In crystallography, a general form is a crystal face — or, more precisely, the complete set of equivalent faces generated by applying every symmetry operation of a crystal's point group to a single starting face — that intersects all three crystallographic axes at different, non-zero intercepts and lies on no symmetry element whatsoever. It is denoted by the Miller index notation (hkl), where h, k, and l are all non-zero and mutually unequal. Because the general form is subject to every symmetry operation in the point group without being left unchanged by any of them, the number of faces it contains equals the order of the point group — that is, the total count of symmetry operations. This property makes the general form the single most diagnostic crystal form for identifying which of the 32 crystal classes a given specimen belongs to, and it underpins much of the practical crystal-system identification work carried out in gemmological laboratories.

Crystallographic Background

Crystals are classified into seven systems — cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, and triclinic — each subdivided into point groups (crystal classes) defined by their combinations of rotation axes, mirror planes, and inversion centres. A crystal face is described by its Miller indices (hkl), which encode the reciprocals of the fractional intercepts the face makes with the unit-cell axes a, b, and c. When all three indices are non-zero and unequal, the face cannot coincide with any mirror plane or lie along any rotation axis, so the full symmetry machinery of the point group acts on it without redundancy. The result is the maximum possible number of equivalent faces for that class.

In the cubic system, for example, the holosymmetric class m3̄m has an order of 48, so the general form {hkl} in that class produces 48 equivalent faces — the hexoctahedron. In the triclinic system, the class 1̄ has an order of only 2, so the general form yields a mere pair of parallel faces (a pinacoid-like pair, though still technically a general form by index). This dramatic range illustrates why the general form is so informative: it encodes the symmetry order directly in the face count.

General Form versus Special Forms

The contrast with special forms is fundamental. A special form is one whose faces lie on one or more symmetry elements — a mirror plane, a rotation axis, or an inversion centre. Because those symmetry elements map the face onto itself rather than onto a new, distinct face, the total number of faces in a special form is smaller than the order of the point group. Common special forms include:

• Pinacoid — two parallel faces, each lying on a mirror plane or perpendicular to a 2-fold axis.
• Prism — faces parallel to one crystallographic axis, intersecting the other two.
• Dome (sphenoid) — faces intersecting two axes and parallel to the third, but related only by a 2-fold axis or mirror.
• Pedion — a single face with no symmetry-related equivalent, found only in the class 1.

In practice, a crystal habit may display both general and special forms simultaneously. A well-formed corundum crystal, for instance, commonly shows the hexagonal prism {1010} (a special form) in combination with rhombohedral faces that, depending on the exact indices, may approach the general-form condition within the trigonal system.

Significance in Gemmology

For the practising gemmologist, the concept of the general form is not merely academic. Crystal habit — the characteristic external shape a mineral adopts during growth — is one of the primary identification criteria used before any optical or spectroscopic testing begins. Recognising which forms are present, and counting the number of equivalent faces, allows an experienced observer to assign a specimen to a crystal class and thereby narrow the list of candidate minerals considerably.

Diamonds crystallising in the cubic class m3̄m frequently display the octahedron {111}, which is itself a special form (eight faces, not 48), but twinned or distorted crystals may develop vicinal faces close to general-form orientations, producing the rounded, striated surfaces characteristic of natural octahedra. Topaz, orthorhombic in symmetry (class mmm, order 8), can display the general form {hkl} as a set of eight non-equivalent-looking faces that nonetheless obey the point-group symmetry precisely. In tourmaline, which belongs to the trigonal class 3m (order 6), the general form yields six faces; the hemimorphic nature of the class — lacking a horizontal mirror — means the top and bottom of a tourmaline crystal are crystallographically distinct, a fact with direct bearing on its piezoelectric and pyroelectric behaviour.

Emerald and other beryls (hexagonal, class 6/mmm, order 24) typically grow as hexagonal prisms terminated by basal pinacoids — both special forms — but inclusions and growth-sector boundaries within the crystal record the underlying point-group symmetry that the general form expresses. Gemmological laboratories examining growth zoning and inclusion patterns in beryl use this symmetry framework to distinguish natural from synthetic stones, since flux-grown and hydrothermal synthetics may develop growth sectors corresponding to general-form faces that are absent or poorly developed in natural crystals.

Miller Indices and the (hkl) Notation

The Miller index system, introduced by the British mineralogist William Hallowes Miller in 1839, provides a compact and unambiguous notation for crystal faces. Each index is an integer derived by taking the reciprocal of the fractional intercept a face makes with the corresponding unit-cell axis, then clearing fractions. A face cutting the a-axis at one unit, the b-axis at two units, and the c-axis at three units would carry indices proportional to 1/1 : 1/2 : 1/3, giving, after clearing fractions, (6 3 2) — a general form in any system with three non-equivalent axes. The curly-bracket notation {hkl} denotes the complete set of symmetry-equivalent faces, while the round-bracket notation (hkl) refers to a single specific face within that set.

In hexagonal and trigonal systems, a four-index notation (hkil) — the Bravais–Miller system — is often preferred, where i = −(h + k). The general form in these systems requires that h, k, i, and l all be non-zero and that no special relationship holds among them.

Practical Identification Notes

When examining an unknown crystal, the following approach draws on general-form theory:

• Count the number of faces in the dominant form. If the count matches the order of a known point group, the form is likely a general form for that class.
• Check whether any face is parallel to another (suggesting a centrosymmetric class) or whether top and bottom terminations differ (suggesting a hemimorphic or polar class).
• Measure interfacial angles with a contact or reflecting goniometer. General-form faces in a given class will reproduce characteristic angular relationships dictated by the unit-cell geometry.
• Cross-reference with known habits: corundum in 3̄m (order 12), spinel in m3̄m (order 48), zircon in 4/mmm (order 16), and so on.

Further Reading

• GIA Gems & Gemology — Crystallography resources
• International Gem Society — Crystallography and Gems