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Dürer's Solid


Melencolia I
DuerersSolidDuerersSolidFace

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 72 degrees, 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"].

DuerersSolidSkeleton

The skeleton of Dürer's solid is the Dürer graph (i.e., generalized Petersen graph P_(6,2)).

Starting with a unit cube oriented parallel to the axes of the coordinate system, rotate it by Euler angles psi=pi/4 and theta=sec^(-1)sqrt(3) to align a threefold symmetry axis along the z-axis. The stretch factor needed to produce rhombic angles of 72 degrees is then

 s=sqrt(1+3/(sqrt(5))).
(1)

The azimuthal points are a distance d=s/2 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 (3-sqrt(5))/2 along the edge from one of the azimuthal points, which corresponds to a height

 h=sqrt((23)/(sqrt(5))-1/4).
(2)

The resulting solid has six 126-108-72-108-126 degrees pentagonal faces and two equilateral triangular faces, and the lengths of the sides are in the ratio

 1:1/2(3+sqrt(5)):sqrt(1/2(5+sqrt(5))).
(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).

DuererCanonicalSolid

The canonical polyhedron of Dürer's solid is a cube truncated at two opposite corners leaving two equilateral trianglular faces whose edges are sqrt(2)-1=0.4142... times the length of the full cube edges (E. Weisstein, Apr. 19, 2022). This solid is implemented in the Wolfram Language as PolyhedronData["DuererCanonicalSolid"].


See also

Dürer Graph, Dürer's Magic Square

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References

Burton, D. M. Cover illustration of Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, 1989.Federico, P. J. "The Melancholy Octahedron." Math. Mag., pp. 30-36, 1972.Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). Appendix, Plate 19. VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, 1989.Hart, G. "Durer's Polyhedra." http://www.georgehart.com/virtual-polyhedra/durer.html.Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, pp. 140-141, 2002.Lynch, T. "The Geometric Body in Durer's Engraving Melencolia I." J. Warburg and Courtauld Inst., 226-232, 1982.MacGillavry, C. H. "The Polyhedron in A. Durer's 'Melencolia I': An Over 450 Years Old Puzzle Solved ?" Nederland Akad. Wetensch. Proc. 1981.Panofsky, E. The Life and Art of Albrecht Durer. Princeton, NJ: Princeton University Press, 1955.Schreiber, P. "A New Hypothesis on Dürer's Enigmatic Polyhedron in His Copper Engraving 'Melencolia I.' " Historia Math. 26, 369-377, 1999.Schröder, E. Dürer--Kunst und Geometrie. Berlin: Akademie-Verlag, 1980.Sharp, J. "Durer's Melancholy Octahedron." Math. in School, 18-20, Sept. 1994.Walton, K. D. "Albrecht Durer's Renaissance Connections Between Mathematics and Art." Math. Teacher, 278-282, 1994.

Cite this as:

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

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