Lab? Tuesday 8-10 am, HGH 105; Wednesday 8-11am : SCIA 364 .

Orientation preserving :

Rotations

Glide reflections

prepare for Video : Isometries

The video introduces the four isometries we have discussed:

reflections, rotations, translations, and glide reflections.

It shows that 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

What about 3 reflections?

Any plane isometry is either a reflection or the product of two or three reflections.

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.

Two reflections = rotation or translation.

Three reflections = reflection or glide reflection

How to figure out what isometry you have.... match features.

Preserve

OrientationReverse

OrientationNo Fixed points Translation Glide reflection Fixed Point(s) Rotation Reflection

Using Isometries to create variations of tilings

Kali: Symmetry group180 degree RotationsTranslations Space: How do we understand objects in space?

How can the Flatlander experience the sphere and space?

Cross sections

shadows

fold downs- flattened figures

analogue... point... line.... polygon.... polyhedron......

Some Issues we'll consider in space:

Platonic (regular convex polyhedra) Solids

Why are there only 5?

> Regular polygons around a vertex.All vertices are "the same".Symmetries (Isometries) in the plane compared to those in space- an introduction:

TranslationsRotations: Center point - central axisReflection : across line - across plane

Symmetries of the cube:

Rotationsreflectionsrotation- reflection

Isometries in space: products of reflections in space:

Rotations and translations>Applications to dance