Chapter 7 Projects

Section 7.1 Individual Research

Individual Research Project 7.1

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

(From Problems 1-3 of Section 7.1.)
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

Individual Research Project 7.2

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

Individual Research Project 7.3

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

A second basic aestheometric design (see Project 7.3) 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.

Individual Research Project 7.4

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.gap-system.org/~history/HistTopics/Trisecting_an_angle.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.