20040525 news articles on origami: BostonGlobe and CNN.

Shukong Ou <email: shukong@ourigami.com> <fax & voicemail: 9784770074> 

Friday, May 14^{th}, 2004, at St John’s Preparatory School, Danvers, MA A Talk
about Geometry and Origami 
What is Geometry ? Let’s look at a Circle 
Geometry is “looking in one of the windows at the House of Mathematics”. The other windows may be algebra, trigonometry, calculus, differential equations, … What are some other windows in this house ? This simple shape can be viewed in many different ways 1) in geometry, we can construct a circle with a compass (point and pencil) 2) in algebra, x^{2} + y^{2} = 0, or (r=1, θ) in Polar Coordinates 3) astronomy, orbits of moons, planets, comets, stars (not quite circles, are they?) 4) ripples on a pond 5) {cone ∩ “plane ┴ axis of symmetry of cone” } What other circleviews can you think of ? 
Geometry & origami Traditional Geometry à Origami à 
Geometry is naturally an intimately related to origami – 1) folding à 3 dimensional result (make a cone !) 2) edges of paper à represent lines & “constructions 3) folds easily establish straight lines 4) things are more fun in 3 dimensions What are some other ways where origami and geometry are related ? Requires 3 tools: straightedge, compass, protractor Requires just the paper (none of the traditional tools) 2 folds to construct a common angle – (demonstration) Fold a piece of paper so Xaxis motion (side /side) causes Yaxis motion (up /down) 
Three Impossible Things (before breakfast) 
1) A piece of paper can have one side. 2) Fold a piece of paper so Xaxis motion (side /side) causes motion in a perpendicular direction, Yaxis (up /down) 3) 2 π R = 2 π r …and… A fancier proof (and a challenge). 
Time permitting… Continue after school ? 
Unit origami – repeated “units” to construct “Platonic Solids” Tetrahedron (4f, triangle), hexahedron (what’s the other name?) (6f, square), octahedron (8f, triangle), dodecahedron (12f, pentagon), icosahedron (20f, triangle). Units generally have “handedness” – right hand shakes right hand, but right hand does not “fit” left hand. 

