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A laser is suspended 1 metre above an infinite plane. The laser swivels on its aperture, which remains at point A. The aperture is directly above point B. The laser shines a infinitely small dot onto point C. We'll call the angle BAC θ, and the distance BC d. Therefore, d = tan θ.

As θ approaches 90°, d approaches infinite. In other words, the dot starts from B, and keeps moving along this plane. Since it moves an infinite distance, it is moving forever away from B. Since it is moving forever on this plane, θ never reaches 90°, and the laser dot never comes off the plane (after all, why would the dot continue going along this plane and suddenly come off it?).

From the above 'proof', the laser could never go past 90°. Yet reality contradicts this: anyone can grab a laser pointer and lift it past 90°; physically a laser wouldn't have a purely theoretical force stop it from moving past 90°.

So, what is wrong with the above proof? If there is a fallacy, please make close reference to exactly when and how the dot parts with the plane and ceases to exist.

Note: Assume the light emitted from the laser instantaneously arrives at its endpoint on the plane.

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This is very much Zeno's paradox. – InterestedGuest Feb 2 '11 at 7:37
I don't even see the paradox. If you think light can move infinitely fast, why can't dots? (There is actually much more of a paradox if you ask yourself how this experiment works if you believe in special relativity: doesn't the dot move faster than the speed of light at some point?) – Qiaochu Yuan Feb 2 '11 at 11:30
I'd appreciate a drawing, and I think I'm not alone, because the description is not clear at all. – Raskolnikov Feb 2 '11 at 17:17
up vote 10 down vote accepted

Of course, this is not a math problem because it has to do with physical objects, which math can only model, and only with varying degrees of appropriateness. In particular, there is obviously no such thing as an infinite plane in reality.

The answer is: The dot is off the plane when $\theta=90^\circ$. The dot is on the plane when $\theta<90^\circ$. One can choose real numbers that are arbitrarily close to $90^\circ$, without being equal; but this no more prevents the motion of the laser than the fact that there are real numbers arbitrarily close to 1 prevents the motion of a point moving with constant speed along the interval $[0,1]$ from reaching 1 in exactly the amount of time you would expect. This is just Zeno's paradox, only involving an angle, instead of a distance.

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