CS 39R: Symmetry & Topology
Lecture #3 -- Mon. 2/11, 2013.

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Preparation:

Bring along an object with some higher-order 3D symmetry.
Try to find all its symmetry operations: rotation axes, mirror- and glide-planes.
Determine what symmetry group from these two charts it belongs to: chart 1, chart 2.

Warm-up:

Identify the symmetry classes for the following depicted objects:

Also: Analyze the object that you brought along!

Try to find all its symmetry operations: rotation axes, mirror- and glide-planes.
Determine what symmetry group from these two charts it belongs to: chart 1, chart 2.

Discuss this with your neighbors!

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

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;
the 3 mirror planes transform one (right-handed) tetrahedron into the other (left-handed) one.

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

It has oriented icosidodecaheral symmetry; no mirror planes.

Let's review the elements of some symmetry groups:

-- The simplest possible frieze (==> integer numbers);
-- The 2D square;
-- The thick 3D square plate;
-- The octahedron (==> see plexiglass model).
==> Find all symmetry operations; determine the size of the group; check for the for 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 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.

Discussion Points:

Why does an ordinary wall mirror reverse left and right, but not up and down ?

What is "Antisymmetry" ?

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 ?

A First Glimpse of Topology:

The genus of an object;

Visualization of Symmetry Groups Using Shape Generator Programs

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

7 families of rotational groups based on the 7 friezes wrapped around a cylinder: Cn, Dn, S2n, Dnd, Cnh, Cnv. Dnh.

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

7 groups of "really 3D" symmetries based on the Platonic and Archimedean solids.

Jane Yen and C. H. Séquin: "Escher Sphere Construction Kit" Presentation at I3D conference (PPT)

New Homework Assignments:

Due: Feb. 25, 2013

Think about how you would extract structural or approximate symmetry from "almost symmetrical" objects such as those:

How would would you describe algorithmically what you are doing intuitively for such a task?
How could you instruct a computer to do such a task for you?
What kind of a user interface would you like to see?

Postponed till March 4, 2013

Make a real (physical) model of a genus-6 object of high symmetry;
use: clay, paper, styro-foam, pipe cleaners, . . .