f38 math illusions

About This PresentationEdition Released Revised Revised This presentation is part of the “Math + Fun!” series devised by Behrooz Parhami, Professor of Computer Engineering at University

Mathematical Illusions

A Lesson in the “Math + Fun!” Series

About This Presentation

Edition Released Revised Revised

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

Things May Not Be What They Seem

1 Which of the two

vertical lines is longer?

Activity 1: Appearances May Be Deceiving

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”

2 Which of the two white

squares is larger?

Explain how the area of the figure

increased from 99 to 100

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?

Hiding Pictures Within Pictures

What’s hiding in the trees?

Find the man’s face.

Morphing and Mutation

Morphing Images on a Computer

See “Tiger Face Morph” and other video clips at: http://creativesam.tfcentral.com/Animations.html

Illusion of Motion

Ambiguous Images

Do you see a vase

or two faces?

A

B

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

Man playing

saxophone

or girl’s face?

Stairs: top side

or bottom side?

Rabbit

or old woman?

What Do You See in These Pretty Patterns?

Fix your stare at a specific point and describe what you experience

Impossible Designs

Impossible Objects

How many legs does this elephant have?

Swedish stamp

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?

3D Illusions

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

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

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?

Activity 5: Making People Appear and Disappear

1 Count the people in the picture to

the right and write the number down

1

3

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)

4 Recount the number of people

Next Lesson

Late May or early June 2007