Overview
This page offers a general overview of the world of symmetries, having a central role in crystallography.
Repeating patterns found in two dimensions are the 17 Wallpaper Groups.
Repeating patterns found in three dimensions are the 219 distinct Space Groups
(or 230 when considering chirality).
Those arise from the combination of 32 crystallographic point groups with the 14 Bravais lattices,
which themselves are categorized into seven fundamental lattice systems.
There exists also 65 Finite Point Groups, which describe the symmetries of objects like molecules and polyhedra.
Note that this introduction aims to provide a broad display of the key concepts
and classifications within symmetries and crystallography and
is not intended to be an exhaustive treatment of each topic.
Basic Geometric Symmetries
Each geometric symmetry is defined as being the characteristic, of a geometric object,
to be invariant in shape under related transformation.
- Reflectional
- 🪞 Mirroring by a line/plan
- Rotational
- 🛞 Rotating by a point/line
- Translational
- 👀 Translating by a vector
- Helical
- 🧬 Translating/Rotating by screw axis
- Scale
- 🛍️ Expanding/Contracting by scalar
- Glide Reflection
- 🧤 Reflecting/Translating by hyperpl.
- Roto-Reflection
- 🪭 Rotating/Reflecting by point
Symmetry Notations
There are multiple ways to note symmetries. Each are roughly equivalent,
given alongside an example.
- Schönflies
- e.g. S2n
- Intl/IUCr
- e.g. p4m
- Coxeter
- e.g. [5,3] to note the icosahedron (Ih, H3).
- Orbifold
-
Positive integers and { infinity,
asterisk, wonder/handle, miracle }.
e.g. 5•
For example, Orbifold *442 is equivalent to IUCr. p4m