  Proper vertebrae formation relies on a tissue-wide oscillator called the
segmentation clock. Individual cellular oscillators in the presomitic mesoderm
are modulated by intercellular coupling and external signals, leading to the
propagation of oscillatory waves of genetic expression eventually stabilizing
into a static pattern. Here, we review 4 decades of biophysical models of this
process, starting from the pioneering Clock and Wavefront model by Cooke and
Zeeman, and the reaction-diffusion model by Meinhardt. We discuss how modern
descriptions followed advances in molecular description and visualization of
the process, reviewing phase models, delayed models, systems-level, and finally
geometric models. We connect models to high-level aspects of embryonic
development from embryonic scaling to wave propagation, up to reconstructed
stem cell systems. We provide new analytical calculations and insights into
classical and recent models, leading us to propose a geometric description of
somitogenesis organized along two primary waves of differentiation.
