**discuss
basic idea:
Reflection is related to "perpendicular bisector" of PP'
With a triangle the 3 vertices ABC -> A'B'C' may be related to at
most
3 lines of reflection.**

Proof: Click here!

How can the Flatlander experience the sphere and space?

The simplest three dimensional figure has 4 points not all in the same plane: three point determine a plane- so a fourth point not in that plane will need "space" to make sense. These four points determine a tetrahedron.

The simplest three dimensional figure has 4 points not all in the same plane: three point determine a plane- so a fourth point not in that plane will need "space" to make sense. These four points determine a tetrahedron.

Fold downs- flattened figures:

Consider how the tetrahedron can be assembled from folded down triangles.

Consider how the cube can be assembled from folded down squares in two different configurations: a cross or a "zig-zag."

Consider how the tetrahedron can be assembled from folded down triangles.

Consider how the cube can be assembled from folded down squares in two different configurations: a cross or a "zig-zag."

What does a folded down flattened torus look like?

A rectangle with opposite sides resulting from cutting the torus open making a cylinder and then cutting the cylinder along its length.

**A torus **

Proof starting with Euclid.

Euclid's tools. Proposition 1 and Proposition 2

Models for possibilities and impossibilities.

A model for the plane geometry of Euclid:

The basis for this model is understanding

Lines: Sets of points (x,y) that satisfy an equation Ax + By = C where A,B, and C are rational numbers and not all are 0.

Circles: Sets of points (x,y) that satisfy an equation of the form (x-A)

Basic Facts:

Examples: (i) There is a rational number (fraction) which will measure the hypotenuse of an isosceles right triangle with a unit length for the side. (ii)There is a rational number (fraction) which will measure the side of the hypotenuse of a right triangle with a unit length for one side and a hypotenuse of length 2 units.