Geometry Introduction
Before we start on geometry, I would like to give you a
little back on what geometry is all about separate from what
we have seen in school geometry courses.
Chances are when you took geometry in school it was
about axioms, theorems, postulates, and proofs. But is
what geometry is really all about and is this how geometry
really started?
Clearly, no, or I would not have asked the question.
Geometry is one of the two subjects of ancient, nearly prehistoric mathematics; the other is numbers.
Literally geometry was known to the cave man who
understood obtuse triangles.
We always associate geometry with the ancient Greeks,
Pythagoras of Ionia, comes to mind; but he was not even
the first Greek involved with the subject. The person who
is given this honor is Thales of Miletus who is credited
with given the first proofs. Pythagoras may have been a
student of Thales, but he certainly went to both Egypt and
Babylon to student the subject.
China, India, and the Arabs have their own rich geometric
histories.
The questions that they were trying to solve had to do with
measuring shapes, setting markers for cities and farm land,
and the like.
Today, Geometry continues to be a vital and growing area
of mathematics with major applications in many areas
including physics, medicine, and computers. The ability to
understand geometric structure in many dimensions is at
the foundation of string theory which is at the forefront of
understanding the structure of the universe from the largest
galacy to the smallest particle. In cancer research, the
workings of cell mechanisms parallel the workings of
computer networks and the internet whose analysis uses
geometry tools.
With this as background, what are we going to study in
geometry. First, we will spend sometime understand the
structure of geometry and geometric reasoning. Second,
we will look at traditional Euclidean geometry in the plane.
Finally, we will look at coordinate geometry which is the
linkage between geometric ideas and geometry. This is an
important area which you should explore further because of
its importance, but we will just scratch the surface of this
form of analytic geometry.
First, precision and structure are two of the keys to
geometry. It is an exact discipline based on building from
the ground up.
What is a point?
What is a line?
What is a line segment?
What is a plane?
Geometry is built on definitions, postulates, axioms, and
theorems. Proofs are key to the building process.
Postulates: Statements that are assumed to be true without
proof.
1) Unique Line: Through any two distinct points in the
plane, there exist exactly one line.
2) Dimension: Given a line in the plane, there exists a
point not on that line.
Given a plane in space, there exists a line or a point in
space not on the plane.
3) Number line: Every line is a set of points that can be
put into one-to-one correspondence with the real numbers
with any point being 0 and any other point being 1.
4) Distance: On a number line, there is a unique distance
between two points.
5. Plane: If two points lie on a plane, the line connecting
them lies in the plane.
What are collinear points? Points on the same line.
6) Through three non-collinear points, there exists exactly
one plane.
7) If two different planes have a point in common, their
intersection is a line.
What is Euclidean geometry?
Are there non-Euclidean geometries?
Give an example.
Definition: 2 distinct lines in the same plane are parallel if
they do not intersect, and a line is parallel to itself.
Definition: 2 planes in space are parallel if they do not
intersect, and a plane is parallel to itself.
Given a line in space and point not on the line, how many
lines can be drawn through the point parallel to the given
line?
a) 1
b) 0
c) Many
It depends:
In a Euclidean space: 1.
On a sphere: 0
On a sphere, lines are great circles which all intersect with
each other so no parallel lines.
On a saddle: Many as the lines are paths on the saddle.
Euclid’s Postulates
1) Two points determine a line.
2) A segment can be extended infinitely along a line.
3) A circle can be drawn with a center and any radius.
4) Right angles are congruent.
(What is the definition of a right angle?)
5) If two lines are cut by a transversal and the interior
angles on the same side total to less than 180 degrees, the
lines will meet on that side of the transversal.
Non-Euclidean spaces are important today. The entire
structure of Einstein’s General Theory of Relativity which
explains gravity is built on Non-Euclidean geometry which
he learned as a student. It talks about space not being flat
but being curved by gravity so that bodies move based on
falling into gravity wells. Such a well without a bottom is a
black hole where a body spirals into the bottom of the well.