Bring pictures of hubcaps and company logos to class or send
them to me: sequin@cs.berkeley.edu
Which friezes in the two
panels correspond to one another?
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^{-1}A = 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.
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 } .
Understanding the symmetries of 1D friezes is crucial to the understanding of the symmetries of 2D figures, 3D objects, and 2D and 3D tilings.
*
(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