Square Billiard Geometry
Credit
I did not create this activity myself. I tweaked Daniel Mentrard's interactive polygonal billiards.
https://www.geogebra.org/u/daniel+mentrard
https://www.geogebra.org/classic/cygr7nby
Billiard geometry
Lines in billiard geometry are the paths a ball makes if it could keep moving and bouncing perfectly forever.
In the microworld below:
- Press "Start". What happens?
- Press "Stop", then "Reset".
Square Billiard Geometry microworld
Exploring lines in square billiard geometry
When in a digital environment, always play around. See what you can change and what appears to stay constant.
When you move the "First bounce" point, what happens?
When you move the "Departure" point, what happens?
What points/features can you move? What points/features stay fixed?
What happens when you move the "Departure" point while the line is still moving? What happens when you move the "Departure" point after you press stop, but before you press reset?
Always question the model
Is this simulation a perfect representation of billiard geometry? Is the path shown the complete line?
in the microworld below:
- Find 3 different examples of entire lines shown by the simulation.
- Find 3 different examples of incomplete representations of lines.
Finding patterns
When you can see entire lines, what do those lines have in common? When lines only have an incomplete representation, what do those lines have in common?
Line Segment Axiom?
The Line Segment Axiom states: Given any two distinct points, there is a unique line segment whose endpoints are those points. Is this assumption ALWAYS, SOMETIMES, or NEVER true in square billiard geometry?
Line Segment Axiom? Why?
What examples did you use to select your answer above?
Paralle Postulate?
The Parallel Postulate states: Given a line L and a point P not on L, there exists a unique line M such that M is parallel to L and M goes through P. Is this postulate ALWAYS, SOMETIMES, or NEVER true in square billiard geometry?
Parallel Postulate? Why?
What examples did you use to select your answer above?
Circles??
What would circles look like?
Where does the idea of a "circle" break down?
Is there a way to fix it?