# Holoedry

The point group of a crystal is called holoedry (full form) if it matches the point group of its crystal lattice . Crystals of these crystal classes develop the full number of faces. This term is therefore mainly used in mineralogy to describe the crystal costume.

## Explanations

The structure of a crystal is described by the lattice and the base . In general, the base lowers the symmetry of the lattice so that the point group of the crystal is a true subgroup of the point group of the crystal lattice.

In those cases, however, in which the base does not lower the symmetry of the lattice, one speaks of a holoedry. The point group of the crystal is equal to the point group of the lattice. The crystal forms the full number of faces. In all other cases the form is called Meroedrie (partial form). Depending on the ratio of the order of the point group of the lattice to the order of the point group of the crystal, one can subdivide the merohedrons into hemiedrien (half order), tetartohedria (quarter order) and ogdoedrien (eighth order). There are seven holohedrals in three dimensions.

## Holohedrons in three-dimensional space

The seven holohedrons correspond to seven grid systems (also known as Bravais systems or axis systems). Each of these lattice systems has a corresponding axis cross, which can be described by conditions on the crystal axes.

Holoedry Grid system Lattice parameters
Surname abbreviation Basis vectors angle
1 triclinic / anorthic a a  ≠  b  ≠  c α  ≠  β  ≠  γ  ≠ 90 °
2 / m monoclinic m a  ≠  b  ≠  c γ  ≠ 90 °,  α  =  β  = 90 °; 1st setting
β  ≠ 90 °,  α  =  γ  = 90 °; 2nd setting
mmm orthorhombic O a  ≠  b  ≠  c α  =  β  =  γ  = 90 °
4 / mmm tetragonal t a  =  b  ≠  c α  =  β  =  γ  = 90 °
3 m rhombohedral r a  =  b  =  c α  =  β  =  γ  ≠ 90 °
6 / mmm hexagonal H a  =  b  ≠  c α  =  β  = 90 °,  γ  = 120 °
m 3 m cubic c a  =  b  =  c α  =  β  =  γ  = 90 °

Since the unit cell of the rhombohedral lattice system is not a conventional cell (the cell edges do not run parallel to the symmetry axes), this lattice system is also described as a hexagonal lattice system with rhombohedral centering. The lengths and angles are to be understood as restrictions. In the monoclinic crystal system, for example, the angle β (in the 2nd setting) can assume any value. It can therefore also be 90 ° by chance within the scope of the measurement accuracy.

## Classification of the crystal classes according to Holoedrien and Meroedrien

All point groups that are not holoedries can be assigned to a holoedry as meroedra. It should be noted that the trigonal point groups are both holohedrons and merohedrons of the rhombohedral and merohedrons of the hexagonal lattice system.

Grid system Holoedry Hemiedrie Tetartoedry Ogdoedrie
triclinic / anorthic 1 1 - -
monoclinic 2 / m m , 2 - -
orthorhombic mmm mm 2, 222 - -
tetragonal 4 / mmm 4 2 m , 4 mm , 422, 4 / m 4 , 4 -
rhombohedral 3 m 3 m , 32, 3 3 -
hexagonal 6 / mmm 6 m 2, 6 mm , 622 6 / m ; 3 m 6 , 6; 3 m , 32, 3 3
cubic m 3 m 4 3 m , 432, m 3 23 -

## Further subdivisions

The Meroedrien can be further subdivided depending on the type of the omitted symmetry elements:

• Hemimorphism: removal of a plane of symmetry perpendicular to the main axis; the corresponding crystal body is also referred to as hemieder (half surface).
• Paramorphy: Removal of a plane of symmetry parallel to the main axis
• Enantiomorphism: removal of all planes of symmetry and the center of inversion: only axes of rotation occur
• Hemiedry 2nd type: removal of the inversion center, existence of n with n even
• Tetartoedrie 2nd kind: removal of m or 2 in hemiedrie 2nd kind; the corresponding crystal body is also referred to as a tetartoeder (quarter surface ).

This results in the following detailed assignment:

Grid system Holoedry Hemimorphism Paramorphy Enantiomorphism Hemiedry 2nd Art Tetartoedry Tetartoedry 2nd kind
triclinic / anorthic 1 - - 1 - - -
monoclinic 2 / m - - 2 m - -
orthorhombic mmm mm 2 - 222 - - -
tetragonal 4 / mmm 4 mm 4 / m 422 4 2 m 4th 4th
rhombohedral 3 m 3 m 3 32 - 3 -
hexagonal 6 / mmm 6 mm 6 / m 622 6 m 2 6th 6th
cubic m 3 m - m 3 432 4 3 m 23 -