Figure 1: (a) ${\text{Ca}}_3{\text{Mn}}_2{\text{O}}_7$ structure in its distorted ferroelectric state. $\text{Mn}$ ions (green) are surrounded by oxygen octahedra (green with red $\text{O}$ ions) with $\text{Ca}$ ions (blue) interspersed. (b) Octahedral rotation ($R$) in ${\text{Ca}}_3{\text{Mn}}_2{\text{O}}_7$, showing the oxygen (blue) and manganese (red) sites. The undistorted structure is shown in grayscale. (c) Octahedral tilt ($T$) of the same. The combination of the tilt and rotation combine to produce ferroelectric order.