In crystallography, the trigonal crystal system is one of the seven crystal systems, and the rhombohedral lattice system is one of the seven lattice systems. They are often confused with each other: crystals in the rhombohedral lattice system are always in the trigonal crystal system, but some crystals such as quartz are in the trigonal crystal system but not in the rhombohedral lattice system. The rhombohedral lattice system consists of the rhombohedral lattice, while the trigonal crystal system consists of the five point groups of the seven space groups with a rhombohedral lattice. There are 25 space groups whose point groups are one of the five in the trigonal crystal system, consisting of the seven space groups associated with the rhombohedral lattice system together with 18 of the 45 space groups associated with the hexagonal lattice system.
"Rhombohedral crystal system" is an ambiguous term that confuses the trigonal crystal system with the rhombohedral lattice system and may mean either of them (or even the hexagonal crystal family).
In the classification into 6 crystal families, the trigonal crystal system is combined with the hexagonal crystal system and grouped into a larger hexagonal family.^{[1]}
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A lattice system is described by three basis vectors. In the rhombohedral system, the crystal is described by vectors of equal length, none of which are orthogonal. The rhombohedral system can be thought of as the cubic system stretched diagonally along a body. a = b = c; . In some classification schemes, the rhombohedral lattice system is combined with the hexagonal lattice system and grouped into a larger hexagonal family.
There is only one rhombohedral Bravais lattice.
The seven space groups associated with the rhombohedral lattice system are listed below, with their international number and notation, followed by their point groups in name and international notation (HermannMauguin notation) and Schoenflies notation, and example crystals. (All these point groups are also associated to some space groups not in the rhombohedral lattice system.)
Number  Space group  Point group  International  Schoenflies  Examples 

146  R3  rhombohedral tetartohedral  3  C_{3}  carlinite 
148  R3  rhombohedral tetartohedral  3  S_{6}  dolomite 
155  R32  trapezohedral  32  D_{3}  abhurite 
160  R3m  rhombohedral hemimorphic  3m  C_{3v}  schorl 
161  R3c  rhombohedral hemimorphic  3m  C_{3v}  cerite 
166  R3m  rhombohedral holohedral  3m  D_{3d}  antimony 
167  R3c  rhombohedral holohedral  3m  D_{3d}  hematite, corundum 
The trigonal crystal system is the only crystal system whose point groups have more than one lattice system associated with their space groups: the hexagonal and rhombohedral lattices both appear.
The 5 point groups in this crystal system are listed below, followed by their representations in international notation (HermannMauguin notation) and Schoenflies notation, and example crystals.^{[2]}^{[1]}
Class name  international  Schoenflies  examples 
Hexagonal Scalenohedral  D_{3d}  calcite, corundum, hematite  
Ditrigonal Pyramidal  3m  C_{3v}  tourmaline, alunite 
Rhombohedral  S_{6}  dolomite, ilmenite  
Trapezohedral  32  D_{3}  quartz, cinnabar 
Pyramidal  3  C_{3}  jarosite 

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In crystallography, the rhombohedral (or trigonal) crystal system is one of the seven lattice point groups, named after the twodimensional rhombus. A crystal system is described by three basis vectors. In the rhombohedral system, the crystal is described by vectors of equal length, none of which are orthogonal. The rhombohedral system can be thought of as the cubic system stretched diagonally along a body. a = b = c; $\backslash alpha,\; \backslash beta,\; \backslash gamma\; \backslash neq\; 90^\backslash circ$.
In some classification schemes, the rhombohedral system is grouped into a larger hexagonal system.
There exists only one rhombohedral Bravais lattice.
The point groups which fall under this crystal system are listed below, followed by their representations in international notation (HermannMauguin notation) and Schoenflies notation, and example crystals.
name  international  Schoenflies  examples 
rhombohedral holohedral  $\backslash overline\{3\}m$  D_{3d}  calcite, corundum, hematite 
rhombohedral hemimorphic  $3m$  C_{3v}  tourmaline, alunite 
rhombohedral tetartohedral  $\backslash overline\{3\}$  S_{6}  dolomite, ilmenite 
trapezohedral  $32$  D_{3}  quartz, cinnabar 
rhombohedral tetartohedral  $3$  C_{3}  none verified 

