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Have you ever noticed that big windows on buildings are bendy and wavy - it's like the stereotypical look of modern architecture - and have you ever looked at those wavy windows and wondered… if the reflections are so wobbly and distorted from the outside, why can people see just fine looking out from the inside? In fact, when I gently touch this window from the outside, I can easily distort the image in the window, but nothing of the sort happens when looking out from the inside. It turns out this wavy window paradox is explained by a key difference between the physics of looking out and looking in.
The fact that windows work at all is a minor miracle of physics: light behaves like a wave, so when it passes from a material where it's traveling one speed (like air) to a material where it's traveling another (like glass), it ends up bending towards the slower material. There's a mathematical formula called Snell's law for the amount light bends - it depends on the incoming angle and the relative speed of light in the materials, and while the details are important if you're, say, designing a camera lens, they aren't important at all for our window question - what matters to us is
that the formula is completely symmetric - if the materials are reversed, then the angles reverse. Which means, for light passing through a glass window, the light first passes from air to glass and is bent inwards, then passes from the glass back to the air and is corrected back outwards the exact same amount… light passing through a window ends up traveling the exact same direction that it started, just offset a little bit. So as long as you have air sandwiching both sides of a layer of glass, the glass can rotate or bend quite a lot and not really affect your ability to see through it! In simple terms, you can see through a window. If the glass surfaces aren't perfectly parallel,
then the relative angles do change a bit: if the surfaces are, say, a degree off from one another, then the image you see will be displaced by about half a degree, so if the window varies in thickness you will see some distortion, like what you see looking out of old windows. However, when you're looking at the wavy reflections of trees from the outside of a building, you're of course not looking at light passing through the glass - you're looking at light bouncing off the glass, and that makes all the difference. Reflections only involve one surface, so they don't have a chance to get "corrected" the way
light does on its way out through the back of a window. A beam of light coming in at an angle relative to the glass bounces off at the same angle, just flipped, so a 1 degree deflection of the glass results in a 2 degree change in the apparent position of the light source. For example, if you're 10 meters away from a window, a 1 degree deflection of the glass will move the apparent position of the sun vertically upwards 70cm! You can see the same effect by tilting a laptop screen towards you - the apparent positions of reflections move upwards. So if window glass is a little bit wavy or curved the light passing through it will
be essentially fine, but the images reflecting off of it will be moved around inconsistently. It doesn't take much to move the reflection - gently touching a window can result in surprisingly large distortions of the reflections in it! You can try it - gently - the next time you're in front of a window or mirror! So the solution to the wavy window paradox is that the path of light passing through a window is miraculously resilient to deviations of the surface, while reflections off of it are not resilient. And during the day when it's bright outside, if we're outside a window looking in, we're typically looking at reflections, while if
we're inside we're looking out through the window. At nighttime, the situation is reversed and we're more often looking at easily distorted reflections when we're inside, and we're looking in through the window from the outside. And that's it: the next time you successfully look through a window, whether in or out, remember to say thank you to the symmetry of Snell's Law!
