# CS 39R:  Symmetry & Topology Lecture #2 -- Mon. 2/04, 2019.

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## Preparation:

Bring pictures of hubcaps and company logos to class or send them to me: sequin@cs.berkeley.edu

## Warm-up -- Classifying Frieze Symmetry Patterns:

A Key Point:
Exploiting symmetry is a great way to reduce the amount of design work that needs to be done
-- and, possibly, to increase the quality of a resulting design.
Fortunately the numbers of all possible symmetries can be nicely catalogued ...

## Only a Finite Number of Frieze Symmetry Groups!

How many are there ... ?   ---  Why ... ?

Fill in Worksheet_7-Friezes ...

A little bit of Math:
Remember the definition of "symmetry" ...   ==> Transformations
Symmetry operations
(the transformations that map an artifact back onto itself) form groups.
The key characteristics that make something a "group":
Closure: A,B ==> AB, BA;   --- All combinations of operations are also elements of the group.
Associativity
:  (AB)C = A(BC);  --- The order in which elements are combined may matter, but the sequence in which the combinations are calculated does not.
Identity: IA = AI = A;  --- The identity element makes no change.
Inverse:  A ==> A-1:  AA-1 = A-1A = I };  --- for every element there is also an inverse element; an element may be its own inverse.

As an example: consider all integer numbers under addition.  Check the four rules above ...

DEMONSTRATE: Symmetry operations (rotation and mirroring) on a cardboard square ...

Friezes have Translatory Symmetry

Where might you find artifacts with (finite) translatory symmetry ? . . .
An example of  translatory symmetry.

Examples of Friezes on the web ...
An example of purely translatory symmetry:
Translatory Symmetry in Musical Scales (keyboards):

To what degree do keyboards have translatory symmetry?

### Symmetry in 2D Space:

• 7 frieze symmetries.
• 2 families of rotational groups Cn, Dn.
• 17 lattice groups = "Wallpaper" (later!)
• A 2D pattern with a rotational center is either Cn-symmetric  { n }, if a rotation through 360/n degrees places it back onto itself;
or it is Dn-symmetric  { *n }  if in addition it also has reflective mirror symmetry on n axes going through the center.

A pattern with only one mirror-symmetry axis would be D1
{ *1 }  .

### Symmetry in 3D Space:

Understanding the symmetries of 1D friezes is crucial to the understanding of the symmetries of 2D figures, 3D objects, and 2D and 3D tilings.

## Conway Notation for Symmetries:

* (star)      ==>  mirror plane
*3             ==>  kaleidoscope with 3 mirror planes
• (dot)       ==>  exactly one point remains in place under all symmetry operations
7•              ==>  gyration point with 7 angle positions in 360 degrees
X "miracles"        ==>  a glide-symmetry axis
O "wonder-ring"  ==>  two pure translation vectors

## For the next lecture, think about the following issues:

#### What are the symmetries of the (finite) objects that you encounter in your daily life ? How do you determine the symmetry group of an object: (follow the same approach as for: frieze symmetries!): FIRST:  Find a maximal-valence rotation axis, make it the z-axis, go to chart 1,  look for C2 axes perpendicular to it, also for mirror planes, ... THEN:  If you find more than one rotation axis with valence >= 3, go to chart 2;  5-fold axes ==> icosa/dodeca;  4-fold axes at right angles ==> cube/octa, ... NEW:  Bring along TWO  3D objects: one of it with symmetry from chart 1, and the other one with symmetry from chart 2. Send me pictures, annotated with the appropriate symmetry groups in both Conway and Schoenflies notation. Also include a brief answer to the following Puzzling issue:  Why does an ordinary wall mirror reverse left and right, but not up and down ?

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