The rod-shaped bacterium Escherichia coli proliferates by a process of elongation, followed by constriction at its centre to create new cell poles. Despite intense study, some apparently simple questions about the dynamics of growth and division in E. coli continue to be debated - these include whether the cell length increases exponentially or with some other dynamics, and whether E. coli continues to elongate along its cylindrical length during the constriction phase of growth, when the new cell poles are formed. I will discuss these questions from a geometrical perspective. I will show using a simple theoretical model that cell morphology, division initiation and cell wall synthesis are not independent processes, but they are connected by strong constraints arising from the requirement for stable growth as a spherocylinder. I will also discuss how antibiotics can change the shape of bacterial cells, and what implications this has for antibiotic efficacy under different growth conditions.