CS 39R: Symmetry & Topology
Lecture #10 -- Mon. 4/15, 2013.

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Find matching pairs with the same symmetries (rows 1, 2, 3; columns A, B, C, D, E).

For each pattern make a list of all symmetries found in the linear friezes and in the round hub caps; compare those lists!

Which symmetries are found in both classes? Which are found in only one of them? (Make a Venn diagram)
             HUBCAPS                                               BOTH                                                     FRIEZES
   (         C_n>2, D_n>2,                   (    mirrors, C₂, D₂,              )               glide axes                   )

In the above patterns:   2A == 3D;    1D == 2B;      1C == 2D == 3B == 3C.

Wall-Paper Symmetries

Which of the above symmetries apply to tessellations of the plane?
=> C_n, D_n, only for n = 2,3,4,6.

Are there any new ones? => No.

A wonderful, "must-have" resource (from which the above pictures originated):
"The Symmetries of Things" by John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss (A.K.Peters, Wellesley, MA, 2008)

Let's make a list of all the symmetries that we are looking for!
Introducing the John H. Conway notation for describing symmetries. (JHC and polyhedra)
sym2 sym1 sym3 sym4 sym5 sym6 sym7 sym8

Multiple mirror planes. Kaleidoscopes. Look for different corners!

In how many ways can we combine all these elements in 2 dimensions?
pg42 pg43

There are exactly 7 types of linear friezes.
How many wallpaper types are there?
WpGrps

The Magic Theorem.
Cost for each symmetry item
With the "costs" listed in the above table for each symmetry element in a wallpaper design,
the total cost for every periodic wall paper is always the same: 2 units ($)!

Apply the Conway notation to hubcaps.
Apply the Conway notation to friezes. (friezes)

Intermission: Fill out questionnaire.

How do we capture the symmetry of finite objects (the 14 families that we talked about in Lecture #3 )?
==> Embed the object in the celestial sphere. Thus let's study the symmetries of spherical tessellations!
Which ones of all the symmetry types encountered so far apply to the surface of a sphere?
=> They must accommodate a spherical triangle that can seamlessly tessellate a sphere:
"Lunes" going from pole to pole of any count: 2 to infinity;
"Half-lunes" going from the equator to one pole any count: 2 to infinity;
Triangles that can make up the faces of a Platonic solid.

What is new if we go to 3-dimensional lattices?
List possible new symmetries! => Helical screw axes.
How many 3-dimensional crystal lattice types ?
=> 230 types, in 32 possible crystal classes, within 7 crystal systems.

Your Project Presentations Next Week

Pick one or two interesting issues that make a good story
(rather than a superficial overview of all possible issues that one could address).

Final Homework Assignments:

Work on your projects! ( ... a crucial part of getting a passing grade in this course!)
Prepare for Monday, 4/22/2013:
an 8-minute (15min. for 2-person teams) formal presentation with visuals (and possibly models).
E-mail me your official title and your main findings (in one or two sentences),
before Friday, 4/19/2013, midnight.
(I will use that information to plan the sequence of the presentations.)

Practice your talk (and time it) !