WittSem 100L: Patterns in Nature, Fall 2007
Assignment 1, due Tuesday Aug 28 at the beginning of class
You may work together with anyone in the class, and you may get help from me or
the tutors in the Math Workshop (218 Hollenbeck), but you should write up your
answers in your own words and with your own understanding (for example, don’t
quote from the book or copy someone else’s answers). Your answers should be
written up neatly on separate sheet(s) of paper, and should be correct,
complete, and concise (that is, show your work). Word processing is fine, but
not required (unless your handwriting is unreadable).
Please hand this assignment sheet in with your answers.
1. Exploration and experiment:
a) Use the Tyler applet demonstrated in class (http://www.superliminal.com/geometry/tyler/Tyler.htm)
to make one of the semi-regular mosaics. Make enough of it to show the pattern
clearly. When you have one you like, copy and paste it into a Word document and
print it out (black and white is OK). To copy it, use Alt-Prt
Screen (while holding down the Alt key, hit the Print Screen key) to copy the
window to the clipboard. Then you can paste it into another document (like a
Word document).
b) Use the applet to make a mosaic out of three different types of regular
polygons: say, squares, triangles, and dodecagons (12-sided). Make enough of it
to show the pattern clearly. Again, copy it to a Word document and print it
out. Are all the joints identical? Explain your answer.
2. Close reading and summarizing:
One good thing to do when reading technical material is to try to organize it
in your own mind if the author doesn't do it for you. So, based on the reading
from Ch. 2, tabulate the total length and average length of each of the
following patterns on a square grid. We'll explore these numbers further in
class on Tuesday.
Pattern:
Total
length Average
length
Spiral (30a)
Random meander (29)
Explosion (31a)
Branching (4-way stem) (32c)
Branching (triple junction) (33)
3. Estimation:
On p. 28, F.W. Went is cited as saying that if the Empire State Building were
as slender as a stalk of wheat, it would only be six feet wide at the base. Is
this true? Estimate the length and diameter of a stalk of wheat. Then find the
height of the Empire State Building (cite your source), and apply your estimate
of the dimensions of the stalk of wheat to estimate what the diameter of the
Empire State Building would be if it were “as slender as a stalk of wheat.” Do
you agree with F.W. Went?
4. Mathematical reasoning:
In the formula on p. 49, the … indicates that the formula can be continued to include five-way joints, six-way joints, etc. Think about what the pattern of the continued formula is. For example, if a figure has only 1-way, 2-way, and 9-way joints, how would you write the formula? (hint: what’s the pattern of the coefficients in front of the J’s in the equation?) Explain your reasoning.
5. Cultural literacy: Look up the following people or things
quoted or mentioned in the reading and answer the questions about them. Give
your source in each case (and be prepared to explain why you thought the source
was reliable enough to cite).
a) D'Arcy Thompson (p. 23): When and where did he live? What did he do? What is
he known for?
b) Buckminster Fuller (p. 31): When and where did he live? What did he do? What
is he known for?
c) Sagrada Familia (p. 42):
What is it? Where is it?
Extra credit challenge: There are models of the 4 regular concave polyhedra in the science building. Where are they?