Intuitive dictionary of unknown terms used for all six surfaces: - Minimal surface. Take any point and a small enough patch around it. That patch has the smallest area among all surfaces with the same boundary curve. A soap film on a wire is the physical picture. An equivalent test: at every point the surface curves up in one direction exactly as much as it curves down in the perpendicular direction, so it is either flat or saddle-shaped there, never dome-shaped. - Complete. You can't walk off the surface in finite time. Every path that leaves every bounded piece of the surface, with length measured along the surface, is infinitely long. A plane is complete, but a disk with its edge cut off is not, since you reach the edge in finite distance. - Embedded. The surface never passes through itself. Immersed means it may pass through itself, but every small piece is still a smooth surface. - Bounded. It fits inside some ball. - Genus. The number of handles, as on a doughnut or a coffee mug. **Simply connected** means every loop on the surface can be shrunk to a point without leaving it, so there are no handles or holes. - Ends. Cut away a huge ball. The unbounded pieces left outside are the ends, the separate ways the surface runs off to infinity. A flat end looks more and more like a plane far away. A catenoid end looks like the flaring part of a catenoid. - Finite topology. Finitely many handles and finitely many ends. **Properly embedded** means embedded, and every ball contains only a finite-area, compact piece of it, so the surface doesn't keep crowding back into a bounded region. - Catenoid. The shape you get by spinning the curve y=cosh x around an axis, like a soap film between two rings. - Helicoid. A spiral ramp, swept out by a horizontal line that rotates at a constant rate while it rises at a constant rate.