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We’d like to know why you’re staring at yourself in a spoon in the first place. But we can at least answer the question of why you look upside-down when you do.

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Sources: http://www.quora.com/Why-is-a-reflection-upside-down-in-a-spoon
https://van.physics.illinois.edu/qa/listing.php?id=1985
http://www.whyzz.com/why-is-my-reflection-upside-down-when-i-look-in-a-spoon
Why am I upside-down in a spoon?   I would like to answer this question with another question, which is: Why are you all staring at yourselves in spoons?   And how much time are you spending doing this?   I have to wonder, because we’ve been asked this particular question pretty often.    Anyway, if you look at yourself on the inside of a spoon, you will see yourself upside-down- also reflected so that if you raise your right hand it, like, goes up on the left.   But flip the spoon around so that you’re looking at the outside, and you will see yourself right-side-up, though a little distorted. And, like, super-long.   It seems kind of odd that flipping over a spoon should completely change your reflection, but there is a perfectly reasonable explanation for it.    And, of course, it’s because of physics.   Everything we see is just a result of photons of light hitting our retinas. Our brains translate all of that activity into information about things like color and brightness and distance.   If you stand in front of a regular flat mirror, you’re just seeing all those photons bouncing directly off of the mirror into your eyes.   The way those photons bounce depends on the surface that’s reflecting them.   So let’s just imagine instead of a photon it’s just a ball, and you’re throwing the ball at the wall. If you live in a hypothetical universe without gravity, that ball should come straight back at you.   But if the wall were a concave surface -- that is, tilted toward you on the top and bottom, so that it’s curved like a kind of giant spoon -- the ball would bounce differently.    If the ball flew straight at the top of the curved wall, it would bounce back down toward the floor, smacking you in the legs.    If it shot straight at the bottom, the ball would bounce up, and suddenly you got a bloody nose.   These same rules apply for light and reflective surfaces -- you’ve just got to substitute photons for balls.   If a mirror is concave, like the inside of a spoon, the photons will bounce back at an angle, just like the balls did.   The ones that hit the top deflect downward, and the ones that hit the bottom deflect upward. And the ones that hit the right deflect left and the ones that hit the left deflect right. And all those deflected paths cross at what’s known as the focal point.    If you’re further away from the spoon than the focal point, which you definitely are unless your spoon is very very large, or you are extraordinarily tiny, then by the time the photons hit your eyes, they form an upside-down image.   Now, the back of the spoon doesn’t have this problem. That surface is convex, so the center of the surface is closer to you than the edges.    Even though the photons are still being deflected at an angle, that angle just stretches out the picture instead of flipping it.    So if you’re really desperate for a way to admire your reflection -- or just super bored at dinner -- the back of the spoon is probably the more reliable picture.   Thanks for asking, this episode was brought to you by our patrons on Patreon. If you’d like to learn how you can help us keep making more great SciShow, you can go to Patreon.com/SciShow!