Reminder: Project Proposals and 1or 2 portfolio sample entries "due Thursday"

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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 ssame type... and also twice as far, and
three times as far , and more....

The same is true for the following pattern:

The same is true for the following pattern:

The following pattern also has glide reflection symmetry, for example taking p to b and q to d, etc.

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.

we discussed the group of symmetries of the seven possible 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

Video : Isometries

The video introduced the four isometries we have discussed:

reflections, rotations, translations, and glide reflections.

It was shown 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? How to figure out.... match features.

Every plane isometry is the product of at most three reflections.

Two reflections = rotation or translation.Three reflections = reflection or glide reflection

Preserve

OrientationReverse

OrientationNo Fixed points Translation Glide reflection Fixed Point(s) Rotation Reflection Using Isometries to recognize symmetries of a figure or tiling.