#2) Incidence axioms for laternate geometry
- Author:
- Bill OConnell
- Topic:
- Geometry
#2) Incidence axioms for laternate geometry
1st incidence axiom (Does Not Hold)
For every line, l, and for every line, m, not intersecting l, there exists a unique plane P incident with l and m.
The model of Euclidean 3-space where lines replace points and planes
replace lines does not satisfy the first incidence axiom. Consider the
line through the points (0, 0, 0) and (1, 1, 1) and the line through the
points (-1, -1, 0) and (0, -1, 1). The lines represented by these
points are called skew lines and do not define a plane.
2nd incidence axiom (Holds)
For all planes, P, there is at least two distinct lines incident with P.
There are an infinite number of parallel lines that are distinct and incident with a given plane.
3rd incidence axiom (Holds)
There are 3 distinct non-coplanar lines.
Certainly 3 parallel lines could be arranged to be non-coplanar.