A Toy Model of the Birth of the Universe: Geometric Transitions and Visual Metaphors

We introduce a two--dimensional toy model of the Universe based on a time--dependent level--set function $U(x,y,t)$. The boundary evolves through distinct geometric phases---point, ellipse, parabola, and flat line---mirroring the Big Bang, inflation, matter domination, and dark--energy flattening. Numerical simulations reveal an unexpected transient gray (or purple) flash during negative times, interpretable as a visual analogue of the cosmic microwave background. While not intended as a physical cosmology, the model provides a simple geometric metaphor for cosmic evolution and connects naturally to ideas from phase transitions and machine learning.