Mathematical Illusions A Lesson in the “Math + Fun!” Series Apr 2007 Mathematical Illusions Slide 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 Things May Not Be What They Seem Do you see circles or spirals? Are some of the squares bigger? Apr 2007 Mathematical Illusions Slide Activity 1: Appearances May Be Deceiving Which of the two vertical lines is longer? Which of the two white squares is larger? 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 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 Hiding Pictures Within Pictures Find the man’s face What’s hiding in the trees? Apr 2007 Mathematical Illusions Slide Morphing and Mutation Apr 2007 Mathematical Illusions Slide Morphing Images on a Computer See “Tiger Face Morph” and other video clips at: http://creativesam.tfcentral.com/Animations.html Apr 2007 Mathematical Illusions Slide Illusion of Motion Apr 2007 Mathematical Illusions Slide 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 Count the people in the picture to the right and write the number down 2 The upper half of the picture is redrawn below Cut out the rectangular pieces and by cutting along the dashed lines Now, put the two cut-out pieces in reverse order on the picture above, covering the original pieces (piece on the right and piece on the left) Recount the number of people and explain what happened Apr 2007 Mathematical Illusions Slide 19 Next Lesson Late May or early June 2007 Apr 2007 Mathematical Illusions Slide 20 [...]... 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... 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... 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... 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