27 lines.


Many people know that a one-sheet hyperboloid is a ruled surface: it can be covered (even twice) by straight lines that are fully contained in it. A hypeboloid is a surface of degree 2, that is, it can be described by an equation of degree 2, like x^2+y^2-z^2=1. You may think that you know all about surfaces of degree 2. We will discuss some day later, whether this is true. But this time we will look at surfaces of degree 3. They are not ruled, in general. But any surface of degree three in space contains at least one straight line. Actually, if you count the lines in a proper way (will not neglect complex solutions of various equations, etc.), then the number of these lines, is almost always the same. What is this number? 27. Why? You will see.