Geometry in actionBicycle Tracks

Which way did the bicycle go?

Geometry and SymmetryClass 1

A geometer sees … and dreams

Some of my wildest dreams are about geometry!

Geometry with ... symmetry

Pierre Curé’s Law:

The symmetry of the input matches the symmetry of the output.

Hang a weight on a string

What do you expect to see?

Why?

Balance of forces:

symmetry

Why is a point in the middle not moving?

Geometry with symmetry

• Use dynamic geometry in my own work Geometer’s SketchPad (GSP)

• Mix visual and kinesthetic:

Some dynamic images

• Exploring ‘equal lengths’, and ‘folding’ in GSP

• ‘Equal length’ is about transformation to superimpose end points!

GSP Exploration

Geometry with symmetry

• Ways of testing a proposed straight line:

• Carpenter’s test:

Symmetry as the core of geometry.

Does no gaps mean the two boards are straight?

Gaps means the two boards are curved.

Goemetry not proof!

• Axioms and proofs differ from mathematics as medicine differs from food.

Paraphrase of Gian-Carlo Rota, 1997

• Only professional mathematics learn anything from proofs. Other people learn from explanations. Raol Boas, 1980

• Any fool can know. The point is to understand. Albert Einstein

Quotes on Proof and the practice of Mathematics