见
jiàn
看 is aiming your eyes; 见 is what lands in them
The picture Your eyes and ears are windows left open. 见 is the moment something actually comes in through one. 看 is pointing your eyes at it; 看见 is it landing inside them — which is why 看了半天也没看见 makes sense.
Where the picture shows up in 见
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reaching the ear
你听见了吗？有人敲门。
nǐ tīng jiàn le ma? yǒu rén qiāo mén
Did you hear that? Somebody's knocking.
A knock coming in through your ear and turning your head.
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not reaching the ear
说大声点，我听不见。
shuō dàshēng diǎn, wǒ tīng bu jiàn
Speak up a bit, I can't hear you.
A voice buried in noise, breaking up before it can get over the sill of your ear.
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reaching the eye
我刚才看见他进电梯了。
wǒ gāngcái kàn jiàn tā jìn diàntī le
I saw him getting into the lift just now.
A back stepping into the lift landing inside your eye and sticking there.
看 is aiming your eyes; 看见 is the thing landing in them. That is why 看了半天也没看见 makes sense.
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reaching the nose
我闻见煤气味儿了。
wǒ wén jiàn méiqì wèir le
I can smell gas.
Gas drifting in through your nose and stopping you mid-step.
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reaching each other's eyes
那明天见！
nà míngtiān jiàn
See you tomorrow, then!
Tomorrow, each of your faces coming in through the other's eyes.
All of it in one picture
见 is one picture: something actually getting in through a window that is always open. A knock that gets in through the ear is heard; a voice blocked by noise never gets in at all. A back that gets in through the eye is seen; gas that gets in through the nose is smelled. Even 明天见 is a promise that tomorrow your faces will get in through each other's eyes. So 见 never means aiming a sense at something — it means the something got through.