# Wednesday  March 5

Discuss assignment for this week : Projection and inference (what could happen in the Cave).
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!

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

The assignment: shadows for the frame of a cube.

Review Briefly activity on recognizing symmetries in frieze and planar patterns.
Show video on symmetries from FAPP.

Based on the group of symmetries for these patterns,
there are seven possible distinct types of frieze or
Border Patterns:
 translation horizontal reflection vertical reflection
 reflection + reflection glide reflection rotation reflection + glide reflection

Show video on symmetries from FAPP.

More on the symmetries of a tiling: There are 17 distinct symmetry groups for tiling the plane. They can be described by the following diagrams indicating the symmetries of the figures  as below:

or with figures as illustrated by the following

 translations reflections reflections + reflections glide reflections reflections + glide reflections rotations (2) reflections + rotations (2) rotations (2) + glide reflections rotations (2) + reflections + reflections rotations (4) reflections + rotations (4) rotations (4) + reflections rotations (3) reflections + rotations (3) rotations (3) + reflections
 rotations (6) reflections + rotations (6)

Do Symmetry/tessellation activity [Connect to Escher tilings. using Miras to reflect and create a tiling.]