Symmetry/tessellation activity [Connect to Escher tilings.]
of Isometries - Symmetries
An isometry is a trwenasformation that preserves the distance between
We have discussed four
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
the reflection are parallel).
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
What about 3 reflections?
Three reflections = reflection or
Visual Proof discussion from Math 371 (HSU Geometry Course): Key idea-
product of two reflections is "flexible."
How do we understand objects in space?
How can the Flatlander experience the sphere and space?
Pick up templates to make Platonic
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
Cross sections: Look at the tetrahedron with cross sections :
Triangles, what if the tetrahedron starts through Flatland with an edge
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
We considered how the tetrahedron might case shadows. Sometimes a
triangle, sometimes a quadrilateral.
shadows and cross sections for a cube.
Fold downs- flattened figures: Consider how the cube can be assembled
folded down squares in two different configurations: a cross or a
What does a folded down
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.