One theory about wormholes — those speculative cosmological structures that tunnel between distant points in the universe — holds that spacetime can be folded like a piece of paper, bringing the near and far into proximity. Examining the manuscript pages from Augustin Hirschvogel’s Geometria (1543), collected below, we are thrust into an analogous encounter with early solid geometry; suddenly the distant past is thrown up before our eyes on screen, and we glimpse another form of interdimensional folding, one much less hypothetical.

Visualizing various polyhedra in three dimensions, Hirschvogel unfolds these bodies on the page, creating flat “nets” (as mathematicians call them) that resemble blueprints for paper sculpture. Dürer was the first to use this method, in Underweysung der messung (1525), and Hirschvogel expresses his debt to him in Geometria, a volume with grand and humorous ambitions, judging by the title page’s inscription: “The Book of Geometry is my name. / All liberal Arts were originally derived from me. / I reunite Architecture and Perspective”. Standing on his master’s shoulders, Hirschvogel drew an Archimedean solid that Dürer’s hands never touched, the rhombicosidodecahedron, which has thirty square faces, twenty triangular faces, twelve pentagonal faces, sixty vertices, and more than one hundred edges. Hirschvogel was evidently proud of this accomplishment, for it adorns his title page too, refashioned as a perch for an owl, the ancient avian symbol for wisdom — here mobbed by lesser birds, perhaps enemies of knowledge. When not drawing shapes, Hirschvogel made solid objects, specifically “Tyrolese owl jugs” (or so some scholars have conjectured), which may have been awarded for feats involving archery or drunkenness.

In Ancient Greek geometry, the Platonic solids were connected to the foundational elements of matter: earth, water, fire, and air. Born in Nürnberg and living the majority of his life in Vienna, Hirschvogel witnessed a fifth association come on the scene in the sixteenth century, the association of dodecahedrons with heaven. Hirschvogel took this sacred geometry further, mapping the Latin alphabet’s vowels onto polyhedra. What purpose this may have served, we do not know, but it hints toward a divine, mathematical language, in which sounds and solids cohere in the eye and on the tongue. The abstractions on the pages of this manuscript, held by Dresdon’s Saxon State and University Library, have gained depth as they’ve aged — smudgy stains litter the paper, interacting with its complex geometry. This decay seems to drag the geometer down from the lofty world of mathematical forms, back into the irregular geometry of embodied life.

For more on Dürer and Renaissance geometry, see Noam Andrews’ essay “The Polyhedral Perspective”. For some lovely stellated and uniform polyhedra, see our post on Max Brückner’s collection.