Tuesday,  September 23
Reminder: Project  Proposals  and   1or 2 portfolio sample entries  "due  Thursday"
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What about the symmetries of a Frieze Pattern on a Strip....
Translations and Glide Reflections  [180 degree Rotations and Reflections]

...|p|q|p|q|p|q|p|q|p|q|p|...

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:

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

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.

...|p|q|p|q|p|q|p|q|p|q|p|...
...|d|b|d|b|d|b|d|b|d|b|d|...

Next time: more on symmetries of frieze patterns .... and

What about the symmetries of a tiling?

Next time: more on symmetries of frieze patterns .... and
we discussed the group of symmetries of the seven possible frieze or
Border Patterns translation horizontal reflection vertical reflection reflection + reflection glide reflection rotation reflection + glide reflection

What about the symmetries of a tiling? There are 17 distinct symmetry groups for tiling the plane. They can be decribed by the following diagrams or with figures as below: Field Patterns translations reflections reflections + reflections glide reflections reflections + glide reflections rotations (2) reflections + rotations (2) rotations (2) + glide reflections rotations (2) + reflections + reflections rotations (4) reflections + rotations (4) rotations (4) + reflections rotations (3) reflections + rotations (3) rotations (3) + reflections rotations (6) reflections + rotations (6)

We did the activity on recognizing symmetries in frieze and planar patterns.

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 preserving
• Translations
• Rotations
• Orientation reversing
• Reflections
• Glide reflections

• Classification of Isometries
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 Orientation Reverse Orientation No Fixed points Translation Glide reflection Fixed Point(s) Rotation Reflection
• Using Isometries to recognize symmetries of a figure or tiling.