abstract: The endoplasmic reticulum (ER) has long been considered an exceedingly important and complex organelle in eukaryotes. It is a membrane structure, part folded lamellae, part tubular network, that both envelopes the nucleus and threads its way outward, all the way to the cell’s periphery. Recently, refined imaging of the ER has recently revealed beautiful and subtle geometrical forms – reminiscent of minimal surfaces and Riemann sheets – that play a crucial role in the organelle's global structure. I’ll review the discovery and physics of Terasaki ramps and discuss their relation to cell-biological questions. Rather than being a footnote in a textbook on differential geometry, these structures suggest answers to a number of the ER’s structure-function problems.