A crystal is a lattice repeated over and over, and a lattice looks the same from more than one point of view. Rotate a cubic crystal by ninety degrees about a cube axis and nothing has changed that any measurement could detect. The set of all such motions is the crystal's symmetry, and it is the single most consequential fact about working with crystal data.
The consequence is this: inside a crystal there is no such thing as one direction. Ask for the [111] direction of a cubic crystal and you have named eight directions at once, because symmetry cannot tell them apart. Every angle, every average, every distance computed later has to respect that, or it computes something the crystal does not know about.
cs = crystalSymmetry('m-3m');
% the eight directions that the label (111) names in a cubic crystal
plot(Miller(1,1,1,cs).symmetrise('unique'),'labeled','grid','backgroundColor','w')
Note the 'unique' above. Without it symmetrise returns one entry per symmetry operation - 48 for this point group - and the eight distinct directions simply appear repeatedly.
Symmetry is also why a direction in a crystal and a plane in a crystal are written differently and behave differently. In a cubic lattice the [111] direction happens to be perpendicular to the (111) plane, which makes cubic a bad place to learn the difference; in a monoclinic lattice they are not perpendicular at all, and confusing the two silently gives the wrong answer.
Point groups, space groups, Laue groups
Three classifications turn up constantly and are easy to mix up. The point group collects the symmetry operations that leave one point fixed - rotations, mirrors, inversion - and there are 32 of them. The space group additionally allows translations, screw axes and glide planes, giving 230. The Laue group is the point group with an inversion centre added, giving 11.
Under Friedel's law, conventional diffraction intensities cannot distinguish a reflection from its opposite, so diffraction symmetry is commonly described by the Laue group. Dynamical and resonant diffraction can reveal departures from Friedel's law. MTEX stores point groups and accepts a space-group symbol or number by reducing it to the corresponding point group.
Where to start
Crystal Symmetries is the foundation - how to declare a phase, from a name, from lattice parameters or from a CIF file.
Miller Indices covers directions and planes in a crystal and the difference between them raised above. Lattice Metric and Plane Geometry covers the unit cell, reciprocal basis, physical lengths, interplanar spacings and the constraints of the seven crystal systems. Operations is the arithmetic - angles, symmetric equivalents, multiplicities, incidence tests and zone axes.
Two pages then separate conventions that are easily confused. A point group says which operations exist; it does not say how the crystal axes are laid onto Cartesian ones. Nor does it say which physical lattice vectors a particular source calls \(\vec a\), \(\vec b\) and \(\vec c\). Reference System and Crystal Axes Alignment are where that choice is made explicit. If data imported from two sources disagrees by a rotation that looks like nothing physical, this is almost always why.
Crystal Shapes and Advanced Crystal Shapes build the little polyhedra used to draw a crystal in a map, which is the most direct way to see what an orientation means.
Fundamental Sector is the counterpart of the opening figure: since symmetry makes many directions equivalent, only a patch of the sphere is needed to hold every distinct one. That patch is what an inverse pole figure is drawn on.
Quasi Symmetries covers symmetries that no periodic lattice can have.
Next
A crystal placed in a specimen is an orientation, Orientations. Directions without a crystal attached are Vectors. Physical properties that depend on crystal direction are Tensors.
Geometry used in deformation is developed in Slip Systems and Dislocation Systems. Twin relationships are Twinning, and geometry connecting two phases starts with Parent-Child Variants.