^{1,a)}and Jeremy W. Holt

^{2,b)}

### Abstract

We present a complete formulation of the two-dimensional and three-dimensional crystallographic space groups in the conformal geometric algebra of Euclidean space. This enables a simple new representation of translational and orthogonal symmetries in a multiplicative group of versors. The generators of each group are constructed directly from a basis of lattice vectors that define its crystal class. A new system of space group symbols enables one to unambiguously write down all generators of a given space group directly from its symbol.

I. INTRODUCTION

II. GEOMETRIC ALGEBRA

III. POINT GROUPS WITH THE VECTOR SPACE MODEL OF

IV. THE EUCLIDEAN GROUP IN CONFORMAL GA

V. SPACE GROUPS IN CONFORMAL GA

A. Planar space groups

B. Space groups in 3D

C. Alternate presentations and notations for the space groups

VI. CONCLUSIONS

### Key Topics

- Algebras
- 15.0
- Parity
- 4.0
- Crystallographic space groups
- 3.0
- Crystallography
- 3.0
- Crystal structure
- 2.0

## Figures

Reflection of vector through the plane with normal vector .

Reflection of vector through the plane with normal vector .

Rotation of vector through the angle about an axis perpendicular to vectors and . Note that the rotation is through twice the angle between and .

Rotation of vector through the angle about an axis perpendicular to vectors and . Note that the rotation is through twice the angle between and .

Examples of one-, two-, and three-dimensional lattices and their lattice vectors. The lattice vectors are given by one-half the distance between neighboring sites (shaded), so that in the one-dimensional lattice we have .

Examples of one-, two-, and three-dimensional lattices and their lattice vectors. The lattice vectors are given by one-half the distance between neighboring sites (shaded), so that in the one-dimensional lattice we have .

Examples of the six types of symmetry transformations relevant to the crystallographic groups.

Examples of the six types of symmetry transformations relevant to the crystallographic groups.

The two-dimensional Bravais lattices and the associated symmetry vectors for each of the five crystal systems. For the trigonal system we have included the nonstandard lattice.

The two-dimensional Bravais lattices and the associated symmetry vectors for each of the five crystal systems. For the trigonal system we have included the nonstandard lattice.

The three-dimensional Bravais lattices and their symmetry vectors. Although not shown in the figure, the symmetry vectors for the nonprincipal lattices are the same as in the principal lattices. For the trigonal/hexagonal system we have introduced two new lattices labeled and .

The three-dimensional Bravais lattices and their symmetry vectors. Although not shown in the figure, the symmetry vectors for the nonprincipal lattices are the same as in the principal lattices. For the trigonal/hexagonal system we have introduced two new lattices labeled and .

Examples of different types of glide reflections. In each case the reflection is generated by the vector and the translational component is along the dashed vector through one-half its length. (a) An axial -glide reflection, (b) a diagonal glide, and (c) a diamond glide.

Examples of different types of glide reflections. In each case the reflection is generated by the vector and the translational component is along the dashed vector through one-half its length. (a) An axial -glide reflection, (b) a diagonal glide, and (c) a diamond glide.

Figures exhibiting the different types of screw displacement symmetries found in the 3D space groups. This figure is based on Fig. 2 in Chapter 8 of Ref. 13.

Figures exhibiting the different types of screw displacement symmetries found in the 3D space groups. This figure is based on Fig. 2 in Chapter 8 of Ref. 13.

## Tables

The ten two-dimensional point groups and the crystal systems to which they belong. Both the international and geometric symbols are given for comparison.

The ten two-dimensional point groups and the crystal systems to which they belong. Both the international and geometric symbols are given for comparison.

The 32 three-dimensional point groups and the crystal systems to which they belong. Listed are both the international and geometric symbols for the groups.

The 32 three-dimensional point groups and the crystal systems to which they belong. Listed are both the international and geometric symbols for the groups.

Geometric point group symbols and their generators. The angles between the generating vectors are related to and as described in the text.

Geometric point group symbols and their generators. The angles between the generating vectors are related to and as described in the text.

The 17 two-dimensional space groups and their generators. Pure translation generators are omitted but can be obtained from Fig. 5. The 13 symmorphic space groups are listed in bold font.

The 17 two-dimensional space groups and their generators. Pure translation generators are omitted but can be obtained from Fig. 5. The 13 symmorphic space groups are listed in bold font.

The 230 three-dimensional space groups and their generators. Pure translation generators have been omitted but can be obtained from Fig. 6. Note that some space groups in the cubic system have the pure translation symmetry even for lattices. The 73 symmorphic space groups are listed in bold font.

The 230 three-dimensional space groups and their generators. Pure translation generators have been omitted but can be obtained from Fig. 6. Note that some space groups in the cubic system have the pure translation symmetry even for lattices. The 73 symmorphic space groups are listed in bold font.

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