A rhombohedron is a six-faced solid whose faces are identical rhombi — parallelograms with all sides equal but with angles other than ninety degrees. It is the characteristic crystal form of the trigonal system, and is most familiar in mineralogy as the cleavage form of calcite, in which the mineral splits cleanly into rhombohedral fragments along three planes. The form also occurs as a crystal termination on corundum, dolomite, and other trigonal species, often in combination with prismatic and pyramidal forms. The Greek rhombos (a spinning top, by extension a rhombic shape) and hedra (face) supply the name.

Geometry

A rhombohedron may be visualised as a cube stretched along one of its body diagonals. The six faces remain identical, but they are no longer squares — they are rhombi, with two of the original four-sided angles obtuse and two acute. The body diagonal aligned with the stretching becomes the threefold rotational axis of the form. The depth of the rhombohedron — whether it is steep or flat compared with its width — is described by an axial ratio that varies between species and between forms within a single species.

Some minerals show characteristic rhombohedron geometries that are useful for identification. Calcite's primary cleavage rhombohedron has interfacial angles of approximately 105 degrees and 75 degrees. Dolomite shows a similar but slightly different ratio. Corundum's terminating rhombohedra are typically much steeper than calcite's, contributing to the barrel-shaped habit common in ruby and sapphire crystals.

In calcite cleavage

Calcite's most familiar form is not a true crystal habit but a cleavage rhombohedron — the shape produced when a calcite crystal of any form is broken along its three perfect cleavage planes. The cleavage is so clean that fragments from a single broken crystal are essentially identical small rhombohedra. The phenomenon is the basis for the classic gemmological demonstration of birefringence: a clear calcite rhomb placed over text shows the doubled image that defines the optical property, and is named for the form, the Iceland spar rhomb, after the historic source of optically pure calcite in Helgustadir, Iceland.

In other trigonal species

Rhombohedral terminations appear in corundum, where they typically modify the principal hexagonal prism to give barrel-shaped or doubly terminated tabular habits. Quartz crystals are terminated by paired rhombohedra (the major and minor rhombohedra) in proportions that distinguish the left-handed and right-handed enantiomorphs of the species. Tourmaline shows rhombohedral terminations on prismatic crystals. Dolomite forms classic rhombohedral crystals, often saddle-shaped from the warped growth of the rhombohedron faces.

In gemmology and lapidary work

Recognising rhombohedral form helps in rough-stone work — in identifying the orientation of an unfaceted crystal, in predicting cleavage in calcite or rhodochrosite, and in locating the optic axis in corundum and tourmaline. Cutters of soft trigonal species respect the rhombohedron-related cleavage planes to avoid catastrophic failure during cutting. For corundum, the rhombohedral termination on the parent crystal indicates the optic-axis direction, which the cutter must orient correctly to manage colour, pleochroism, and parting; ruby and sapphire are typically cut with the optic axis perpendicular or close to perpendicular to the table, balancing the requirements of colour appearance and weight retention against the parting tendency that aligns with rhombohedral planes.

Iceland spar and the historic role of rhombohedral calcite

Optically pure calcite from Helgustadir in eastern Iceland — historically the principal source of clear calcite cleavage rhombs — played a foundational role in the history of optics. The Danish mathematician and physicist Erasmus Bartholin published the first description of double refraction in Iceland spar in 1669, and the phenomenon led to the development of polarised optics over the following centuries. The Nicol prism, a polarising element constructed from two calcite cleavage rhombs cemented together with Canada balsam, was the standard polarising device in microscopy and optical instruments through the early twentieth century. Modern synthetic polarising materials have largely replaced the calcite Nicol, but the rhombohedral form remains a teaching tool in mineralogy and optics.

The rhombohedron in mineralogical descriptions

Mineralogical descriptions of trigonal species use a standardised notation for the various rhombohedron forms that may appear on a crystal. The fundamental rhombohedron is denoted r, with steeper rhombohedra denoted with appropriate Miller indices. Calcite, with its enormous variety of crystal habits, has been described with hundreds of different rhombohedron forms over the centuries of mineralogical study, and serves as the textbook example of the system's geometric possibilities.

For practical mineral identification, the most important rhombohedra are the fundamental cleavage rhombohedron of calcite and the dolomite cleavage rhombohedron, which differ slightly in their interfacial angles and serve to distinguish the two carbonate species in routine work. Tools as simple as a contact goniometer and a hand loupe permit field identification of the rhombohedral cleavage angle to within a degree, which is sufficient to distinguish the two minerals.

Rhombohedral cleavage in carbonate gem species

The carbonate gem species — calcite, dolomite, magnesite, siderite, smithsonite, and rhodochrosite — all share rhombohedral cleavage as a defining physical property. The cleavage is perfect in three directions, and a cleavage rhomb fragment is therefore a reliable starting point for identification of any of these species. The cleavage angles differ slightly between species, reflecting the small differences in unit-cell geometry that arise from the different cation chemistry, and a precise interfacial-angle measurement (the contact-goniometer reading at one of the rhombohedral edges) can distinguish calcite, dolomite, and rhodochrosite where chemical or physical testing is unavailable.

For the cutter, the rhombohedral cleavage is a constraint and a hazard. A faceted carbonate stone must be oriented with the table well away from cleavage planes, and the cutting and polishing routines avoid the impacts and thermal shocks that could initiate cleavage. The challenge is part of why faceted material in these species is uncommon and is generally limited to collector stones rather than commercial production.

Notational conventions for rhombohedron faces

Crystallographic notation describes individual rhombohedron faces using Miller indices in either three-axis (rhombohedral) or four-axis (hexagonal-Bravais) form. The same physical face can therefore appear in mineralogical descriptions under different index notations depending on the convention used. The fundamental rhombohedron of calcite, for example, is described as (1011) in the four-axis hexagonal notation and as (100) in the three-axis rhombohedral notation. Modern mineralogical references generally use the four-axis notation for trigonal-system descriptions, with the three-axis notation reserved for specialised structural analysis.

For practical mineral identification, the indices themselves rarely matter; what matters is the recognition of the rhombohedral form and the measurement of its interfacial angle. The contact-goniometer reading of the angle between adjacent faces is the standard practical measurement, and reference works provide tables of these angles for trigonal species.

Further reading

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