Tuesday,  February 14

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Do Symmetry/tessellation activity [Connect to Escher tilings.]

Classification of Isometries - Symmetries
An isometry is a trwenasformation that preserves the distance between points.
We have discussed four isometries:
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).

Wingeometry demonstration for reflection- one and two reflections

Any plane isometry  is either a reflection or  the product of two or three reflections.
Two reflections = rotation or translation.

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

Space: How do we understand objects in space?
How can the Flatlander experience the sphere and space?
Pick up templates to make Platonic solid models for next class!

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.

Analyze shadows and cross sections for a cube.

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."

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