Three Dimensional Geometry Three Dimensional GeometryRemember your high school geometry class, where you studied points, lines, and triangles through the use of straightedges (rulers), compasses, and axioms? Well, we are going to do a quick review of all that right now, and while we are at it, we'll throw in planes and polyhedra and all the other three dimensional stuff.
Geometry is based on three primitive notions: containment, congruence, and segment. Containment is the same as the set theory notation of "is a subset of", and the following all mean the same thing: A contains x, x is a subset of A, x lies on A, A goes through x. In all that follows, all our variables and formula will refer to sets, thought of as subsets of three dimensional space. We assume that A always contains A (we say "properly contains" if we want to avoid this), that two sets with the same subsets are equal, that we can take the union and intersection of two sets, etc.
Definition A set that contains only itself and the empty set is called a point.
Definition Two sets intersect if they have a point in common.
Luckily, there is more to geometry than just points. There are also segments. Given any two distinct points P and Q, we can draw the segment (which we call ) using our straight edge.
Axiom 0. For every two points A and B, contains A and B.
Axiom 1. equals if and only if {A,B} = {C,D}.
Axiom 2. If you have two segments and , and if , then .
Axiom 4. There are an infinite number of different points on every segment.
Using segments, we can define lines.
Definition A set of points is called bounded colinear if there is some segment which contains them all.
Definition Given any two distinct points A and B, the set of all points C such that {A,B,C} is bounded colinear is called the line through A and B and is denoted line(A,B).
Definition A set of points is called colinear if there is some line which contains them all.
We can also define planes.
Definition If the points A,B, and C are not colinear, then the set of all points contained on segments of the form , where D is a point on the is called .
Definition A set of points is called bounded coplanar if there is some triangle which contains them all.
Definition Given any three non-colinear points A,B, and C, then the set of all points D such that {A,B,C,D} is bounded coplanar is called the plane through A, B, and C and is denoted plane(A,B,C).
Definition A set of points is called coplanar if there is some plane which contains them all.
Axiom 5. If two planes intersect, their intersection contains at least two points.
Axiom 6. There are an infinite number of points NOT contained in any finite set of planes.
Axiom 7. (Pasch's Axiom): If you have a plane plane(A,B,C) and a coplanar line(D,E) which doesn't contain A, B, or C, but does intersect , then line(D,E) must intersect or .
Axiom 8. If , and , then .
Axiom 9. If properly contains , then they are not congruent.
Axiom 10. For every two segments, and , then either is congruent to segment for some E on or is congruent to for some F on .
Definition The halfplane(A,B,C) is the set of all points D on the plane(A,B,C) such that the does not intersect the line(A,B).
Axiom 11. Given six distinct points A,B,C,D,E,F such that , then there exists an unique point G such that (a) , (b) , and (c) G is contained by the halfplane(D,E,F).
Definition The ray(A,B) is the set of all points C on the line(A,B) such that does not properly contain A.
Definition is the union of ray(B,A) and ray(B,C).
Theorem The projection of a point onto a line or a plane always exists and is unique.
The proof of this theorem depends on the following famous axiom.
Axiom 12. (Parallel Axiom). For every line line(A,B) and every point C not on line(A,B) there is an unique line(C,D) contained on plane(A,B,C) that does not intersect line(A,B).
The parallel axiom is famous because for a long time, mathematicians tried to show that you didn't need to assume it was an axiom: they tried to show it could be proven from the previous axioms. (Try it!) In the nineteenth century, several mathematicians (Gauss, Bolyai, Lobachevsky) showed that the parallel axiom was in fact independent from the other axioms. They did this by showing the consistency of non-Euclidean geometries in which the parallel axiom is false. For example, in some non-Euclidean geometries there are no parallel lines (spherical geometries) and in others there are several lines parallel to a given going through a given point (hyperbolic geometries). These non-Euclidean geometries came of age when Einstein realized that his theory of general relativity implied that the large scale structure of space time doesn't necessarily obey the parallel axiom.
We still assume the parallel axiom, though, because it is an excellent approximation for the small scale structure of space time (including all the lines and planes you ever likely to draw!).
The following two axioms are almost never used, but are necessary for completeness:
Axiom 13. (Archimedes' Axiom). For every two segments and , there exist a finite number of points A1, ... An on line(A,B) such that AND such that contains B.
[Stripped of the geometric language, Archimedes' axiom just says that for every two positive real numbers a and b, there is a positive integer n such that n a > b.]
Axiom 14. (Limit Axiom). Let A1, A2, ... be an infinite sequence of points all contained on the such that whenever n > m. Then there is a point A such that (1) , (2) for every i, and (3) there is no point B properly contained by which contains for every i.
[Again, this axiom has a very simple non-geometric meaning: an increasing and bounded sequence of real numbers has a limit.]