# Form

### From Online Dictionary of Crystallography

Forme (Fr). Kristallform (Ge). Forma (It). 結晶形 (Ja).

## Definition

For a point group P a form is a set of all symmetrically equivalent 'elements', namely:

• in vector space, a crystal form or face form is a set of all symmetrically equivalent faces;
• in point space, a point form is a set of all symmetrically equivalent points.

The polyhedron or polygon of a point form is dual to the polyhedron of the corresponding face form, where 'dual' means that they have the same number of edges but the number of faces and vertices is interchanged. The inherent symmetry of a form is a point group C which either coincides with the generating point group P or is a supergroup of it.

Forms in point groups correspond to crystallographic orbits in space groups.

## Wyckoff positions of forms

The number of possible forms of a point group is infinite. They are easily classified in terms of Wyckoff positions of point groups.

• A Wyckoff position of a crystal form consists of all those crystal forms of a point group P for which the face poles are positioned on the same set of conjugate symmetry elements of P.
• A Wyckoff position of a point form consists of all those point forms of a point group P for which the points are positioned on the same set of conjugate symmetry elements of P.

## Classification of forms

Forms are classified on the basis of their symmetry properties and of their orientation with respect to the symmetry elements of the point groups in which they occur.

### General vs special forms

A face is called general if only the identity operation transforms the face onto itself. Each complete set of symmetrically equivalent general faces is a general crystal form. The multiplicity (number of faces of the form) of a general form is the order of the point group P. In the stereographic projection, the poles of general faces do not lie on any symmetry element of P.

A point is called general if its site symmetry is 1. A general point form is a complete set of symmetrically equivalent general points.

A face is called special if it is transformed into itself by at least one symmetry operation of P, in addition to the identity. Each complete set of symmetrically equivalent special faces is called a special crystal form. The face symmetry of a special face is the group of symmetry operations that transforms this face onto itself; it is a subgroup of P. The multiplicity of a special form is the multiplicity of the general form divided by the order of the face-symmetry group. In the stereographic projection, the poles of special faces lie on symmetry elements of P.

A point is called special if its site symmetry is higher than 1. A special point form is a complete set of symmetrically equivalent special points. The multiplicity of a special point form is the multiplicity of the general from divided by the order of the site symmetry group and is the same as that of the corresponding special crystal form.

### Characteristic vs non-characteristic forms

A form is called characteristic if its inherent symmetry coincides with the generating point group P.

A form is called non-characteristic if its inherent symmetry is a supergroup of the generating point group P.

### Basic vs limiting forms

In a Wyckoff position, forms of different inherent symmetries may occur.

• Forms with the lowest inherent symmetry are called basic forms.
• Forms of higher inherent symmetry are called limiting forms.

Limiting forms always have the same multiplicity and oriented symmetry as the corresponding basic forms because they belong to the same Wyckoff position. The face poles (for face forms) or points (for point forms) of a limiting form lie on symmetry elements of a supergroup of the point group that are not symmetry elements of the point group itself.

### Example

In the point group 4mm, the pyramid {h0l} has inherent symmetry C = 4mm and multiplicity 4; its face symmetry is .m. In the same group, the prism {100} has inherent symmetry C = 4/mmm and multiplicity 4; its face symmetry is again .m.

• For the pyramid, C = P and thus the form is characteristic. For the prism, CP and thus the form is non-characteristic.
• Both forms lie on the mirrors perpendicular to the secondary symmetry directions; both forms are special.
• The prism can be seen as the limiting result of opening the pyramid at its vertex: the pyramid is the basic form, whereas the prism is a limiting form (the only one in this case). The face poles of the prism stay also on the mirror plane perpendicular to the fourfold axis which belong to C, supergroup of P, but not to P.

## List of crystal (face) and point forms

The list of face forms includes 47 forms.

### Open face forms and their dual point forms

Face form Point form Inherent symmetry
Pedion Single point $\infty$m
Pinacoid Line segment through origin $\infty$m /m
Dihedron Line segment mm2
Rhombic prism Rectangle through origin mmm
Rhombic pyramid Rectangle mm2
Trigonal pyramid Trigon 3m
Tetragonal pyramid Square 4mm
Hexagonal pyramid Hexagon 6mm
Ditrigonal pyramid Truncated trigon 3m
Ditetragonal pyramid Truncated square 4mm
Dihexagonal pyramid Truncated hexagon 6mm
Trigonal prism Trigon through origin $\bar62m$
Tetragonal prism Square through origin 4/mmm
Hexagonal prism Hexagon through origin 6/mmm
Ditrigonal prism Truncated trigon through origin $\bar62m$
Ditetragonal prism Truncated square through origin 4/mmm
Dihexagonal prism Truncated hexagon 6/mmm

### Closed face forms and their dual point forms

Face form Point form Inherent symmetry
Rhombic disphenoid Rhombic disphenoid 222
Rhombic dipyramid Rectangular prism mmm
Trigonal dipyramid Trigonal prism $\bar62m$
Tetragonal dipyramid Tetragonal prism 4/mmm
Hexagonal dipyramid Hexagonal prism 6/mmm
Ditrigonal dipyramid Edge-truncated trigonal prism $\bar62m$
Ditetragonal dipyramid Edge-truncated tetragonal prism 4/mmm
Dihexagonal dipyramid Edge-truncated hexagonal prism 6/mmm
Tetragonal disphenoid Tetragonal disphenoid $\bar42m$
Rhombohedron Trigonal antiprism $\bar3m$
Tetragonal scalenohedron Tetragonal disphenoid cut off by pinacoid $\bar42m$
Ditrigonal scalenohedron* Trigonal antiprism sliced off by pinacoid $\bar3m$
Tetragonal trapezohedron Twisted tetragonal antiprism 422
Trigonal trapezohedron Twisted trigonal antiprism 32
Hexagonal trapezohedron Twisted hexagonal antiprism 622
Tetartoid or pentagono-tritetrahedron Snub tetrahedron 23
Pentagon-dodecahedron Irregular icosahedron $m\bar3$
Diploid or Didodecahedron Cube & octahedron & pentagon-dodecahedron $m\bar3$
Gyroid or Pentagon-trioctahedron Cube & octahedron & pentagon-trioctahedron 432
Tetrahedron Tetrahedron $\bar43m$
Tetragon-tritetrahedron Cube & two tetrahedra $\bar43m$
Trigon-tritedrahedron Tetrahedron truncated by tetrahedron $\bar43m$
Hexatetrahedron Cube truncated by two tetrahedra $\bar43m$
Cube Octahedron $m\bar3m$
Octahedron Cube $m\bar3m$
Rhomb-dodecahedron Cuboctahedron $m\bar3m$
Trigonotrioctahedron Cube truncated by octahedron $m\bar3m$
Tetragonotrioctahedron Cube & octahedron & rhomb-dodecahedron $m\bar3m$
Tetrahexahedron Octahedron truncated by cube $m\bar3m$
Hexaoctahedron Cube truncated by octahedron and by rhomb-dodecahedron $m\bar3m$

* The special case of ditrigonal scalenohedron where the dihedral angles are 60º is a hexagonal scalenohedron.

Note: the disphenoids are sometimes improperly called 'tetrahedra'.

## Discussion

Some texts show 48 forms instead of 47 because the dihedron is separated into sphenoid and dome depending on the handedness of the faces. This is however inconsistent and has been repeatedly criticized, because such a splitting has to be applied to all the forms (leading to 130 affine forms), or to none (47 geometric forms).