Math 302:     The Axioms for Straight Lines

  1. The "incidence axiom":
    1. There is at least one line between any two given points.
    2. There is at most one line between any two given points.
  2. The "ruler axiom":
    1. Along any given line, you can travel an infinite distance forwards or backwards.
    2. As you travel forwards along a line, you never pass over the same point twice.
  3. The "protractor axiom":
    1. There is at least one line through any given point in any given direction.
    2. There is at most one line through any given point in any given direction.
  4. The "halfplane" axiom:
    1. If you cut the surface along a line, you get exactly two pieces.
    2. If H is one such piece, and x and y are two points in H , then:
      1. There is a line segment from x to y which is contained in H .
      2. Every line segment in the surface from x to y is contained in H .
  5. The "mirror axiom":
    1. There is a local reflection across every line.
    2. There is a global reflection across every line.


Straight lines could be defined by means of axioms like these: we would take "lines" to mean those paths in a surface which satisfy the specified axioms. In this course we will take a different approach. We will use various criteria to determine which paths should be called "lines" on the surfaces we study, eventually arriving at a definition of "line". Then, for each surface (or "space") that we study, we will check which of the axioms above are true. For most spaces, some of these axioms are true but others are false.