Biologists have long considered the endoplasmic reticulum (ER) to be an exceedingly important and complex intracellular organelle in eukaryotes. It is a membrane structure, part folded sheet, part branching network, that both envelopes the nucleus and threads its way outward, all the way to the cell’s periphery. Microscopic images attest to its convoluted geometry, but can the complexity of its architecture be understood in a precise, mathematical way? Recently, refined imaging of the ER has revealed beautiful and subtle geometrical forms – "Terasaki ramps'' -- suggestive of Riemann sheets and helical minimal surfaces. What is the physics of these structures, and how do these architectural motifs connect to biological function?