An isometry is a trwenasformation that preserves the distance between points.

reflections, rotations, translations, and glide reflections.

The Product of isometries:

The product of two reflections is either a rotation (if the axes of the reflection intersect) or a translation (if the axes of the reflection are parallel).

**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!

What about 3 reflections?

Visual Proof discussion from Math 371 (HSU Geometry Course): Key idea- The product of two reflections is "flexible."

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.

Cross sections: Look at the tetrahedron with cross sections : Triangles, what if the tetrahedron starts through Flatland with an edge first?

Shadows: A sphere might cast a circular shadow, but more typical the sphere's shadow is a cone in space and thus casts an elliptical shadow!

We considered how the tetrahedron might case shadows. Sometimes a triangle, sometimes a quadrilateral.

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