Long before the classification of finite simple groups, the sporadic groups were already proving to be a thorn in the side of ardent group theorists. Many worked on studying these uncomfortable beasts in the hope of unlocking their secrets and some experienced great success. However, group theory by itself can only get you so far.
In this talk I will outline my work on an alternative approach to better understanding the sporadic simple groups. That is, by investigating discrete geometries associated with these groups and how the two interact.