This applet shows one geometric representation of the expansion [math]e^{i\theta}=1+i\theta+\frac{\left(i\theta\right)^2}{2!}+\frac{\left(i\theta\right)^3}{3!}+...[/math] Note that each term's numerical coefficient has the same color as the corresponding vector. The blue vector for example has a length of 1 unit and the red vector has a length of [math]\theta[/math] units.

Based on 3Blue1Brown's video: https://www.youtube.com/watch?v=ZxYOEwM6Wbk