Name __________________________________________
MAT 102 – Survey of Contemporary Mathematical Topics
Professor Pestieau
Extra Credit Assignment – Non-Euclidian Geometry & Fractals
Due by Monday, November 21
PRESENT YOUR WORK NEATLY ON THE PROBLEMS BELOW AND JUSTIFY
YOUR ANSWERS TO RECEIVE FULL CREDIT
Problem 1 – A Finite Geometry
[20 pts]
Consider a finite geometry with the six points A, B, C, D, E, and F.
In this geometry a line is defined as any two of the six points. Parallel lines are defined as two
lines that do not share a common point.
a)
How many lines are in this geometry? List them all.
b)
How many of the lines are parallel to line AB? List them all.
Suppose we now define parallel lines as lines for which the numerical equivalent of their letters
have equal sum.
For example, line BC is parallel to line AD because for both lines the numerical equivalent of
their letters add up to 5:
B+C=2+3=5
and
A + D = 1 + 4 = 5.
c)
Which lines have no other parallel lines?
d)
Which lines have exactly one other parallel line?
e)
Which lines have more than one parallel line?
Problem 2 – Sierpinski’s Fractals
[30 pts]
Two fractals associated with the Polish mathematician Waclaw Sierpinski are the Sierpinski
triangle and the Sierpinski carpet. Both are shown on pages 550-551.
On 2 separate pieces of paper, draw each of these fractals up to stage 5.
Warning: You will need to start with a very large initiator (stage 0) in order to have a neat result.
Problem 3 – Minkowski’s Fractal
[20 pts]
The fractal associated with the German mathematician Hermann Minkowski has been widely
used in the design of antennas. This fractal is shown in exercise 6 on page 559.
a)
Draw the first 3 stages of this fractal on a separate piece of paper.
b)
Compute the similarity dimension of this fractal using the formula on p. 554.
c)
How can you interpret this result?