Dürer's solid is the 8-faced solid depicted in an engraving entitled Melancholia I by Albrecht Dürer (The British Museum, Burton 1989, Gellert et al. 1989), the same engraving in which Dürer's magic square appears, which depicts a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Although Dürer does not specify how his solid is constructed, Schreiber (1999) has noted that it appears to consist of a distorted cube which is first stretched to give rhombic faces with angles of , and then truncated on top and bottom to yield bounding triangular faces whose vertices lie on the circumsphere of the azimuthal cube vertices.

The skeleton of Dürer's solid is the generalized Petersen graph .

Starting with a unit cube oriented parallel to the axes of the coordinate system, rotate it by Euler angles and to align a threefold symmetry axis along the -axis. The stretch factor needed to produce rhombic angles of is then

(1)

The azimuthal points are a distance away from the origin, and in order for the vertices of the triangles obtained by truncation to lie at this same distance, the truncation must be done a distance along the edge from one of the azimuthal points, which corresponds to a height

(2)

The resulting solid has six pentagonal faces and two equilateral triangular faces, and the lengths of the sides are in the ratio

(3)

Examination of this solid shows it to be identical to the dimensions of the solid reconstructed from its perspective picture (Schröder 1980, p. 70; Schreiber 1999).

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Schreiber, P. "A New Hypothesis on Dürer's Enigmatic Polyhedron in His Copper Engraving 'Melancholia I.' " Historia Math. 26, 369-377, 1999.

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Sharp, J. "Durer's Melancholy Octahedron." Math. in School, 18-20, Sept. 1994.

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Weisstein, Eric W. "Dürer's Solid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DuerersSolid.html