Math 302:     The Axioms for Straight Lines

Given any space of points (probably a surface in three-space, but perhaps something more abstract) and given a notion of which curves in that space are to be considered lines (which always means straight lines), we can consider whether or not each of the following statements is true.

In other contexts, straight lines could be defined by means such axioms: we would take lines to mean those paths in a surface which satisfy the specified axioms. In this course we instead take the approach or using various other criteria to determine which paths should be considered lines in the spaces we study. (For a smooth surface in three-space, the lines will be the geodesics.) Then, for each space that we study, we will check which of the following axioms are true. For most spaces, some of these axioms are true but others are false. (We break each axiom down into several parts so that if it fails we can tell exactly which part fails.)

  1. The "incidence axiom":
    1. There is at least one line through any two given points.
    2. There is at most one line through 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.