Table of Contents
Module 3, continued
A3: Every two distinct lines have a least one point
on them both.A4: Every line has at least three points on it.
Read and format the information create models look
for familiar objects and look for any differences…can you “bend” familiar
definitions (triangle, parallel, quadrilateral) to fit the new space?
Sketch and Model time
More Definitions:
Module 4
Model
Axiomatic Systems
Metamathematics
Metamathematics provides the framework upon which
we have built all of the structures that allow mathematics
Properties of Axiomatic Systems
Consistency
Independent
Completeness
More Metamath
Euclidean Geometry
From algebra
A function, f, is said to be one-to-one if, for any
choice of two distinct domain set elements, the image of these is two distinct set
elements from the range.
Definition
All models of Euclidean geometry using modern
axioms are isomorphic.
Geometry of Example 1, p30
Axioms: A1: There is at least one point
A2: Every point is on exactly two lines A3: Every line is on exactly three
points Explore the geometry with sketching and models. Check to see which models are
isomorphic and which are not.
Definition
Taxicab distance:
The taxicab distance shows one very subtle flaw in
the Euclid’s Postulates…he assumed that the traditional definition of distance
was the only feasible one…it’s not.
Example 4
Construct a spherical line segment
To draw a straight line from any point to any
point.
Use the construction from Proposition I-1:
Independent Statements
I-1 is in the section of propositions that use only
the first four postulates and, if you use all of Euclid’s assumptions, it does hold.
Notebook Problems
In-class problems
Sketchpad Demo |