In geometry, the truncated triangular trapezohedron is the first in an infinite series of truncated trapezohedra. It has 6 pentagon and 2 triangle faces.
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 bipyramid |
Properties | convex |
This polyhedron can be constructed by truncating two opposite vertices of a cube, of a trigonal trapezohedron (a convex polyhedron with six congruent rhombus sides, formed by stretching or shrinking a cube along one of its long diagonals), or of a rhombohedron or parallelepiped (less symmetric polyhedra that still have the same combinatorial structure as a cube). In the case of a cube, or of a trigonal trapezohedron where the two truncated vertices are the ones on the stretching axes, the resulting shape has three-fold rotational symmetry.
This polyhedron is sometimes called Dürer's solid, from its appearance in Albrecht Dürer's 1514 engraving Melencolia I. The graph formed by its edges and vertices is called the Dürer graph.
The shape 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. Despite this agreement, the exact geometry of this rhombohedron is the subject of several contradictory theories:
{{citation}}
: CS1 maint: location missing publisher (link). As cited by Weitzel (2004).{{citation}}
: CS1 maint: location missing publisher (link). As cited by Lynch (1982).{{citation}}
: CS1 maint: location missing publisher (link). As cited by Weitzel (2004).