictorial whole, the loss of labor and time of such intense and misspent concentration was wont to drive an artist to melancholy.

For although these are ingenious and beautiful, yet if a man pursues them beyond measure he does nothing but waste his time, exhausts his powers, fills his mind with difficulties, and often transforms its fertility and readiness into sterility and constraint … not to mention that very often he becomes solitary, eccentric, melancholy, and poor.⁸

Perhaps such resignation weighs down Dürer’s iconic angel in Melencolia I (1514), one of the best-known yet elusive images in the history of art, whose posture hints at the difficulty of reconciling the world of abstract geometrical knowledge with the concrete reality of objects, models, and visualizations. Eluding definitive analysis as it captures the hermetic spirit of an age in a claustrophobic panoply of symbolic artifacts — the magic square, the starving dog, the bat carrying the engraving’s title, the scattered tools — Melencolia I remains unique by virtue of the sheer density and labyrinthine structure of its self-referential ambiguity. It manages to convey Dürer’s complete mastery over the medium and somehow to represent access to his unconscious, as if, in its delirious precision and masterful array of intellectual conceits, the image had opened up a vista to a primal scene of memory and loss in excess of its own aspirations.

Melencolia I has garnered many readings, but central to most of them are the limits of geometry to describe both the world and artistic self-understanding. Surrounded by accouterments borrowed from the traditional allegory of geometry as a woman engaged in acts of measurement, a truncated rhombohedron looms enigmatically in the middle distance, dividing the foreground from a placid sea that vanishes into the horizon, ostensibly the type of natural territory the figure purports to measure.⁹ After Dürer, many of the allegorical staples made famous by Melencolia I would remain in iconographic circulation in similar works by Virgil Solis, Abraham Bloemaert, Giovanni Benedetto Castiglione, and Hans Sebald Beham (1500–1550), whose own Melancholia (1539) is replete with a sphere, workshop tools, and the acquiescent expression on the face of the burly figure as she distractedly toys with a compass. Nevertheless, with the exception of the imitative copy of Melencolia I (1602) by the Antwerp-based artist Johannes Wierix (1549–1615), the corpus irregulatum from the 1514 original was always conspicuously avoided.

The visual history of polyhedra is littered with false starts, poignant failures, and allegories unable to convey the weight of their subject matter. Polyhedra appeared in more or less proficient varieties, but artists overall became steadily acquainted with the solids without having to know the specifics of Euclid. That this familiarity would be beholden to technologies of their reproduction and thus to questions of representation signified a stark departure from eons of learning and thinking in the classical mold. Whereas Euclid trafficked in the absolute precision born from abstract and idealized quantities, the impetus to visualize mathematical concepts for general consumption had the counterintuitive impact of opening up geometrical-cum-theological reasoning to the peculiarities of aesthetics — to the condition of a line, to the quality of a print, and above all to the exercise of perspective and judgment. While these tensions had always been present in the “speculative sciences”, the gap between geometry as text and as image had not heretofore been plumbed by artists, certainly not by artists of the caliber of Leonardo and Dürer, not to speak of de’ Barbari and Neufchâtel. The strangeness of confronting the solids in vivid three dimensions, not as diagrams but as fully realized objects, cannot be underestimated. Distilling philosophical concepts to the level of tangible things consolidated models of thinking based upon the demand to witness what could no longer be taken for granted. Hereafter, human perception would continue to intervene in what had once been the “divine nature” of geometry.