Crystal Form

A crystal form is a collection of crystal faces that are all equivalent to one another under the symmetry operations of the mineral. ^[1] The key word is “equivalent” - not just similar in size or shape, but genuinely interchangeable by mirror reflection, rotation, or inversion. Apply the symmetry operations of the crystal class to any one face in the form, and you reproduce every other face in that same form. Faces that belong to different forms may look similar but are not related by the crystal’s own symmetry, and they therefore belong to separate forms on the same crystal.

Each form is labelled with braces around the Miller index of one of its member faces: {hkl}. So {011} names the form consisting of all faces equivalent to the (011) face, however many that turns out to be under the mineral’s symmetry. ^[1]

Open and Closed Forms

Every crystal form is either open or closed. A closed form encloses a volume entirely on its own - a cube, for example, seals off space on all six sides without needing any other faces to complete it. An open form does not enclose a volume: a prism is a tube that is open at both ends, and a pyramid comes to a point at one end but is open at the base. ^[1]

This distinction has a direct consequence for how crystals are built. A mineral whose only forms are closed can present any one of them as a complete crystal. But a mineral that develops only open forms must combine at least two - sometimes three or more - to close off the volume that the crystal occupies. ^[1] A prism, for instance, requires at least a pair of terminal faces (a pinacoid or pyramid) at its ends to form a complete, bounded crystal.

One common mistake is to treat a multifaced form as if it were assembled from simpler ones. A pinacoid is not two pedions stacked together - it is a single form consisting of two and only two parallel faces that the mineral’s symmetry requires to coexist. ^[1] Similarly, a cube is a single six-faced form - not three pinacoids combined. ^[1]

The Named Forms

The following forms occur across the non-isometric crystal systems. Open forms are marked (O) and closed forms (C).

Pedion (O): A single face with no geometrically equivalent partner anywhere else on the crystal. No symmetry operation repeats it. ^[1] A pedion can exist only in crystal classes that lack a center of symmetry - because inversion would immediately generate a second, opposite face, turning the pedion into a pinacoid.

Pinacoid (O): A matched pair of faces situated on exactly opposite flanks of the crystal, brought into alignment either by an inversion center or mirror plane. Also called a parallelohedron. ^[1] The basal pinacoid {001}, side pinacoid {010}, and front pinacoid {100} are the three principal pinacoids, each parallel to one pair of crystal axes.

Dihedron (O): Two non-parallel faces. When the two faces are related only by a mirror plane the form is called a dome; when related by a 2-fold rotation axis it is called a sphenoid. ^[1]

Prism (O): A set of 3, 4, 6, 8, or 12 faces whose intersections form a set of mutually parallel edges - geometrically, a tube. The faces that may cap the ends of the tube are not part of the prism form itself. Prisms are named for the shape of their cross-section: a tetragonal prism has a square cross-section, a hexagonal prism a hexagonal one, and so on. ^[1]

Pyramid (O): A set of 3, 4, 6, 8, or 12 non-parallel faces that, if extended, could all meet at a single point. Named for cross-section shape using the same scheme as prisms. Pyramids can occur on either the top or the bottom of a crystal. ^[1]

Dipyramid (C): Two pyramids joined at their bases, with 6, 8, 12, 16, or 24 faces total. The upper and lower pyramids are related by a horizontal mirror plane between them. ^[1] Because it is closed, a dipyramid can form a complete crystal on its own.

Trapezohedron (C): A closed form of 6, 8, or 12 faces, each shaped as a trapezoid. The faces on one end of the crystal are offset relative to those on the other end - they do not sit directly above or below each other. ^[1] Tetragonal trapezohedrons have eight faces (four per end), trigonal trapezohedrons have six (three per end), and hexagonal trapezohedrons have twelve (six per end). ^[1]

Scalenohedron (C): A closed form of 8 or 12 faces, each a scalene triangle (three sides of unequal length). The faces appear in pairs. ^[1]

Rhombohedron (C): A six-faced closed form where every face is rhomb-shaped. It resembles a cube that has been either stretched or compressed along the axis running from one corner through the center to the opposite corner. ^[1] Rhombohedrons occur only in the trigonal division of the hexagonal crystal system, where a single 3-fold axis is present.

Tetrahedron / Disphenoid (C): A four-faced form with triangular faces. In the isometric system all four faces are equilateral triangles. In the tetragonal system they are identical isosceles triangles. In the orthorhombic system two pairs of different isosceles triangles occur. Non-isometric tetrahedra are also called disphenoids. ^[1]

Isometric Forms

The isometric crystal system permits 15 forms - all of them closed - more than any other system. ^[1] The shapes seen most often in nature include the six-faced cube {001}, the eight-faced octahedron {111}, the four-faced tetrahedron {111}, and the twelve-faced rhombic dodecahedron {110}. ^[1] Note that both the octahedron and tetrahedron share the same form symbol {111}. This apparent ambiguity is resolved by crystal class: octahedra and tetrahedra appear in minerals with different point group symmetries, so there is no practical confusion in assigning the symbol to a real specimen. ^[1]

Zones

A zone is a group of crystal faces that all share a common parallel direction - meaning every face in the group is parallel to a single line called the zone axis, identified by its crystallographic direction index [uvw]. ^[1] The practical consequence is that all faces in a zone intersect each other along edges that are mutually parallel. A tetragonal prism is a straightforward example: all four vertical faces are parallel to the c axis, making [001] the zone axis, and all the edges between those faces run parallel to c as well. ^[1]

