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

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 Revised First Apr. 2007 Apr. 2007 Mathematical Illusions

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

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

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

Hiding Pictures Within Pictures What’s hiding in the trees? Find the man’s face. Apr. 2007 Mathematical Illusions

Mathematical Illusions Morphing and Mutation   Apr. 2007 Mathematical Illusions

Morphing Images on a Computer See “Tiger Face Morph” and other video clips at: http://creativesam.tfcentral.com/Animations.html Apr. 2007 Mathematical Illusions

Mathematical Illusions Illusion of Motion Apr. 2007 Mathematical Illusions

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

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

Mathematical Illusions Impossible Designs Apr. 2007 Mathematical Illusions

Impossible Objects How many legs does this elephant have? Swedish stamp Apr. 2007 Mathematical Illusions

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 Apr. 2007 Mathematical Illusions

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

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

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

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

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