Staring at Infinity – The Disk Model of the Projective Plane Jim Hatton September 2012

The Plan Projective Plane– 40 minutes Operations Research - 5 minutes Jim’s Fractured Career - 5 minutes Jim Hatton September 2012

Staying alive John Travolta Rule

Looking for Infinity Jim Hatton September 2012 Interactive Graph Can’t Get There from Here

Cool Book A guide to Plane Algebraic Curves by Keith Kendig Jim Hatton September 2012

What do we Expect? Jim Hatton September 2012 Intersecting Lines One “Point” – A “Loop”

Map the Number Line onto [-1,1] Jim Hatton September 2012 Map Number Line Take 1 “captured” point at infinity. Gave up “normal” distance metric

Map the Number Line onto [-1,1] Jim Hatton September 2012 Map Number Line Take 2 All “points” the same. Identify all antipodal points

Map the Number Line onto [-1,1] Jim Hatton September 2012 Map Number Line Take 3 But we want to be looking at only one point

Map the Number Line onto [-1,1] Jim Hatton September 2012 Map Number Line Take 4 So far, so good Extend to Mapping the plane to the disk

The 3D Version – Plane onto a Sphere Jim Hatton September 2012 Plane on a Sphere We can draw a plane curve on a sphere Now extend by mapping the plane to the disk

The 3D Version – Plane onto a Sphere – Part II Jim Hatton September 2012 Plane on a Sphere- Part II We can draw a plane curve on a sphere onto a disk

The Disk Model of the Projective Plane Jim Hatton September 2012 Antipodal Points are Identified!

The Disk Model of the Projective Plane Jim Hatton September 2012 We just stared at infinity! Now compare behavior at infinity. Parabola

The Disk Model of the Projective Plane Jim Hatton September 2012 Even degree Polynomials

The Disk Model of the Projective Plane Jim Hatton September 2012 Odd degree Polynomials

The Disk Model of the Projective Plane Jim Hatton September 2012 Inverse Function

The Disk Model of the Projective Plane Jim Hatton September 2012 Rational Function I

The Disk Model of the Projective Plane Jim Hatton September 2012 Rational Function II

Tools used for this Demonstration Jim Hatton September 2012 Analytic Geometry

The End – Now OR Jim Hatton September 2012

Operations Research Sometimes called Industrial Engineering, Management Science, Quantitative management… Some of the tools used are statistics, optimization, probability, queuing theory, game theory, graph theory, decision analysis, simulation, and computer applications. (Wikipedia) Types of problems solved: project planning, factory layout, efficient reliable telecommunications networks, traffic flow, bus routes, computer chip layout, supply chains, robotics, freight transportation, scheduling, blending … (Wikipedia) Jim Hatton September 2012

Classes Hatton took: Linear Programming Microeconomics Linear Algebra Game Theory Algorithmic Processes Network Flows Macroeconomics Mathematical Programming Stochastic Processes Elementary Statistical Inferences Reliability Models Inventory-Production Control Queuing Theory Mathematical Systems Theory Dynamic Programming Advanced Probability Large Scale Systems Decision Theory Combinatorial Analysis Applied Probability Modern Algebra, Advanced Stochastic Processes Combinatorial Optimization. Jim Hatton September 2012

Now, Jim’s Fractured Career- Kirkwood High School - St. Louis, Missouri Rice University - Houston, Texas Stanford University - Palo Alto, California U.S. Army - St. Louis, Missouri Stanford University Massachusetts; Mt Shasta, California College of the Siskiyous - Weed, California SOU Jim Hatton September 2012