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.
Martin Gardner,"Mathematical Games," Scientific American,
Richard Gregory, "Visual Illusions," Scientific American,
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:
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
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
2. Square a circle:
Using only a straightedge and compass, construct a square
with an area equal to the area of a given circle.
This site has an interesting
interactive component to help you to understand the
problem. There are also links to other sites. Other
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