Truncated triangular trapezohedron

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Truncated triangular trapezohedron
Dürer's solid
Triangular truncated trapezohedron.png
Type Truncated trapezohedron
Faces6 pentagons,
2 triangles
Edges18
Vertices12
Symmetry group D3d, [2+,6], (2*3)
Dual polyhedron Gyroelongated triangular bipyramid
Propertiesconvex

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.

Contents

Geometry

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.

Dürer's solid

Melencolia I. Durer Melancholia I.jpg
Melencolia I .

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:

See also

Notes

  1. See Weitzel (2004) and Ziegler (2014), from which much of the following history is drawn.

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Truncated icosahedron Archimedean solid

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Regular dodecahedron

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In the mathematical field of graph theory, the Dürer graph is an undirected graph with 12 vertices and 18 edges. It is named after Albrecht Dürer, whose 1514 engraving Melencolia I includes a depiction of Dürer's solid, a convex polyhedron having the Dürer graph as its skeleton. Dürer's solid is one of only four well-covered simple convex polyhedra.

<i>Melencolia I</i> 1514 engraving by Albrecht Dürer

Melencolia I is a large 1514 engraving by the German Renaissance artist Albrecht Dürer. The print's central subject is an enigmatic and gloomy winged female figure thought to be a personification of melancholia – melancholy. Holding her head in her hand, she stares past the busy scene in front of her. The area is strewn with symbols and tools associated with craft and carpentry, including an hourglass, weighing scales, a hand plane, a claw hammer, and a saw. Other objects relate to alchemy, geometry or numerology. Behind the figure is a structure with an embedded magic square, and a ladder leading beyond the frame. The sky contains a rainbow, a comet or planet, and a bat-like creature bearing the text that has become the print's title.

Chamfer (geometry)

In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.

Icosahedron Polyhedron with 20 faces

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