Project 6.1

What do the following people have in common?
Eamon de Valera, Prime minister and President of the Republic of Ireland
Tom Lehrer, songwriter-parodist
Edmund Husserl, the "Father of Phenomenology"
Frank Ryan, quarterback for the Cleveland Browns

Project 6.2

What are optical illusions?
What is the following optical illusion:
                 

(From Problem 1 of Section 6.1.)
Find some unusual optical illusions and illustrate with charts, models, advertisements, pictures, or illusions.

References:

Martin Gardner, "Mathematical Games," Scientific American, May 1970.
Richard Gregory, "Visual Illusions," Scientific American, November 1968.
Lionel Penrose, "Impossible Objects: A Special Type of Visual Illusion," The British Journal of Psychology, February 1958.
Jim Meador, "Pool Illusions," web site found at:
http://www.billiardworld.com/puzzles.html

Project 6.3

Many curves can be illustrated by using only straight line segments. The basic design is drawn by starting with an angle, as shown below.

Procedure for basic angle design for aestheometry Step 1: Draw an angle with two sides of equal length


Step 2: Mark off equally distant units on both rays using a compass
Step 3: Connect #1 to #1; connect #2s, #3s, ...


The result is called aestheometry and is depicted below. Make up your own angle design.


a. Angle design	b. Angle design	c. Circle design
Aestheometry designs

Project 6.4

Many curves can be illustrated by using only straight line segments. The basic design is drawn by starting with an angle, as shown below.


A second basic aestheometric design (see Project 6.2) begins with a circle as shown:


a. Draw a circle and mark off equally spaced points.
b. Choose any two points and connect them.
c. Connect succeeding points around the circle.

Construct various designs using circles or parts of a circle.

Project 6.5

In 1928 Frank Ramsey, an English mathematician, showed that there are patterns implicit in any large structure. The accompanying diagram (which resembles the aestheometric design of Projects 6.3 and 6.4) typifies the problems that Ramsey theory addresses.


"How many people does it take to form a group that always contains either four mutual acquaintances or four mutual strangers?" In the diagram, points represent people. A red edge connects people who are mutual acquaintances, and a blue edge joins people who are mutual strangers. In the group of 17 points shown, there are not four points whose network of edges are either completely red or completely blue. Therefore, it takes more than 17 people to guarantee that there will always be four people who are either acquaintances or strangers. Write a report on Ramsey theory.

What is Ramsey Tic-Tac-Toe?

Reference:

Ronald L. Graham and Joel H. Spencer, "Ramsey Theory," Scientific American, July 1990, pp. 112-117

Project 6.6

Euclid clearly made a distinction between the definition of a figure and the proof that such a figure could be constructed. Two very famous problems in mathematics focus on this distinction:
1. Trisect an angle using only a straightedge and compass.

Reference:

Trisecting:
http://www.geocities.com/CapeCanaveral/Launchpad/5577/musings/trisect.html

2. Square a circle: Using only a straightedge and compass, construct a square with an area equal to the area of a given circle.

Reference:

http://mathforum.org/isaac/problems/pi3.html
This site has an interesting interactive component to help you to understand the problem. There are also links to other sites. Other sites are:

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html

These problems have been proved to be impossible (as compared with unsolved problems that might be possible). Write a paper discussing the nature of an unsolved problem as compared with an impossible problem.

Project 6.7

References:

Arthur F. Smith, "Angles of Elevation to the Pyramids of Egypt," The Mathematics Teacher (February 1982, pp. 124-127).
Kurt Mendelssohn, The Riddle of the Pyramids, New York: Praeger Publications, 1974.

Project 6.8

Do some research on the length-to-width ratios of the packaging of common household items. Form some conclusions. Find some examples of the golden ratio in art. Do some research on dynamic symmetry.

References:

Philip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Houghton Mifflin Company, 1981), pp. 168-171
H. E. Huntley, The Divine Proportion: A Study in Mathematical Beauty (New York: Dover Publications, 1970).

Project 6.9

In Example 2 Section 6.6, we assumed the width of the Parthenon to be 101 ft and found the height to be 62.4 (assuming the golden ratio). If you worked Problem 8, you assumed the height to be 60 ft and found the width using the golden ratio to be 97 ft. Are the numbers from Example 2 and from Problem 8 consistent? Can you draw any conclusions?

References:

George Markowsky, "Misconceptions About the Golden Ratio," The College Mathematics Journal, January 1992.

Project 6.10

The German artist Albrecht Durer (1471-1528) is not only a Renaissance artist, but also somewhat of a mathematician. Do some research on the mathematics of Durer.

References:

Charles Lenz "Modeling the Matrix of Albrecht Durer," Modeling and Simulation, 1979, pp. 2149-2161.
Francis Russell, The World of Durer, 1471-1528 (New York: Time, 1967).
Karen Walton, "Albrecht Durer's Renaissance Connections between Mathematics and Art," The Mathematics Teacher, April 1994, pp. 278-282.

Project 6.11

Write a paper on perspective. How are three-dimensional objects represented in two dimensions?

References:

Jan Garner, "Mathematics and Perspective Drawing," http://forum.swarthmore.edu/sum95/math_and/perspective/perspect.html
Morris Kline, Mathematics, a Cultural Approach (Reading, MA: Addison-Wesley, 1962), Chapters 10-11.
C. Stanley Ogilvy, Excursions in Geometry (New York: Oxford University Press, 1969), Chapter 7.

Project 6.12

The discovery and acceptance of non-Euclidean geometries had an impact on all of our thinking about the nature of scientific truth. Can we ever know truth in general? Write a paper on the nature of scientific laws, the nature of an axiomatic system, and the implications of non-Euclidean geometries.

References:

Philip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Houghton-Mifflin, 1981).