Rhombohedron

Rhombohedron as a special case of a parallelepiped

A rhombohedron is a polyhedron bounded by six diamonds . It is a parallelepiped with equally long edges.

volume

elongated and flattened rhombohedra

The volume of a rhombohedron with the side length and the angle ( ) is given by ${\displaystyle a}$ ${\displaystyle \theta }$ ${\displaystyle \theta \leq 90^{\circ }}$

${\displaystyle V=a^{3}(1-\cos \theta ){\sqrt {(1+2\cos \theta )}}}$.

The surface surrenders to

${\displaystyle A=6a^{2}\sin \theta }$.

Applied rhombohedra

Melencolia I , copper engraving (1514)

Art and nature

• In his partly mathematically inspired graphic Melencolia I, Albrecht Dürer depicts a specially trimmed rhombohedron , which through this modification would lie with all of its corner points on a spherical surface.
• Maurits Cornelis Escher also used various rhombohedra for his impossible figures in preliminary considerations and structural development.

Crystallography

The rhombohedron can be found in nature as a crystal form and at the atomic level in crystal structures . It is the general planar shape of the rhombohedral crystal class ( 3 ), a border shape of the trigonal-trapezoidal ( 32 ) and a special shape of the ditrigonal-scalenohedral crystal class ( 3 m ). It is also the basic shape of the rhombohedral Bravais lattice and the (no longer in use) rhombohedral crystal system . The rhombohedron as a crystal form only exists in the trigonal crystal system ; it has nothing to do with the (ortho-) rhombic crystal system - despite the similar name.

• 

  Calcite.

• 

  Calcite crystal

• 

  yellow calcite

• 

  Rhombohedral crystals

The color thrombohedron

According to Harald Küppers, the color rhombohedron fulfills the geometric solution for his theory of colors . Each point within the body corresponds to a color valence . This means that each of these (color) points is defined by its three vector potentials. The color rhombohedron can be converted into an RGB or a CYM color space through compression and distortion , naturally with different ratios (vectors) between the color values.

A rhombohedron, in which the short diagonal of the outer surfaces is as long as the rhombohedron sides, represents a symmetrical parallelepiped. There are two outer surfaces parallel to each other. Each diamond-shaped outer surface consists of two equilateral triangles. If you cut a rhombohedron along the short diagonal of the outer surfaces, you get three parts: two tetrahedra and one octahedron. Again, these three bodies are completely symmetrical. All of the outer surfaces of these three new geometric bodies are equilateral triangles.

• Küppers' color rhombohedron
• 

  Color bodies according to Harald Küppers

• 

  remarkable cuts

• 

  Axes, diagonals and edges

Web links

• Blue calcite rhombohedron
• Diamond body
• Eric W. Weisstein : Rhombohedron . In: MathWorld (English).

Individual evidence

1. ↑ from Augsburg Naturmuseum, found Goslerwand, East Tyrol
2. ↑ Museo civico di storia naturale a Milano, location Kazakhstan
3. ↑ Location China: rhombeohedral yellow transparent crystal: Calcite jaune
4. ^ Illustration from Encyclopædia Britannica (1911), article CALCITE.
5. ↑ Küppers' Theory of Colors ( Memento of the original from January 26, 2012 in the Internet Archive ) Info: The archive link was inserted automatically and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. 
6. ↑ W: white, S: black, N: neutral gray, B → M → R → Y → G → C: six bright colors (blue, magenta, red, yellow, green, cyan)