# Form

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

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

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 transform the face onto itself. Each **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

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

### 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 4*mm*, the pyramid {*h*0*l*} has inherent symmetry C = 4*mm* 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, C ⊃ P 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 or 48 forms, depending on whether the dihedron is separated into sphenoid and dome or not.

### Open face forms and their dual point forms

Face form | Point form | Inherent symmetry |
---|---|---|

Pedion | Single point | m |

Pinacoid | Line segment through origin | 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 | |

Tetragonal prism | Square through origin | 4/mmm |

Hexagonal prism | Hexagon through origin | 6/mmm |

Ditrigonal prism | Truncated trgion through origin | |

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 bipyramid | Quad | mmm |

Trigonal bipyramid | Trigonal prism | |

Tetragonal bipyramid | Tetragonal prism | 4/mmm |

Hexagonal bipyramid | Hexagonal prism | 6/mmm |

Ditrigonal bipyramid | Edge-truncated trigonal prism | |

Ditetragonal bipyramid | Edge-truncated tetragonal prism | 4/mmm |

Dihexagonal bipyramid | Edge-truncated hexagonal prism | 6/mmm |

Tetragonal disphenoid | Tetragonal disphenoid | |

Rhombohedron | Trigonal antiprism | |

Tetragonal scalenohedron | Tetragonal disphenoid cut off by pinacoid | |

Ditrigonal scalenohedron* | Trigonal antiprism sliced off by pinacoid | |

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 | |

Diploid or Didodecahedron | Cube & octahedron & pentagon-dodecahedron | |

Gyroid or Pentagon-trioctahedron | Cube & octahedron & pentagon-trioctahedron | 432 |

Tetrahedron | Tetrahedron | |

Tetragon-tritetrahedron | Cube & two tetrahedra | |

Trigon-tritedrahedron | Tetrahedron truncated by tetrahedron | |

Hexatetrahedron | Cube truncated by two tetrahedra | |

Cube | Ochtaedron | |

Octahedron | Cube | |

Rhomb-dodecahedron | Cuboctahedron | |

Trigonotrioctahedron | Cube truncated by octahedron | |

Tetragonotrioctahedron | Cube & octahedron & rhomb-dodecahedron | |

Tetrahexahedron | Octahedron truncated by cube | |

Hexaoctahedron | Cube truncated by octahedron and by rhomb-dodecahedron |

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

**Note**: the disphenoids are sometimes improperly called "tetrahedra"

### See also

- Chapter 10 in the
*International Tables for Crystallography*, Volume A