The relationship between art and geometry is among the most prominent themes of the histories of art and knowledge in early modern Europe. The literature on linear perspective alone encompasses a baffling array of tomes, many of which—such as Martin Kemp’s The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat (Yale University Press, 1990) and Samuel Y. Edgerton’s The Heritage of Giotto’s Geometry: Art and Science on the Eve of the Scientific Revolution (Cornell University Press, 1991)—assess early modern artists’ application of geometry towards representational ends as episodes in the history of science. Over the past two decades, amidst the so-called material and practical turns, as art historians have also adopted methods from historians of mathematics and science, and as scholars of the early modern period outside the discipline of art history have focused their studies on images and objects, our understanding of the rapport between art and geometry in early modern Europe has gained new dimensions. For instance, Alexander Marr’s Between Raphael and Galileo: Mutio Oddi and the Mathematical Culture of Late Renaissance Italy (University of Chicago Press, 2011) and Richard Oosterhoff’s Making Mathematical Culture: University and Print in the Circle of Lefèvre d’Étaples (Oxford University Press, 2018) have both revealed the intersecting material cultures and visual practices of mathematical and artistic inquiry. Another prominent tradition, exemplified in works such as Geometrical Objects: Architecture and the Mathematical Sciences 1400–1800, edited by Anthony Gerbino (Springer, 2014), has explored the practical application of geometry in sacred and secular architecture of the early modern period as well as the theoretical architectural treatises published in this time. Yet despite this wealth of research, no book until now has spotlighted a ubiquitous force in artistic engagements with geometry in early modern Europe: corpora irregulata (irregular bodies), in other words, those polyhedra that do not number among the five solids described in Euclid’s Elements and Plato’s Timaeus.  

Enter Noam Andrews’s The Polyhedrists: Art and Geometry in the Long Sixteenth Century. The Polyhedrists mingles investigative tactics from the history of science (the discipline in which Andrews trained) and the history of art to scrutinize the rise of corpora irregulata as bodies whose allure came to rival that of the “regular,” Platonic solids famously vaunted by classical geometers and Renaissance artists. True to its name, The Polyhedrists revolves around the personalities who catalyzed and sustained the ascent of corpora irregulata in the visual culture of early modern Europe. The cast ranges from mathematician Luca Pacioli (ca. 1445–1517) to artist Albrecht Dürer (1471–1528), printmaker Augustin Hirschvogel (1503–53), goldsmith Wenzel Jamnitzer (1508–85), draftsman and painter Lorenz Stör (ca. 1530–1621) and ivory turner Egidius Lobenigk (d. 1595), to name a few. Andrews’s visual evidence encompasses a variety of media, from drawings to printed, illustrated treatises, intarsia cabinets, and turned ivory objects.  

Andrews wields this cornucopia of material to argue that, with the revival of Euclid’s Elements and Plato’s Timaeus from the later fifteenth century on, artists in Europe, who at first fixated on the “regular” Platonic solids, began to generate forms using Archimedean and other “irregular” polyhedra. Andrews notes that the study of irregular polyhedra germinated not in the rarefied atmosphere of the universities, but in artists’ workshops—a phenomenon with several ramifications. Geometry transformed from an elite academic topic to a more accessible form of knowledge, while artistic craft became an indisputable forum for the formation of mathematical knowledge. The upshot for art was a new visual genre. The geometric image became an autonomous form of art.

Andrews develops the case for the polyhedrists as harbingers of a new mathematic visuality across six chapters, each featuring one or two artists. The introduction orients us in the intellectual and visual culture of classical geometry in Europe during the long sixteenth century. Chapter one covers artists’ initial forays into picturing polyhedra and the proliferation of Archimedean and irregular solids in European visual culture through the publication of Pacioli’s De divina proportione, derived from the notes of Piero della Francesca (ca. 1415–1492). Andrews argues that De divina proportione and the other illustrated geometry texts that circulated in its wake overcame the substantial challenges of verbalizing abstract concepts related to polyhedra to spark the new genre of polyhedral images. Chapter two examines Dürer’s transformative reception of such texts. In his Underweysung der Messung [. . .] of 1525, the artist eschewed the convention of describing polyhedra through perspectival representations and instead depicted the solids in plan, as paper objects that could be cut from his treatise and assembled by artist-readers in three dimensions. The chapter contends that Dürer thereby reimagined polyhedral images as tactile media and bodies inherently open to artistic elaboration.  

