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

Apr. 2007Mathematical IllusionsSlide 2 About This Presentation EditionReleasedRevised FirstApr 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

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

Apr. 2007Mathematical IllusionsSlide 4 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.

Apr. 2007Mathematical IllusionsSlide 5 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. 2007Mathematical IllusionsSlide 6 Hiding Pictures Within Pictures What’s hiding in the trees? Find the man’s face.

Apr. 2007Mathematical IllusionsSlide 7 Morphing and Mutation

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

Apr. 2007Mathematical IllusionsSlide 9 Illusion of Motion

Apr. 2007Mathematical IllusionsSlide 10 Ambiguous Images Do you see a vase or two faces? ABAB 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?

Apr. 2007Mathematical IllusionsSlide 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.

Apr. 2007Mathematical IllusionsSlide 12 Impossible Designs

Apr. 2007Mathematical IllusionsSlide 13 Impossible Objects How many legs does this elephant have? Swedish stamp

Apr. 2007Mathematical IllusionsSlide 14 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. 2007Mathematical IllusionsSlide 15 3D Illusions

Apr. 2007Mathematical IllusionsSlide 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. 2007Mathematical IllusionsSlide 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. 2007Mathematical IllusionsSlide 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?

Apr. 2007Mathematical IllusionsSlide 19 Activity 5: Making People Appear and Disappear 1. Count the people in the picture to the right and write the number down The upper half of the picture is redrawn below. Cut out the rectangular pieces 1 and 2 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 1 on the right and piece 2 on the left). 4. Recount the number of people and explain what happened.

Apr. 2007Mathematical IllusionsSlide 20 Next Lesson Late May or early June 2007