If the Miller indices of two faces that intersect within a zone are known, the zone axis index can be calculated directly using a cross-multiplication procedure. ^[1] Write out the index of the first face twice in a row, then write the index of the second face twice immediately below it. Cross off the first and last columns of each row, leaving only the four middle columns. Then cross-multiply the remaining two-by-two blocks to obtain u, v, and w: u is found by multiplying the first remaining column of the top row by the second column of the bottom row and subtracting the reverse product; v and w are found the same way for the subsequent column pairs. Divide through by any common factor to reduce to the smallest integers. ^[1]

For the faces (11̄0) and (110) on a tetragonal prism, the procedure yields u = ((-1)×0) - (0×1) = 0, v = (0×1) - (1×0) = 0, and w = (1×1) - ((-1)×1) = 2. Dividing through by 2 gives [001] - confirming that the zone axis is the c axis, as expected for a prism whose faces are all parallel to c. ^[1] This procedure works for any two faces that share a zone, and it is the standard method for identifying zone axes when they are not immediately obvious from inspection.

Enantiomorphous Forms

Some forms come in left-handed and right-handed versions that are mirror images of each other, like a pair of gloves. These are called enantiomorphous forms, and they arise in crystal classes that possess no center of symmetry, no mirrors, and no rotoinversion axes. ^[1] Because no operation in such a crystal class can convert a left-handed arrangement into a right-handed one, any given crystal displays only one version - either right or left - never both on the same specimen. Different samples of the same mineral may carry either version. ^[1]

The trigonal trapezohedron is the clearest example: the right-hand {211} and left-hand {31̄1} versions are geometrically identical except for their handedness, and the individual trapezoidal faces on one are mirror images of the corresponding faces on the other. ^[1]

Enantiomorphous forms are directly linked to screw axes in the crystal structure. A screw axis is a repeating helical arrangement that can wind either clockwise or counterclockwise, and it is this internal handedness that determines whether the crystal displays the right- or left-handed form at the surface. ^[1] Quartz is the best-known example: right-handed quartz can develop the right-hand {511} trigonal trapezohedron; left-handed quartz can display the {61̄1} left-hand trapezohedron, though both faces are uncommon on natural specimens. ^[1]

The specific forms capable of enantiomorphism, and the crystal classes in which they occur, are listed below. All of these classes share a common feature: they lack center symmetry, rotoinversion axes, and mirror planes.

An important distinction separates the forms listed above from certain other crystal classes. The 1, 2, 3, 4, and 6 classes also lack center symmetry, rotoinversion axes, and mirrors - so they too can produce crystals that exist in left- and right-handed versions. The difference is that the individual forms present in these classes are not themselves enantiomorphous: a left-handed and right-handed crystal in the 4 class, for example, are built from forms like the {211} pyramid and the {110} prism, none of which carry handedness on their own. ^[1] The handedness of the crystal as a whole emerges from the combination of forms, not from any single form in isolation. This is different from the trapezohedron classes in the table above, where the individual form itself exists in two mirror-image variants.

Positive and Negative Forms

Some forms - rhombohedrons and tetrahedrons especially - occur in positive and negative varieties. The negative form is simply the positive form rotated to a new orientation, so that its faces sit in the gaps between the faces of the positive form. ^[1] For a tetrahedron, the negative {11̄1} version is related to the positive {111} by a 90° rotation about the c axis. ^[1] For rhombohedrons on a 3-fold axis, the negative version is produced by a 60° rotation, so the positive {101} and negative {011} rhombohedrons interleave at 60° intervals around the c axis. ^[1]

The same crystal can carry both the positive and negative version of a form simultaneously. Quartz commonly shows both the positive {101} and negative {011} unit rhombohedrons together, with one set usually having larger faces than the other - by convention the larger set is assigned to the positive form. ^[1]

A practical shorthand for relating positive and negative forms is to reverse the sign of the last digit of the Miller index: the positive {101} rhombohedron pairs with a negative form whose index has the l reversed, giving {101̄}; the positive {111} tetrahedron pairs with {111̄} by the same logic. ^[1] In practice, however, the negative unit rhombohedron is conventionally identified as {011} rather than {101̄}, and the negative tetrahedron as {1̄1̄1} rather than {111̄}, because convention prefers index symbols in which the faces cut the positive c axis. ^[1]

This last-digit rule does not apply universally. In certain crystal classes, the positive and negative versions of a form are related instead by reversing two index digits rather than one - the pyritohedron is the standard example, where the positive form is {102} and the negative is {012}. ^[1] The geometric reason is that the pyritohedron occurs in crystal classes whose symmetry relates faces differently than the threefold or fourfold axes that govern rhombohedrons and tetrahedrons, so the rotational offset between positive and negative is expressed differently in the index notation.

Forms and Crystal System Compatibility

Every form on a crystal must be consistent with the crystal class to which the mineral belongs. An isometric mineral can display only isometric forms; a tetragonal mineral only tetragonal forms; and so on. ^[1] This constraint is not arbitrary - it follows from the definition of a form as faces related by the symmetry operations of the mineral. If the mineral lacks a particular symmetry element, it cannot generate the faces that element would create.

Recognising forms that belong to higher symmetry classes is a practical pitfall. A crystal habit can appear more symmetric than the mineral actually is: nepheline, for example, develops prisms and pinacoids that produce an apparently high-symmetry habit, even though its actual point group symmetry is lower than the habit suggests. Identifying the correct crystal class requires looking for faces that indicate reduced symmetry, not just the faces that dominate the visual impression of the crystal.

References

1. Nesse, W. D. (2018). Introduction to Mineralogy, 3rd ed. Oxford University Press.

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