DMGT

Discussiones Mathematicae Graph Theory 22(1) (2002) 101-109
DOI: https://doi.org/10.7151/dmgt.1160

DÜRER POLYHEDRA: THE DARK SIDE OF MELANCHOLIA

Patrick W. Fowler

School of Chemistry, University of Exeter
Stocker Rd, Exeter EX4 4QD, UK

Peter E. John

Institute of Mathematics
Technical University of Ilmenau
Ilmenau, Germany

Abstract

Dürer's engraving Melencolia I famously includes a perspective view of a solid polyhedral block of which the visible portion is an 8-circuit bounding a pentagon-triple+triangle patch. The polyhedron is usually taken to be a cube truncated on antipodal corners, but an infinity of others are compatible with the visible patch. Construction of all cubic polyhedra compatible with the visible portion (i.e., Dürer Polyhedra) is discussed, explicit graphs and symmetries are listed for small cases ( ≤ 18 vertices) and total counts are given for 10 ≤ vertices ≤ 26.

Keywords: graph theory, geometry, chemistry.

2000 Mathematics Subject Classification: 05C30, 05C90, 52B10, 92E10.

References

[1]	E.g. 1471 Albrecht Dürer 1971. Ausstellung des Germanischen Nationalmuseums Nürnberg 21. Mai bis 1. August 1971 (Prestel-Verlag, München, 1971). Item 270: Die Melancholie.
[2]	E. Panofsky, The Life and Art of Albrecht Dürer (Princeton University Press, Princeton NJ, 1955) 4th Edition, 156-171.
[3]	H. Böhme, Albrecht Dürer Melencolia I. Im Labyrinth der Deutung (Fischer, Kunststuck, 1991). ISBN 3 59623958-3.
[4]	E. Panofsky and F. Saxl, Dürers Melencolia I. Eine quellen- und typengeschichtliche Untersuchung (Studien der Bibliotek Warburg, Band 2), Teubner, Leipzig, 1923.
[5]	The work by Dürer cited in Ref. [3] as containing such a drawing is: Unterweysung der Messung mit dem Zirkel un Richtscheyt in Linien Ebnen unnd Gantzen Corporen (Instruction in the Measurement with Compasses and Straight-Edge of Lines, Planes and Solid Bodies), Nuremberg, 1525, which is available as The Painter's Manual, A. Dürer, trans. W.S. Strauss, Abaris Books, New York, 1978, ISBN 091 387 0528.
[6]	See Ref. [4], Tafel IV, Abb. 7.
[7]	P.J. Federico, The Melancholy Octahedron, Mathematics Magazine, (1972) 20-36.
[8]	M. Deza and V. Grishukhin, l₁-embeddable Polyhedra, in: Algebras and Combinatorics. An International Congress, ICAC '97, Hong Kong. ed., Kar-Ping Shum (Springer-Verlag, Singapore, 1999) 189-210.
[9]	C.H. MacGillavry, The Polyhedron in A. Dürer's Melancolia I: An Over 450 Years Old Puzzle Solved?, Netherlands Akad. Wetensch. Proc. (B) 84 (3) (1981) 287-294.
[10]	T. Lynch, The Geometric Body in A. Dürer's Engraving Melancolia I, Journal of the Warburg and Courtauld Institute 45 (1982) 226-232, doi: 10.2307/750979.
[11]	J. Sharp, Dürer's Melancholy Octahedron, Mathematics in School, 18-20, (Sept. 1984).
[12]	B. Grünbaum, Convex Polytopes (Wiley, London and New York, 1967).
[13]	E. Steinitz, Polyeder und Raumteilungen, Enzykl. Math. Wiss., 3 (Geometrie), Part 3 AB 12, (1922) 1-139.
[14]	E. Steinitz and H. Rademacher, Vorlesungen über die Theorie der Polyeder (Berlin, 1934).
[15]	R.C. Read and R.J. Wilson, An Atlas of Graphs (Oxford University Press, Oxford, 1999).
[16]	The nauty program written by B.D. McKay, including the graph generators makeg and makebg is available from http://www.anu.edu.au/people/bdm/index.html
[17]	D.E. Manolopoulos and P.W. Fowler, Molecular graphs, point groups and fullerenes, J. Chem. Phys. 96 (1992) 7603-7614, doi: 10.1063/1.462413.
[18]	A.J. Stone and D.J. Wales, Theoretical studies of icosahedral C₆₀ and some related species, Chem. Phys. Lett. 128 (1986) 501-503, doi: 10.1016/0009-2614(86)80661-3.
[19]	G. Brinkmann and B. McKay, Version 1.0 of the programme plantri.c and its documentation are obtainable at http://cs.anu.edu.au/people/bdm

Received 22 August 2000
Revised 14 May 2001