f38 math illusions

Mathematical Illusions
A Lesson in the “Math + Fun!” Series

Apr. 2007 Mathematical Illusions

Slide 1


About This Presentation
This presentation is part of the “Math + Fun!” series devised
by Behrooz Parhami, Professor of Computer Engineering at
University of California, Santa Barbara. It was first prepared
for special lessons in mathematics at Goleta Family School
during four school years (2003-07). “Math + Fun!” material
can be used freely in teaching and other educational
settings. Unauthorized uses are strictly prohibited. ©
Behrooz Parhami
Edition

Released

First

Apr. 2007

Revised

Apr. 2007 Mathematical Illusions

Revised

Slide 2


Things May Not Be What They Seem

Do you see circles or spirals?

Are some of the squares bigger?

Apr. 2007 Mathematical Illusions

Slide 3


Activity 1: Appearances May Be Deceiving
1. Which of the two
vertical lines is longer?
2. Which of the two white
squares is larger?
3. Draw a 9” × 11”
rectangle and cut it along
the dashed line, as shown.
Slide the lower piece down
and to the left. Cut out the
small triangle that sticks out
at the lower left and use it to
fill the opening at the upper
right. You get a 10” × 10”
square.

Explain how the area of the figure
increased from 99 to 100.

Apr. 2007 Mathematical Illusions

Slide 4


Activity 2: Does 64 Equal 65?

Cut out an 8” × 8” square as shown and rearrange the pieces to form
a 5” × 13” rectangle, thus proving 64 = 65. What is wrong with this?

Apr. 2007 Mathematical Illusions

Slide 5


Hiding Pictures Within Pictures
Find the man’s face.

What’s hiding in the trees?

Apr. 2007 Mathematical Illusions

Slide 6


Morphing and Mutation

Apr. 2007 Mathematical Illusions

Slide 7


Morphing Images on a Computer

See “Tiger Face Morph” and other video clips at: />
Apr. 2007 Mathematical Illusions

Slide 8


Illusion of Motion

Apr. 2007 Mathematical Illusions

Slide 9


Ambiguous Images

A

Do you see a vase
or two faces?

Man playing
saxophone
or girl’s face?

Stairs: top side
or bottom side?

Rabbit
or bird?

Apr. 2007 Mathematical Illusions

B

Is Point A
closer to us
than B or
further back?

Young woman
or old woman?

Slide 10


What Do You See in These Pretty Patterns?

Fix your stare at a specific point and describe what you experience.
Repeat the process, this time slowly moving your head back and forth.

Apr. 2007 Mathematical Illusions

Slide 11

Impossible Designs

Apr. 2007 Mathematical Illusions

Slide 12


Impossible Objects

Swedish stamp

How many legs does
this elephant have?

Apr. 2007 Mathematical Illusions

Slide 13


Illusion of Distortion

Horizontal lines appear slanted in
the middle portion of the diagram
Vertical lines appear crooked,
but they are perfectly straight

Which line on the left
is a continuation of

the one on the right?

Apr. 2007 Mathematical Illusions

Slide 14


3D Illusions

Apr. 2007 Mathematical Illusions

Slide 15


Illusion of Distortion or Deformation
The figure on the right
spins when you view
this page in a slide show.
Notice how the shape is
deformed as it spins.

Apr. 2007 Mathematical Illusions

Slide 16


Activity 3: Experiencing Distortion
In each of the following diagrams, use a ruler and pencil to connect the
black dots on either side using two horizontal lines. Explain what you see.

Apr. 2007 Mathematical Illusions

Slide 17


Activity 4: Where Does the Sixth Box Go?
Cut along the dashed line . . .

and shift the bottom half to left, as shown below.

What happened to one of the boxes?

Apr. 2007 Mathematical Illusions

Slide 18


Activity 5: Making People Appear and Disappear
1. Count the people in the picture to
the right and write the number down.

1

2

2. The upper half of the picture
is redrawn below. Cut out the
rectangular pieces 1 and 2 by
cutting along the dashed lines.
3. Now, put the two cut-out

pieces in reverse order on the
picture above, covering the
original pieces (piece 1 on the
right and piece 2 on the left).

3

1

2

4. Recount the number of people
and explain what happened.
3

Apr. 2007 Mathematical Illusions

Slide 19


Next Lesson
Late May or early June 2007

Apr. 2007 Mathematical Illusions

Slide 20