Cylindrical Mirror

1. Put the stiff paper backing behind the Mylar and slide both into the transparent page protector.
2. Bend the sandwiched Mylar lengthwise to form a portion of a cylinder. Make sure that the long side is parallel to the axis of the cylinder.

Hold the cylindrical mirror so that its long axis is horizontal. Curve the Mylar slightly and look into the mirror. Position yourself so that you can clearly see a reflection of your face. Notice how the image changes when you move closer to or farther from the mirror. When you move far enough away from the mirror, your image will flip upside down.

Wink your right eye. Which eye does the image wink? The image may wink its left eye or its right eye, depending on how far your face is from the mirror. When you are close to the mirror and your image is right side up, the image winks its left eye. When it’s upside down, the image winks its right eye. (If you have trouble deciding which eye in the image is winking, have someone stand beside the mirror and do what the image does—that is, wink the same eye as the image. Then ask your partner which eye it is. If the image is upside down, your partner’s head will have to be upside down, too. The easiest way for your partner to get this effect is to turn around, bend over as far as possible, and look back at you from there.)

Now orient the cylindrical mirror so its long axis is vertical. Notice how the image changes when you move closer to and farther from the mirror. Wink your right eye and notice how the image in the mirror responds. When you are close, the image will wink its left eye. When you are far away, it will wink its right eye.

You see the world because light gets into your eyes. You see these words, for example, because light from the screen or reflected off the printed page enters your eyes and makes an image on your retina.

When you make a visual picture of the world, you assume that the light entering your eyes has traveled in a straight line to reach you. But mirrors and other shiny objects change the path of the light, bouncing it back in an organized fashion. When you look into a mirror, you see your image because light reflecting from your face bounces off the mirror and back into your eyes (click to enlarge the diagram below). Your eyes and brain assume that the light has traveled in a straight line to reach your eyes, so you see an image of your face out there in front of or behind the mirror.

When you look into an ordinary flat mirror, the image of your face is right side up: Your hair is on top of your head and your chin is underneath. To reach your eyes, the light from your hair hits the mirror at a slight angle and then bounces into your eyes from above—which is why you see your hair on top and your image as right side up.

When you look into a cylindrical mirror with the axis of the mirror horizontal and with your face a foot or more away from the mirror, your image is upside down. That’s because the light from your hair bounces off the curved mirror and comes to your eyes from below.

To make sense of the angle at which the light is entering your eyes, your eyes and brain must see the image of your face as upside down and a little bit in front of the mirror (click to expand diagram below).

As everyone knows, a flat mirror reverses your right side and your left side. How does it do that? Suppose you are standing face to face with someone. If your right ear points toward the east, the other person’s left ear will point toward the east. Now, instead of facing another person, suppose you are facing a flat mirror with your right ear pointing to the east. The light from your right ear will bounce off the flat mirror and enter your eyes from the east. Even though your east ear is the east ear of the image, your right ear has become the left ear of the image! (Yes, this is a little mind-boggling at first reading. But once you get it, it will seem simple.)

Now look into the cylindrical mirror with its axis vertical. Stand at least a foot away from the mirror. Once again, place your right ear so that it points to the east. Light from your right ear bounces off the curved mirror and enters your eyes from the west. Light from your right ear appears to come from the right ear of the image. In this cylindrical mirror, you see yourself as others see you. You see the image of your face just a little bit in front of the mirror.

Here’s a classic, tricky question: “If a flat mirror reverses right and left, why doesn’t it reverse up and down?”

The answer is that a flat mirror actually reverses in and out. That is, light that travels “in” to the mirror is bounced back “out” of the mirror. This reversal does not change up into down, but it does change right into left. Consider the outline of the hand below. Is it a right hand or a left hand? You cannot tell which hand it is unless you know whether the palm of the hand is facing “in” to the page or “out” of the page. So right and left depend on in and out.

This Science Snack is part of a collection that showcases female mathematicians and math educators whose work aids or expands our understanding of the phenomena explored in each Snack.

Source: Wikimedia Commons

Sometimes math is symmetrical: 1 + 2 = 3 | 3 = 2 + 1. Sometimes, it’s not. Chelsea Walton (pictured above) is a mathematician who studies noncommutative algebra, a branch of math where changing the order changes the result. She also studies quantum symmetry, which is impossible to see in the real world. Mathematicians have excellent imaginations, and not always about math! Walton imagined more collaboration between women and nonbinary mathematicians, and she is making that vision come true. Explore symmetry in your own image with our Cylindrical Mirror Science Snack.