The Trigonal System: Quartz, Calcite, Tourmaline, Ruby and Sapphire

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A system of its own, often confused with the hexagonal

The trigonal system is that of crystals whose mandatory symmetry is a three-fold axis: turn them by a third of a turn around that axis and they look identical. Many famous minerals are of this kind, including quartz, which makes this one of the most important systems for anyone interested in stones.

Its crystals resemble those of the hexagonal system, hence the confusion. The difference is a matter of symmetry, not appearance: a quartz prism seems to have six equal faces, but details (large and small faces at the tip, the striations) show that it has only a three-fold axis. See the hexagonal system.

Two ways of describing the unit cell

For the trigonal system, crystallographers use two equivalent “frames”:

Description	Axes	Angles	Numerical example
Hexagonal cell	a = b ≠ c	α = β = 90°, γ = 120°	Calcite: a = 4.989 Å, c = 17.062 Å
Rhombohedral cell	a = b = c	α = β = γ ≠ 90°	Calcite: a ≈ 6.37 Å, angle ≈ 46.1°
Rhombohedral cell	a = b = c	α = β = γ ≠ 90°	Corundum: a ≈ 5.13 Å, angle ≈ 55.3°

We converted the published hexagonal values into rhombohedral values by calculation: they describe the same crystal, simply viewed in a different frame. The trigonal system has 5 symmetry classes and a single Bravais lattice, the rhombohedral lattice. Watch out for a common confusion: “trigonal” is a crystal system, while “rhombohedral” is a type of lattice, and these are not the same thing. A trigonal crystal can have a hexagonal lattice (as is the case for quartz).

Quartz, examined closely

Quartz is the best-known of the stone minerals. Its formula is SiO2, its Mohs hardness 7 and its density about 2.65. It is trigonal at ordinary temperature (see the hexagonal page for its high-temperature form). Measurements on a well-formed crystal show this:

• The prism faces meet at 60° or 120° to one another, giving the look of a six-faced prism.
• The tip is made of six faces, three large (positive rhombohedron) and three small (negative rhombohedron), which alternate. When they are of nearly equal size, the crystal is wrongly taken for hexagonal.
• The angle between a prism face and a tip face is 38°13', and that between the vertical axis and a tip face is 51°47'. These two angles are recovered by calculation from the quartz cell parameters (a = 4.913 Å, c = 5.405 Å), to within a few arcminutes.
• The prism faces carry horizontal striations, useful for telling natural quartz apart from a synthetic crystal or moulded glass.

Quartz is also piezoelectric: when compressed, it produces an electric voltage. This was discovered by Pierre and Jacques Curie in 1880, and it is why a small crystal vibrates at a stable frequency in a quartz watch. This tells us nothing about any effect on the body. There are “right-handed” and “left-handed” crystals, mirror images of one another. See our pages rock crystal and amethyst.

Calcite, its rhombohedra and its double image

Calcite (CaCO3, hardness 3, density 2.71) breaks into rhombohedral fragments, a “distorted cube” whose angles are about 74°55' and 105°05'. We recovered the value by calculation from its cell parameters. It is also the first stone in which double refraction was observed: the Danish scientist Erasmus Bartholin described it in 1669 in Iceland spar, a clear calcite.

A harmless experiment. If you have a rhombohedron of Iceland spar (available from mineral shops), place it on a sheet with a dot or a letter in large print: you will see two images. Turn the crystal: one of the two images stays still while the other turns around it. Quartz, by contrast, is too weakly birefringent to produce such an effect to the naked eye.

Other trigonal minerals to know

Mineral	Simplified formula	Mohs hardness	Density	Reference point
Corundum (ruby, sapphire)	Al2O3	9	3.9 to 4.1	Barrel-shaped or tabular crystals, dichroic
Hematite	Fe2O3	5 to 6.5	about 5.2	Red-brown streak; iron rose
Tourmaline (schorl, elbaite)	complex boron silicate	7 to 7.5	3.0 to 3.2	Prism with a rounded triangular cross-section
Dolomite	CaMg(CO3)2	3.5 to 4	about 2.85	Curved rhombohedra
Rhodochrosite	MnCO3	3.5 to 4.5	3.5 to 3.7	Pink to red
Smithsonite	ZnCO3	4 to 4.5	about 4.4	Often in rounded masses
Phenakite	Be2SiO4	7.5 to 8	about 2.95	Rare, colourless to yellow
Cinnabar	HgS	2 to 2.5	about 8.1	Bright red; contains mercury

Tourmaline is also pyroelectric: when heated, it attracts small light fragments. Dutch merchants had observed this in the early 18th century with stones from Ceylon, hence the name “ash-puller”. See black tourmaline, rhodochrosite, hematite and ruby.

Safety. Cinnabar is a mercury sulphide. Do not put it in your mouth, do not sand it, do not heat it, and wash your hands after holding it. Carbonates such as calcite and rhodochrosite are sensitive to acids, including vinegar: do not clean them with an acidic product.

Why this classification matters

Knowing that a mineral is trigonal helps you interpret what you see. Calcite and dolomite, for example, have the same type of rhombohedral cleavage; calcite is recognised by its double image and tourmaline by its rounded triangular cross-section. Rubies and sapphires show dichroism (two colours depending on direction), which distinguishes them from glass imitations, which show no pleochroism at all. See colour and pleochroism.

Frequently asked questions

Also read

The meanings mentioned belong to tradition and symbolism. They are neither promises of results nor medical advice. Back to Mineralogy for beginners.