The rest of The Polyhedrists assesses the enduring artistic and mathematical consequences of Dürer’s non-perspectival, sculptural renditions of the corpora irregulata and the three-dimensional thinking they inaugurated. Chapter three addresses the reception of Dürer’s novel, polyhedral pedagogy in Lehrbücher, or artisans’ manuals, produced in sixteenth-century South Germany, such as Hirschvogel’s Ein aigentliche vnd grundtliche anweysung, in die Geometria [. . .] (1543) and Heinrich Lautensack’s Des Circkels vnnd Richtscheyts [. . .] (1564). The chapter asserts that such Lehrbücher acted as crucibles of visual and scientific investigation, in which images of polyhedra valorized the craft expertise of their authors by transposing ideal, a priori (first-principle) knowledge into tangible forms. Chapter four investigates the aftermath of the Lehrbüch tradition in Jamnitzer’s Perspectiva corporum regularium [. . .] (1568), a sumptuously etched ode to the elemental qualities of the Platonic solids and the corpora irregulata that evolve from them. Andrews here contends that Jamnitzer’s book and the geometrical drawings he produced alongside it mark a watershed in the history of polyhedral imagery, where artists were now expected to invent novel corpora irregulata. 

Chapter five features a prime site of polyhedral invention—Augsburg intarsia—as well as a key source for the ingenious cabinetry, namely, Stör’s Geometria et perspectiva [. . .] (1567). It proposes that Stör popularized a visual dichotomy between decrepit, all’antica (ancient style) architectures and pristine polyhedra to evoke the relationships between classical geometry and the novel inventions that spring from its ruins. Deepening Andrews’s account of the rise of polyhedra as agents of transformation, chapter six considers the heyday of fantastical, turned ivory corpora irregulata at the courts of Central Europe, especially the Dresden residence of Augustus, Elector of Saxony (1526–86). The chapter claims that the ivory corpora irregulata that such elites collected and even learned to manufacture themselves supported their owners’ political aspirations by both embodying and promoting princely precision and technical prowess. Finally, in an epilogue that features Johannes Kepler’s (1571–1630) anachronistic celebration of regular, Platonic solids as measures of planetary intervals in Prodomus dissertationvm cosmographicarvm [. . .] (1596), Andrews contends that the polyhedrists and their corpora irregulata laid foundations for modern, non-Euclidean geometry as well as the empiricism of the Scientific Revolution. 

There is much to admire in Andrews’s text. The Polyhedrists elucidates obscure mathematical concepts in arresting prose and nimbly strings the book’s disparate case studies into a coherent, compelling narrative. Formidable erudition in the histories of art, science, mathematics, and knowledge undergirds Andrews’s arguments, and yet the author consistently delivers the case with sprezzatura (an apparently effortless grace). Of course, any study so ambitious and ranging is also bound to harbor certain blind spots. While Andrews generally displays meticulous attention to images and objects, discussions of the visual evidence are not always supported by the book’s format—an unfortunate loss, in light of the book’s case for the role of images in forming knowledge. This is most apparent in the absence of figure callouts, as well as various discussions of images and objects (such as the remarkable discovery of an unpublished Lorenz Stör manuscript at Harvard University’s Houghton Library) that, while crucial to the argument, go unillustrated. On another front, while Andrews mentions the violence associated with the acquisition of ivory showcased in chapter six, an opportunity to scrutinize at greater length the ways in which European affinities for turned polyhedra fed into an ecologically pernicious global trade in ivory that operated alongside the trade in enslaved Africans—each of which depended on the geometrical navigation and image-making techniques of cartography—is lost. The omission matters not least because The Polyhedrists is a story about how geometrical explorations brought on modernity, anchored as it is in the global colonialism of the same European princes who collected turned ivory corpora irregulata. 

Despite its imperfections, The Polyhedrists has a great deal to offer art historians. By tracing the visual and material history of the irregular solids across treatises, artist manuals, drawing, print, metalwork, intarsia, and ivory, Andrews convenes previously siloed aspects of the histories of art, architecture, science, and knowledge. Andrews’s synthesis of medial and academic domains shows how geometrical knowledge migrated between artistic media and traversed mathematics and art, while also modeling a way to analyze those phenomena. In so doing, the book substantially enriches the recent research on the material culture and visual practices of early modern geometry, most of which has not dwelt on the relationships between media or transactions between professional geometers and artists. In other words, what art historians can gain from The Polyhedrists is a framework for tracing the interconnected worlds of mathematical and artistic thinking across texts, images, and objects.