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2. Bravais lattice - Wikipedia

   en.wikipedia.org/wiki/Bravais_lattice

   The seven lattice systems and their Bravais lattices in three dimensions. In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (), [1] is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by

3. List of space groups - Wikipedia

   en.wikipedia.org/wiki/List_of_space_groups

   The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2 1 is a 180° (twofold) rotation followed by a translation of ⁠ 1 / 2 ⁠ of the lattice vector. 3 1 is a 120° (threefold) rotation followed by a translation of ⁠ 1 / 3 ⁠ of ...

4. Oblique lattice - Wikipedia

   en.wikipedia.org/wiki/Oblique_lattice

   The oblique lattice is one of the five two-dimensional Bravais lattice types. [1] The symmetry category of the lattice is wallpaper group p2. The primitive translation vectors of the oblique lattice form an angle other than 90° and are of unequal lengths.

5. Crystal structure - Wikipedia

   en.wikipedia.org/wiki/Crystal_structure

   The fourteen three-dimensional lattices, classified by lattice system, are shown above. The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the Bravais lattices.

6. List of planar symmetry groups - Wikipedia

   en.wikipedia.org/wiki/List_of_planar_symmetry_groups

   The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular).

7. Crystal system - Wikipedia

   en.wikipedia.org/wiki/Crystal_system

   These lattices are classified by the space group of the lattice itself, viewed as a collection of points; there are 14 Bravais lattices in three dimensions; each belongs to one lattice system only. They [ clarification needed ] represent the maximum symmetry a structure with the given translational symmetry can have.

8. Rectangular lattice - Wikipedia

   en.wikipedia.org/wiki/Rectangular_lattice

   The rectangular lattice and rhombic lattice (or centered rectangular lattice) constitute two of the five two-dimensional Bravais lattice types. [1] The symmetry categories of these lattices are wallpaper groups pmm and cmm respectively. The conventional translation vectors of the rectangular lattices form an angle of 90° and are of unequal ...

9. Crystallographic point group - Wikipedia

   en.wikipedia.org/wiki/Crystallographic_point_group

   Leave out the Bravais lattice type. Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry. (Glide planes are converted into simple mirror planes; screw axes are converted into simple axes of rotation.) Axes of rotation, rotoinversion axes, and mirror planes remain unchanged.