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Apr. 2007 Mathematical Illusions Slide 1

Mathematical IllusionsA Lesson in the “Math + Fun!” Series


About This Presentation

Edition Released Revised Revised

First Apr. 2007

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


Do you see circles or spirals?

Things May Not Be What They Seem

Are some of the squares bigger?


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” square.

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 bird? Young woman

or old woman?


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.


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 DistortionIn 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.

1

3

2

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 and explain what happened.


Next LessonLate May or early June 2007