Finish FAPP Tape on Symmetry

**Using Symmetries to create variations of tilings
**

Kali: Symmetry group** **
**180 degree Rotations **
** **Translations

Kali: Symmetry group

World of Escher

**ISOMETRIES: Rigid Motions in (or about)
the plane. Also called "Isometries"**

Orientation preserving :

**Translations
**

Rotations

**Orientation reversing:
**

**Reflections
**

Glide reflections

Orientation preserving :

Rotations

Glide reflections

prepare for Video : Isometries (show video on Tuesday)

The video introduces the four isometries we have discussed:

reflections, rotations, translations, and glide reflections.

Show 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

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 translationsApplications to dance