CS 39R:  Symmetry & Topology
Lecture #3 -- Mon. 2/11, 2019.

Preparation:

Warm-up:

Identify the symmetry classes for the following depicted objects:

Also:  Discuss with your neighbors the symmetry of the objects that you brought along.  Check symmetry classification.
(See posters on the wall!)

Symmetries!   The Elements of Symmetry

A Key Point: Any finite physical object falls into one of the 14 symmetry classes described in: chart 1 and  chart 2 (chart 2b).

How to determine the symmetry group of an object:

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, ...
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, ...
the difficult one (for me) is the oriented double tetrahedron; it has a mixture of pure rotation points and kaleidoscope points;
      the 3 mirror planes transform one (right-handed) tetrahedron into the other (left-handed) one.

Let's review (and count) the elements of some symmetry groups:

REVIEW:  the required properties to make this 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 arranged may matter, but the sequence in which they are combined 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.

==> Now let's apply this to some of the symmetry classes we have encountered: 

-- The simplest possible frieze with an asymmetrical fundamental domain (repetitive element); translatory symmetry only:
    This is equivalent to the group formed by all integer numbers!  (e.g., right shift = "+1")  {infinitely many elements!} [DONE last time]

-- The 2D square  or {D4} hubcap -- under rotations and mirroring:  {8 elements};
-- The thick 3D square plate {D4h} -- under rotations and mirroring:  {16 elements};
-- Permutation of 4 books on a shelf -- while switching pairs of books:  {24 elements};  {mathematicians call this "S4"};
-- All symmetry operations on a tetrahedron:  {24 elements};  {when "oriented": only 12 elements};
-- The octahedron (==> see plexiglass model);  == same as a cube:  {48 elements};

"Rainbow-Bits" by George Hart -- a Propellerized Icosahedron

It has  oriented icosidodecaheral symmetry;  no mirror planes.

The Platonic Solids:

Tetrahedron, cube (hexahedron), octahedron, dodecahedron, icosahedron.
(Could you explain to a high-school student why there are exactly (and only) five Platonic Solids ?

Discussion:

Visualization of Symmetry Groups Using Shape Generator Programs

Understanding Chart I:  with "Sculpture Generator I"  {The program for you to experiment with}

Understanding Chart II:  with "Escher Sphere Editor" {The program for you to experiment with}

A First Glimpse of Topology:

The  genus  of simple "handle-bodies"  (g=1;  g=2;  g=3;  g=7).

These are all equivalent definitions of "genus" (in this particular context):
How many tunnels are there through this (volumetric) body?
How many handles are attached to a simple spherical blob?
How many closed boundary loops can be drawn on this surface that do not yet partition it into separate regions?
         (You can still go from any point to any other point without crossing any of those boundary loops).

Perhaps -- if there is time -- "The Secret of a Happy Life" ... something to think about over the next two weeks.

New Homework Assignments:

Due: Feb. 25, 2019 (in two weeks).