Table of Contents
Module 3
Some of the problems were:
Continuity
PPT Slide
The classic example used to demonstrate the
continuity problem is the construction of an equilateral triangle. The construction is
easy to understand and the proof that the sides are congruent is fine, but...
The problem is that Euclid assumed what everybody
knows:
the “common notions”
Recent Axiomatic Systems:
A “lite” example: SPC System
Axioms
3: If there are 3 consecutive P’s, you may
replace them with one C.
A production path that can be obtained by applying
the rules to the initial path is called a valid path.
Conjecture:
BONUS: come up with a theorem and a proof.
Hilbert’s Axioms
In Appendix B
Definitions (start up)
Axioms
GROUP II: Axioms of Order
GROUP III: Axioms of Congruence
GROUP IV: Axiom of Parallels
GROUP V: Axioms of Continuity
New Definitions
Picture Problems
All triangles are isosceles.
Notebook Problems
Active Geometry
Axioms for The Geometry of Pappus:
Pappas, cont.
Pappas, cont.
Pappas, cont.
The Geometry of Pappas has exactly nine ponts and
nine lines. This geometry is a subset of Euclidean geometry and was drawn from a theorem
proven about 340 AD by Pappas of Alexandria
The Three Point Geometry
A3: Not all the points are on the same line.
Theorem 1
Problem 7, page 29
Axioms:
Read and format the information
Sketch and
Convexity, continued
Other Definitions
Compare and Contrast
Another new Geometry
Axioms:
Explore the geometry with sketching. |