The rhombohedral system, more often called the trigonal system in modern crystallographic usage, is one of the seven crystal systems by which all crystalline solids are classified. It is defined by a single threefold rotational axis of symmetry, with the conventional unit cell described either as a rhombohedron — a parallelepiped whose six faces are identical rhombi — or as a hexagonal cell with constraints that make the trigonal sub-symmetry explicit. The system is host to several gemmologically important minerals: corundum (and therefore ruby and sapphire), tourmaline, quartz, calcite, dolomite, and others.

Symmetry and unit cell

A trigonal crystal has one and only one threefold axis of rotational symmetry, distinguishing it from the cubic system (which has multiple threefold axes), the tetragonal system (which has a fourfold axis), and the hexagonal system (which has a sixfold axis). Some references treat the trigonal system as a subdivision of the hexagonal system because the conventional unit cell is hexagonal, but the symmetry is genuinely lower and the distinction matters for the optical and physical behaviour of the crystals.

Trigonal minerals are uniaxial — they have one optic axis along the threefold-rotation axis — and most are anisotropic, showing two refractive indices that vary depending on the direction of light propagation through the crystal. This is the basis for the birefringence visible in calcite, the pleochroism strong in tourmaline and present in corundum, and the optical character that gemmologists use to identify these species.

Habit and form

Rhombohedral or trigonal crystals show characteristic habits that the cutter and the gemmologist learn to recognise. Corundum forms barrel-shaped, tabular, or doubly terminated hexagonal prisms with rhombohedral terminations; tourmaline forms long prismatic crystals with characteristic triangular cross-section and striated faces; quartz forms hexagonal prisms terminated by rhombohedra in either left-handed or right-handed configurations. Calcite shows the widest variety of habits of any common mineral, including scalenohedra, rhombohedra, and a host of intermediate forms.

These habits arise from the underlying threefold symmetry and from the specific ways the crystal grows in its host environment. They are useful identifiers in rough-stone work and in mineral specimen identification.

Optical and physical implications

Trigonal minerals show direction-dependent properties because of their lower symmetry. Birefringence in calcite is high enough to produce visible doubling of features seen through a cleavage rhomb. Pleochroism in tourmaline and in some corundum can be strong enough that orientation of the cut stone meaningfully affects its face-up colour. Cleavage and parting follow the system's planes; calcite has perfect rhombohedral cleavage in three directions, and corundum has a parting (not a true cleavage) that the cutter must respect.

Refractive index, hardness, and other directional properties are typically reported in trigonal minerals as ranges or as ordinary-and-extraordinary pairs, reflecting the variation along and across the optic axis.

In gemmology

Knowledge of the system underlies routine gemmological identification. The optic character (uniaxial), the birefringence value, the pleochroism pattern, and the habit are all consequences of the trigonal symmetry. A gemmologist using a polariscope, a refractometer, and a dichroscope is in effect probing the symmetry of the host crystal, and recognising that a stone is uniaxial and trigonal narrows the candidate species considerably. The standard identification protocol — refractive index, specific gravity, optic sign, pleochroism, and absorption spectrum — relies on the regular relationship between symmetry, structure, and optical behaviour that the trigonal system imposes.

The polariscope test is particularly diagnostic. A trigonal mineral oriented with its optic axis perpendicular to the polariscope's optical path will appear isotropic (no extinction on rotation), while the same mineral oriented with the optic axis parallel will show full extinction every ninety degrees. This single test, combined with the dichroscope reading, can rule in or rule out trigonal candidate species with high confidence in routine work.

Notable trigonal gem species

Corundum (ruby and sapphire) is the most commercially important trigonal species, with refractive indices of approximately 1.762 and 1.770 (uniaxial negative) and birefringence of 0.008. Quartz, including amethyst, citrine, and the rock-crystal variety, is uniaxial positive with refractive indices of approximately 1.544 and 1.553 and birefringence of 0.009. Tourmaline (a complex group with several end-member species) shows refractive indices around 1.62 to 1.65 and stronger birefringence, with strong pleochroism characteristic of the species. Calcite, used decoratively in optical demonstrations and rarely faceted because of its low hardness, has the system's strongest birefringence at 0.172.

Other trigonal species relevant to the trade include dolomite, hematite, smithsonite, dioptase, and proustite. Each shows the underlying threefold symmetry in its physical and optical properties, with details that vary by species but follow the system's general patterns.

Symmetry classes within the trigonal system

The trigonal system contains five point-group symmetry classes, distinguished by the additional symmetry elements that may be present in addition to the defining threefold axis. The classes range from the lowest-symmetry trigonal-pyramidal (no additional symmetry beyond the threefold axis) to the highest-symmetry hexagonal-scalenohedral (with additional reflection planes and twofold axes). Different gem species occupy different point groups within the system: corundum is in the hexagonal-scalenohedral class, quartz is in the trigonal-trapezohedral class, and tourmaline is in the ditrigonal-pyramidal class.

The point-group classification matters for some advanced gemmological tests — the orientation of the optic axis with respect to other symmetry elements affects the appearance of stones under conoscopic illumination, and the presence or absence of mirror planes affects optical activity in quartz. For routine identification, the broader trigonal-system identification is usually sufficient.

Crystallographic notation and the rhombohedral cell

Rhombohedral crystals are described in two equivalent notational systems. The hexagonal-axis system uses four axes — three at 120 degrees in a horizontal plane and one perpendicular — and is the more commonly used in practical mineralogy. The rhombohedral-axis system uses three equal axes at equal angles other than ninety degrees and is closer to the intrinsic symmetry of the unit cell. Conversion between the two notations is straightforward, and reference works typically give crystallographic data in both forms. Practical gemmology rarely needs the conversion, but mineralogists working with structure determination and X-ray diffraction use both systems routinely.

Practical implications for the cutter

For lapidaries, the trigonal symmetry of the host crystal has direct implications for cutting decisions. Optic-axis orientation determines the face-up colour of pleochroic stones — a tourmaline cut with its long axis perpendicular to the table will show one colour face-up and another from the side; a tourmaline cut with its long axis parallel to the table will show a different combination. The cutter's choice of orientation balances pleochroism, weight retention, parting tendencies, and the geometry of the rough. Trained cutters of trigonal-system gem species develop intuitions about these tradeoffs that take years of practice to internalise.

Birefringence in trigonal stones can be visible in the finished gem in some cases — facet-edge doubling in calcite is the most extreme example, but moderate birefringence in zircon (also trigonal in some references, tetragonal in others depending on classification scheme) and in tourmaline can be visible under loupe magnification. The cutter typically cannot eliminate this, but can manage how it presents to the viewer.

Further reading

• GIA Gems & Gemology — articles on crystal systems and gem identification
• International Gem Society — crystal system references
• ICA — gem crystallography references