Truncated triangular trapezohedron
| Truncated triangular trapezohedron | |
|---|---|
| |
| Type | Truncated trapezohedron |
| Faces | 6 pentagons, 2 triangles |
| Edges | 18 |
| Vertices | 12 |
| Symmetry group | D3d, [2+,6], (2*3) |
| Dual polyhedron | Gyroelongated triangular dipyramid |
| Properties | convex |
In geometry, the truncated triangular trapezohedron is the first in an infinite series of truncated trapezohedron polyhedra. It has 6 pentagon and 2 triangle faces.
This polyhedron can be constructed by taking a rhombohedron, trigonal trapezohedron or a cube (with a specified diagonal axis defined) and truncating the polar axis vertices.
Dürer's solid
This polyhedron is sometimes called Dürer's solid, from its appearance in Albrecht Dürer's 1514 drawing Melencolia I. The graph formed by its edges and vertices is called the Dürer graph.
The exact geometry of the solid depicted by Dürer is a subject of some academic debate.[1] According to Lynch (1982), the hypothesis that the shape is a misdrawn truncated cube was promoted by Strauss (1972); however most sources agree that it is the truncation of a rhombohedron. Richter (1957) claims that the rhombi of the rhombohedron from which this shape is formed have 5:6 as the ratio between their short and long diagonals, from which the acute angles of the rhombi would be approximately 80°. Schröder (1980) and Lynch (1982) instead conclude that the ratio is √3:2 and that the angle is approximately 82°. MacGillavry (1981) measures features of the drawing and finds that the angle is approximately 79°. Schreiber (1999) argues based on the writings of Dürer that all vertices of Dürer's solid lie on a common sphere, and further claims that the rhombus angles are 72°. Weitzel (2004) analyzes a 1510 sketch by Dürer of the same solid, from which he confirms Schrieber's hypothesis that the shape has a circumsphere but with rhombus angles of approximately 79.5°.
See also
Notes
- ^ See Weitzel (2004), from which most of the following history is drawn.
References
- Lynch, Terence (1982), "The geometric body in Dürer's engraving Melencolia I", Journal of the Warburg and Courtauld Institutes, 45, The Warburg Institute: 226–232, doi:10.2307/750979, JSTOR 750979.
- MacGillavry, C. (1981), "The polyhedron in A. Dürers Melencolia I", Nederl. Akad. Wetensch. Proc. Ser. B, 84: 287–294. As cited by Weitzel (2004).
- Richter, D. H. (1957), "Perspektive und Proportionen in Albrecht Dürers "Melancholie"", Z. Vermessungswesen, 82: 284–288 and 350–357. As cited by Weitzel (2004).
- Schreiber, Peter (1999), "A new hypothesis on Dürer's enigmatic polyhedron in his copper engraving "Melencolia I"", Historia Mathematica, 26: 369–377, doi:10.1006/hmat.1999.2245.
- Schröder, E. (1980), Dürer, Kunst und Geometrie, Dürers künstlerisches Schaffen aus der Sicht seiner “Underweysung”, Basel
{{citation}}: CS1 maint: location missing publisher (link). As cited by Weitzel (2004). - Strauss, Walter L. (1972), The Complete Engravings of Dürer, New York, p. 168, ISBN 0-486-22851-7
{{citation}}: CS1 maint: location missing publisher (link). As cited by Lynch (1982). - Weber, P. (1900), Beiträge zu Dürers Weltanschauung—Eine Studie über die drei Stiche Ritter, Tod und Teufel, Melancholie und Hieronymus im Gehäus, Strassburg
{{citation}}: CS1 maint: location missing publisher (link). As cited by Weitzel (2004). - Weitzel, Hans (2004), "A further hypothesis on the polyhedron of A. Dürer's engraving Melencolia I", Historia Mathematica, 31 (1): 11–14, doi:10.1016/S0315-0860(03)00029-6.
External links
- Weisstein, Eric W. "Dürer's Solid". MathWorld.
- How to build Dürer's Polyhedron - by DUPLICON (in German)
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