So we can compare objects for symmetries....

how many?

does the multiplication table for the symmetries look the same in some sense?

This pattern has vertical axes for reflective
symmetries between each p and q... and a translation symmetry taking each
letter to the next letter of the same type... and also twice as far, and
three times as far , and more....

The same is true for the following pattern:

**
...|d|b|d|****b****|d|****b****|d|****b****|d|****b****|d|...**

Notice the difference -180 degree Rotations and Translations are symmetries for the next pattern.

The same is true for the following pattern:

Notice the difference -180 degree Rotations and Translations are symmetries for the next pattern.

It also has a 180 degree rotational symmetry with center midway between the vertical reflection axes and the letter p and d or q and b.

there are seven possible distinct types of frieze or

translation |
horizontal |
vertical |

reflection + |
glide |
rotation |
reflection + |

translations |
reflections |
reflections + |

glide |
reflections + |
rotations (2) |

reflections + |
rotations (2) + |
rotations (2) + |

rotations (4) |
reflections + |
rotations (4) + |

rotations (3) |
reflections + |
rotations (3) + |

rotations (6) |
reflections + |

One of the features of almost all we have done so far this term, the proofs of the Pythagorean Theorem, Dissections, Tilings and Symmetry have involved

Rigid Motions in (or about) the plane. Also called "Isometries"Orientation preservingTranslationsRotationsOrientation reversingReflectionsGlide reflections

Next Class...Video : Isometries