WittSem100: Patterns in Nature
Assignment 4 due Tuesday, 9/18/07
0.
Reading (not to hand in) Stewart pp. 17-25, 54-61
the
notation on p. 59: O(2) refers to circular (fully rotational) symmetry; D24
refers to 24-fold rotational and reflectional
symmetry.
You
should be able to:
*State the Extended Curie Principle in your own
words. Give an example of a physical process that illustrates the Extended
Curie Principle, and explain why and how it does.
*Explain
what symmetry-breaking (or symmetry-sharing) is. Relate this concept to the
concept of stability.
*Explain
how symmetry-breaking produces patterns and give some examples.
ALSO,
Reading
on matter, antimatter, and symmetry-breaking
http://www.space.com/scienceastronomy/antimatter_040831.html
http://www.space.com/scienceastronomy/antimatter_sun_030929.html
How
does this example illustrate the Extended Curie Principle (or does it?)
To be handed in at the beginning of class on Tuesday
9/18:
Usual ground rules apply.
Because you may find it helpful to draw on the figures/pictures to illustrate
your answers, you may write your answers on this sheet and hand it in (but
please be neat and clear about your answers—remember that it's your
responsibility to communicate your understanding to me!) You may use additional
sheets if you prefer or need to.
1. For each of the following
images, list all of the (approximate) symmetries in 2D that apply. For
rotations and reflections, be specific about how many there are (for example,
for a starfish you'd say 5 rotations and 5 reflections). For reflections, also
say which axis (or axes) the reflections are about. Include the identity
transformation. In each case, explain briefly why the symmetries are
approximate in each case (for example, symmetries may be approximate because of
irregularities in the object, or in the case of translational symmetry because
the object doesn’t extend off to infinity in all directions). Color versions of
the pictures may be found by following the links.
a) http://coe.west.asu.edu/students/rbailey/Ronni/pic4.gif
b) http://www.scifun.ed.ac.uk/card/images/flakes/honeycomb.jpg
c) http://photos12.flickr.com/17004033_1e935139e9.jpg
d) http://www.dwstroud.com/_mondaymorningphoto/index.php?showimage=3
Consider all of the 3D symmetries for this one. (This is a hard one because
it's tough to see the whole thing. Do the best you can, and explain your
reasoning so I can give you credit for your reasoning.)
2. Are there any symmetries, other than the identity, in the work below (by
the Dutch artist M.C. Escher)? If so, show what they are. Specifically, if there
are translations (approximate, since the picture doesn't go off to infinity),
show the shift necessary. If there are rotations, show the point (or points) of
rotation, and the angle(s) of rotation. If there are reflections, show the
line(s) of reflection. If there are glide reflections, show the line(s) of
reflection and the shift needed. If there are partial symmetries (part, but not
all, of the picture has a certain symmetry), discuss
those too. This is a more open-ended question than #1 is—I'm interested in your
reasoning and communicating about the concept of symmetry in a more complicated
situation.
3
. a) For each of the following transformations of a
two-dimensional, blank, perfectly uniform 8.5” by 11” piece of paper, state
whether any points, lines, or planes in the original object are left unchanged,
and carefully explain your reasoning. If there are any points, lines, or planes
that are unchanged, what are they? (A sketch may help.) You can assume that the
paper starts in the “portrait” orientation, like this:
i) Translation 1 inch to the right
ii) Rotation by 90 degrees
about its center
iii) Reflection with respect
to a diagonal line from the upper right corner to lower left corner of the
paper
b) Is the paper symmetric
under any of the above transformations? If so, which one(s)? Explain your
reasoning.
c) Would an infinite sheet of
perfectly uniform paper (found, of course, in the Platonic mathematical
universe of pure forms) be symmetric under the above transformations? In fact,
would there be any symmetries that the infinite sheet
of perfectly uniform paper would NOT have in 2D (“Flatland”)? Explain your
reasoning.
e) Stewart and Golubitsky say (p. 5) that the human mind “perceives too
much symmetry as a bland uniformity rather than a striking pattern” and that a
loss of symmetry is often necessary to produce patterns (“in the sense of regular
geometric figures”). Comment on these statements in light of your answers
above.