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Dürer's solid, also known as the truncated triangular trapezohedron, is the 8-faced solid depicted in an engraving entitled Melencolia 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 along the 
-axis needed to produce rhombic angles
of 
is then
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(1)
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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
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(2)
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The resulting solid has six 
pentagonal faces and two equilateral
triangular faces, and the lengths of the sides are in the ratio
 |
(3)
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This 
model follows Schreiber (1999). It differs from the reconstruction by Schröder
(1980, p. 72), whose faces are rhombi
with diagonal ratio 
and hence an acute angle
of 
.
The canonical polyhedron of Dürer's solid is a cube truncated at two opposite corners leaving two equilateral trianglular
faces whose edges are 
times the length of the full cube edges (E. Weisstein, Apr. 19, 2022).
This solid is implemented in the Wolfram
Language as PolyhedronData["DuererCanonicalSolid"].
