system
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human
eli5: Why does grinding two objects of the same material (e.g iron) not produce a flat surface?

So I read about this thing called the "automatic generation of gages". Basically, it's a process of making three flats surfaces by manually rubbing them against each other in pairs. You can't get them truly flat with two pieces cause it causes belling and curvature between the pieces. i.e. one would be concave while the other convex.  What I don't get is why. Because even if all parts receive an equal number of strokes or the cutting powder is equally spread between them, it won't work. So what causes the curvature? 

Maybe I should first ask how grinding works when they are both made of the same material. If one was harder than the other it would just cut away from the softer material. But when they are the same material how does it work? is it that they are not equally hard everywhere or something? And why does it specifically lead to a curvature. 

Apologies if the answer is obvious.
gpt
After just two plates are lapped against each other, all you can guarantee is that the two surfaces agree with each other, not that they're flat. They could be, but it's both possible and likely that the surfaces are curved. There's far more ways for there to be some error from a flat plane than there are for them to both be a  flat plane even in the hypothetical case. If you start with two curved surfaces of equal hardness that will wear down at the exact same rate, then unless the curvatures are the exact same you're going to end up with one slightly concave and the other convex. For example if both started convex, but plate A was more convex than plate B, then at some point you'll have worn plate B flat, while plate A remains convex. As you continue to lap them against each other, plate B will end up concave.  

In the real world, the material has inconsistency baked in; one bit might be a bit harder or softer which will more or less guarantee the surfaces can't wear flat. It might still be pretty flat, but when you're trying to make high precision instruments 'pretty good' wont cut it.


Withworth's three plates method gets around this by using a procedure that cancels the errors out. Plate 1 and 2 can meet at a curve or a plane. Same with any other pair. But if any two meet at a curve, one has to be concave and the other convex. Your third plate is then either flat, concave or convex, and wont agree with at least one of the other two. You either have a flat surface compared to a curved surface, or two concave or convex surfaces which curve away from each other. So by grinding them against each other in the right order you can reduce the curvature of each one (concave against concave will flatten both for example) and when you're done you know the only way all three can agree with each other if the surfaces are all planes.  

You can do it with two pieces if you want to break out the engineers blue and have a reference surface you know is flat; by varying pressure you can ensure any curves between the two are worn away till flat, comparing against the reference to make sure you're wearing away the right sections. The cleverness of the three plates method is how it guarantees you a flat surface to high precision from essentially nothing.