1. Mathematical Illusions
A Lesson in the “Math + Fun!” Series
Apr. 2007
Mathematical Illusions

2. 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 ( ). “Math + Fun!” material can be used freely in teaching and other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami
Edition
Released
Revised
First
Apr. 2007
Apr. 2007
Mathematical Illusions

3. Things May Not Be What They Seem
Do you see circles or spirals?
Are some of the squares bigger?
Apr. 2007
Mathematical Illusions

4. 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.
The squares example is akin to a goldfish appearing larger when placed in a small fishbowl.
Explain how the area of the figure increased from 99 to 100.
Apr. 2007
Mathematical Illusions

5. Mathematical Illusions
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

6. Hiding Pictures Within Pictures
What’s hiding in the trees?
Find the man’s face.
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Mathematical Illusions

7. Mathematical Illusions
Morphing and Mutation
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Mathematical Illusions

8. Morphing Images on a Computer
See “Tiger Face Morph” and other video clips at:
Apr. 2007
Mathematical Illusions

9. Mathematical Illusions
Illusion of Motion
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Mathematical Illusions

10. Mathematical Illusions
Ambiguous Images
Do you see a vase or two faces? A B
Is Point A closer to us than B or further back?
Stairs: top side or bottom side?
Man playing saxophone
or girl’s face?
Young woman or old woman?
Rabbit or bird?
Apr. 2007
Mathematical Illusions

11. 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.
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Mathematical Illusions

12. Mathematical Illusions
Impossible Designs
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Mathematical Illusions

13. Impossible Objects How many legs does this elephant have?
Swedish stamp
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Mathematical Illusions

14. Illusion of Distortion
Horizontal lines appear slanted in the middle portion of the diagram
Which line on the left is a continuation of the one on the right?
Vertical lines appear crooked, but they are perfectly straight
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Mathematical Illusions

15. Mathematical Illusions
3D Illusions
Apr. 2007
Mathematical Illusions

16. 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

17. 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

18. 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?
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Mathematical Illusions

19. 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). 1 3 2
4. Recount the number of people and explain what happened.
Apr. 2007
Mathematical Illusions

20. Mathematical Illusions
Next Lesson
Late May or early June 2007
Area and Tilings
Apr. 2007
Mathematical Illusions