In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the truncated triangular trapezohedron is the first in an infinite series of
truncated trapezohedra
In geometry, an truncated trapezohedron is a polyhedron formed by a trapezohedron with Pyramid (geometry), pyramids Truncation (geometry), truncated from its two polar axis Vertex (geometry), vertices.
The vertices exist as 4 in four paralle ...
. It has 6
pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
and 2
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
faces.
Geometry
This
polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
can be constructed by
truncating two opposite
vertices of a
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
, of a
trigonal trapezohedron
In geometry, a trigonal trapezohedron is a polyhedron with six congruent quadrilateral faces, which may be scalene or rhomboid. The variety with rhombus-shaped faces faces is a rhombohedron.
An alternative name for the same shape is the ''trig ...
(a convex polyhedron with six congruent
rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
sides, formed by stretching or shrinking a cube along one of its long diagonals), or of a
rhombohedron
In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a special case of a parallelepiped in which all six faces are congruent rhombi. It can be used to define the rhombohedral lattice system, a honeycomb w ...
or
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square.
Three equiva ...
(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
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...
.
Dürer's solid
This polyhedron is sometimes called Dürer's solid, from its appearance in
Albrecht Dürer
Albrecht Dürer ( , ;; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer or Duerer, was a German painter, Old master prin ...
's 1514 engraving ''
Melencolia I
''Melencolia I'' is a large 1514 engraving by the German Renaissance artist Albrecht Dürer. Its central subject is an enigmatic and gloomy winged female figure thought to be a personification of melancholia – melancholy. Holding her head in ...
''.
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. According to , the hypothesis that the shape is a misdrawn
truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangle (geometry), triangular), 36 edges, and 24 vertices.
If the truncated cube has unit edge length, its dual triak ...
was promoted by ; however most sources agree that it is the truncation of a
rhombohedron
In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a special case of a parallelepiped in which all six faces are congruent rhombi. It can be used to define the rhombohedral lattice system, a honeycomb w ...
. Despite this agreement, the exact geometry of this rhombohedron is the subject of several contradictory theories:
* claims that the
rhombi
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
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°.
* and instead conclude that the ratio is
and that the angle is approximately 82°.
* measures features of the drawing and finds that the angle is approximately 79°. She and a later author,
Wolf von Engelhardt
Wolf Jürgen Baron von Engelhardt (9 February 1910, Tartu – 4 December 2008, Tübingen) was a German geologist and mineralogist.
Baron von Engelhardt was a descendant of a Baltic German noble family Engelhardt.
Biography
In the years 192 ...
argue that this choice of angle comes from its physical occurrence in
calcite
Calcite is a Carbonate minerals, carbonate mineral and the most stable Polymorphism (materials science), polymorph of calcium carbonate (CaCO3). It is a very common mineral, particularly as a component of limestone. Calcite defines hardness 3 on ...
crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
s.
[, .]
* 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°. lists several other scholars who also favor the 72° theory, beginning with Paul Grodzinski in 1955. He argues that this theory is motivated less by analysis of the actual drawing, and more by aesthetic principles relating to
regular pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
s and the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
.
* analyzes a 1510 sketch by Dürer of the same solid, from which he confirms Schreiber's hypothesis that the shape has a
circumsphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...
but with rhombus angles of approximately 79.5°.
* argues that the shape is intended to depict a solution to the famous geometric problem of
doubling the cube
Doubling the cube, also known as the Delian problem, is an ancient geometry, geometric problem. Given the Edge (geometry), edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first ...
, which Dürer also wrote about in 1525. He therefore concludes that (before the corners are cut off) the shape is a cube stretched along its long diagonal. More specifically, he argues that Dürer drew an actual cube, with the long diagonal parallel to the
perspective plane, and then enlarged his drawing by some factor in the direction of the long diagonal; the result would be the same as if he had drawn the elongated solid. The enlargement factor that is relevant for doubling the cube is 2
1/3 ≈ 1.253, but Hideko derives a different enlargement factor that fits the drawing better, 1.277, in a more complicated way.
* classify the proposed solutions to this problem by two parameters: the acute angle and the level of cutting, called the cross ratio. Their estimate of the cross ratio is close to MacGillavry's, and has a numerical value close to the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
. Based on this, they posit that the acute angle is
and that the cross ratio is exactly
.
See also
*
Chamfered tetrahedron, another shape formed by truncating a subset of the vertices of a cube
Notes
References
*.
*.
*.
*. As cited by .
*. As cited by .
*.
*. As cited by .
*. As cited by .
*. As cited by .
*.
*.
External links
* {{MathWorld, urlname=DuerersSolid, title=Dürer's Solid, mode=cs2
How to build Dürer's Polyhedron - by DUPLICON (in German)Open-source 3D models of Dürer's Solid
Polyhedra