Retrieving "Octahedron" from the archives

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1. Ferroelectrics

   Linked via "octahedron"

   Ferroelectrics are broadly categorized based on the mechanism driving the polarization switch:
   Displacive Ferroelectrics: These materials undergo a transition where the structure distortion is driven by the softening of a transverse optical phonon mode as $T \to TC$. The ions shift slightly from their symmetric positions, leading to polarization. A prime example is Barium Titanate ($\text{BaTiO}3$), where the transition involves the displacement of the $\text{Ti}^{4+}$ ion within the $\text{TiO}_6$ [octahedron…

2. Plane

   Linked via "regular octahedron"

   The concept of reflectional symmetry is intrinsically linked to the plane. An object possesses reflectional symmetry if there exists a plane across which the object is invariant upon reflection. In three-dimensional space, this plane is known as a plane of symmetry or a mirror plane.
   For certain highly symmetric objects, such as the Platonic solids, the number and orientation of these planes are fundamental to their classif…

3. Point Group

   Linked via "octahedron"

   Tetrahedral Group ($Td$): Possesses the symmetry of a tetrahedron, including $4C3$ axes, $3C2$ axes, and $6\sigmad$ (dihedral mirror planes). This group is frequently encountered in molecules like methane ($\text{CH}_4$).
   Octahedral Group ($Oh$): Corresponds to the symmetry of a cube or octahedron. It is the largest finite point group, containing 48 symmetry elements, including the improper rotation $S6$ and a [center of inversion](/entries/center-of-inversi…