A New Hypothesis on Dürer's Enigmatic Polyhedron in His Copper Engraving “Melencolia I”

Abstract

A hitherto unnoticed property of Dürer's truncated rhombohedron, a way he could have constructed it, and a possible motivation for this construction are discussed. C 1999 Academic Press Eine bisher nicht beachtete Eigenschaft von Dürers gestutztem Rhomboeder, ein Weg, wie er ihn konstruieren konnte, und eine mögliche Motivation für diese Konstruktion werden diskutiert. C 1999 Academic Press 1991 AMS subject classification: 01A40. There are many books and papers on Albrecht Dürer (1471-1528), especially in relation to his contributions to geometry (for an introduction, see [6-9]) and his masterpiece "Me-lencolia I" (1514, Fig. 1). For this reason, I will dispense with preliminaries and present a hypothesis on the well-known polyhedron that, as far as I know from the literature on Dürer, is new. Let us remember two facts: First, Dürer was a pioneer in solving spatial constructions by the method of plan and elevation, later perfected by Gaspard Monge. (It consists of representing spatial objects by their projections onto two or three mutually perpendicular planes and executing the spatial constructions with these projections.) This is demonstrated not only by about 15 different examples in his Underweysung [1] (1525) but also in his Vier Bücher von menschlicher Proportion [2] (1528, cf. Fig. 2), where he applies this method also to human heads and bodies. There, he shows how the plan and elevation of a cube in a general position can be constructed from a simple initial position by repeated turning (Fig. 2). Unfortunately, the Vier Bücher are less known and less studied by historians of geometry than the Underweysung. For the construction of perspective pictures, Dürer gives in the Un-derweysung only technical proposals by means of mechanical devices and self-explanatory illustrations, without commentaries or geometric reasoning. The enlarged posthumous edition (1538) of the Underweysung merely repeats some ideas on perspective from Italian authors. Dürer thus seemed fully occupied by the powerful plan and elevation method, much more than by the method of central perspective.