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From this video (relevant timestamp included in URL):

https://www.youtube.com/watch?time_continue=768&v=ABRysNzcdvw&feature=emb_title

It shows how rubbing two surfaces together by hand with abrasive powder creates a top concave surface and a bottom convex surface for mirror production.

Could someone explain the physical reasoning (in terms of mechanics) for why this happens and why the shapes for the top and bottom are not the other way round?

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Let's say you start with two identical disks, 1 and 2. You start by slightly tilting disk 1 and grinding it with its center along the edge of disk 2. In this way, the edges of disk 2 as well as the center of disk 1 are ground down. In other words, disk 1 starts becoming slightly concave, while disk 2 starts becoming slightly convex.

What happens when we continue grinding the two disks together? Well, more material will be ground away until the two disks fit perfectly onto one another in every orientation encountered during the grinding process (rotating, tilting). The only surface that has this property is a spherical one. Hence, the end result after grinding for a long time will be two (nearly) perfect spherical lenses, one concave and one convex.

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  • $\begingroup$ Why wouldn't it results in two perfectly flat surfaces though? Because any surface irregularity would be ground away, so wouldn't it result in two flat mirrors? I don't understand what is meant by 'tilting' here. $\endgroup$
    – XXb8
    Mar 31, 2020 at 9:57
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    $\begingroup$ Indeed if you keep the two disks parallel to each other you would just flatten/smooth them, so the tilting is critical. A drawing or two would make things clearer, I will try making some when I have some time. $\endgroup$
    – Jeff
    Mar 31, 2020 at 12:50

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