# Is there a general test for chirality?

by Vorde
Tags: chirality, test
 Sci Advisor P: 1,807 I will give an informal explanation, but I'm open for corrections here. You have three general classes of isometries of a 2-dimensional geometric shape: translations, rotations around a point, and reflections over a line. If your shape X is a subset of $\mathbb{R}^2$, then you may - Translate it by a translation map $t_a : \mathbb{R}^2 \to \mathbb{R}^2$ where $a$ is a vector in $\mathbb{R}^2$, -Rotate it by a rotation map $r_{p,\theta} : \mathbb{R}^2 \to \mathbb{R}^2$ where $p$ is a point and $\theta$ is an angle, - Reflect it by a reflection map $s_l : \mathbb{R}^2 \to \mathbb{R}^2$ where $l$ is a line (defined by some function linear $y = ax+b$). You may also compose these different maps, like first translate it, then rotate it about a point, then reflect it about a line, and then rotate it about another point again. Any arbitrary series of compositions will again yield an isometry. The definition of a chiral shape is that for any line $l$ in $\mathbb{R}^2$, you may not end up with the reflection about this line by translations and rotations alone. Note that which line $l$ is does not matter, as any line may be mapped to another line by a translation or a rotation. Translations and rotations are said to preserve orientation, and reflections are said to reverse orientation. Any isometry preserves or reverses orientation. This stuff is defined in detail here: http://en.wikipedia.org/wiki/Euclidean_plane_isometry