Sources page biographical material

6.AT.8. DÜRER'S OCTAHEDRON
New section -- I know of other articles claiming to 'solve' the problem.
Albrecht Dürer. Melencolia I. 1514. Two impressions are in the British Museum. In the back left is an octahedron whose exact shape is the subject of this section. It looks like a cube truncated at two opposite corners, but the angles do not quite look like 90^o.

Albrecht Dürer. Dresden Sketchbook. Facsimile as The Human Figure, the complete Dresden Sketchbook; Dover, NY, 1972. ??NYS -- cited by Sharp. This has a sketch of the solid with hidden lines indicated, so the combinatorial shape is definitely known and is a hexahedron of six equal faces, truncated at two opposite corners.

E. Schröder. Dürer Kunst und Geometrie. Birkhäuser, Basel, 1980. ??NYS -- cited by Sharp and MacGillavry.

Caroline H. MacGillavry. The polyhedron in A. Dürer's Melencolia I An over 450 years old puzzle solved? Koninklijke Nederlandse Akademie van Wetenschappen Proc. B 84:3 (28 Sep 1981) 287-294. The rhombohedral angle, i.e. the angle between edges at the truncated top and bottom vertices of the rhombohedron, was estimated as 72^o by Grodzinski. She determines it is 79^o ± 1^o. She then built and photographed such a polyhedron and then computed its projection, both of which seem identical to Dürer's picture. Crystallographers believe Dürer was drawing an actual crystal, with a form of calcite having rhombohedral angle of 76^o being the closest known shape, though it is not known to have been studied in Dürer's time, so others have suggested fluorite, though fluorite has two standard forms, neither of this form, but Dürer's 'hybrid' artistic version could have been derived from them.

Terence Lynch. The geometric body in Dürer's engraving Melencolia I. J. Warburg and Courtauld Institutes 45 (1982) 226-232 & plate a on p. 37?. Lots of references to earlier work. Notes that perspective was not sufficiently advanced for Dürer to construct a general drawing of such an object. After many trials, he observes that a parallel projection of the solid fits onto a 4 x 4 grid -- like the magic square in the picture -- and that symmetry then permits the construction with straight edge and compass (which are both shown in the picture). This shows that the original faces are rhombuses whose diameters are in the ratio 2 : 3. And the dihedral angle between the triangular faces and the cut off rhombuses is 30^o Further, the actual drawing can then be made by one of the simplified perspective techniques known to Dürer. However, Dürer has taken a little bit off the top and bottom of the figure and this distortion has misled many previous workers.

John Sharp. Dürer's melancholy octahedron. MiS (Sep 1994) 18-20. Asserts that the shape was first determined by Schröder in 1980 and verified by Lynch.