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Dürer's solid, also known as the truncated triangular trapezohedron, 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.
It is implemented in the Wolfram Language as PolyhedronData["DuererSolid"].
The skeleton of Dürer's solid is the Dürer graph (i.e., 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).
