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> It turns out that if you rub two bars of metal (or pieces of glass) across each other with grinding paste in between, one surface naturally ends up convex, an

> It turns out that if you rub two bars of metal (or pieces of glass) across each other with grinding paste in between, one surface naturally ends up convex, and the other concave. The reason is that when the pieces aren’t directly over each other the same amount of downward force exerts a greater pressure on the portions that are still in contact, causing more material to be ground away at the edges of one piece to produce a convex surface, and from the center of the other to produce a concave surface. Does someone have a deeper explanation than this?

tsomctl 6y ago
Go read up on the three plate method to manufacture surface plates.
- XorNot 6y ago
  Learning about the 3-plate method was mind-blowing for me. I hadn't realized how much I was just accepting this huge gap in my knowledge of how we build modern technology, a ton of which centers on "so how do you get an initially flat surface to make a reference from if you don't have one?" Between that, and the lathe, you can build the world.
DoctorOetker 6y ago
suppose the 2 plates have the same dimensions, and you start grinding, by rubbing them over each other with dilute paste, then part of the time the edges are uncovered, while the centers are nearly always covered, so the centers should grind faster. and both try to go concave, this is however self-limiting: if the centers are ground slightly more than the edges, then the edges protrude and would grind faster again to catch up. the used motions translating and rotating in 2 dimensions, result in spherical surfaces being the only solutions, concave and convex, flat and flat, and convex and concave. its just that because of the initial phenomenon (the centers trying to grind faster than the edges, because the edges are unexposed part of the time) that you get spontaneous symmetry breaking and you get a concave and convex surface pair. this unstable equilibrium at flat can be made stable by using 3 surfaces, A,B,C and going through AB, BC, CA, and repeat, so that the only valid solution becomes an optical flat (but this is not how your window panes are made, they simply float liquid glass on a molten lead bath)
- pfdietz 6y ago
  The float glass process uses molten tin, not molten lead.
- jfengel 6y ago
  I'd have expected that the convex-concave and concave-convex solutions would be unstable, and that they'd average out to flat-flat. If I understand you, you're saying that you get symmetry breaking instead, so that once it starts with either concave-convex or convex-concave, it continues going down that route. Is that right? Is there a way to predict whether you get convex or concave on top, or do you just look at it after a while and say, "Huh, today I got one of these. Guess that's what I'll be looking at for the next N weeks of this project".
gwern 6y ago
Worth noting is https://ericweinhoffer.com/blog/2017/7/30/the-whitworth-three-plates-method https://ericweinhoffer.com/blog/2017/7/30/the-whitworth-thre...

tsomctl 6y ago

Go read up on the three plate method to manufacture surface plates.

XorNot 6y ago
Learning about the 3-plate method was mind-blowing for me. I hadn't realized how much I was just accepting this huge gap in my knowledge of how we build modern technology, a ton of which centers on "so how do you get an initially flat surface to make a reference from if you don't have one?" Between that, and the lathe, you can build the world.

DoctorOetker 6y ago

suppose the 2 plates have the same dimensions, and you start grinding, by rubbing them over each other with dilute paste, then part of the time the edges are uncovered, while the centers are nearly always covered, so the centers should grind faster. and both try to go concave, this is however self-limiting: if the centers are ground slightly more than the edges, then the edges protrude and would grind faster again to catch up. the used motions translating and rotating in 2 dimensions, result in spherical surfaces being the only solutions, concave and convex, flat and flat, and convex and concave. its just that because of the initial phenomenon (the centers trying to grind faster than the edges, because the edges are unexposed part of the time) that you get spontaneous symmetry breaking and you get a concave and convex surface pair. this unstable equilibrium at flat can be made stable by using 3 surfaces, A,B,C and going through AB, BC, CA, and repeat, so that the only valid solution becomes an optical flat (but this is not how your window panes are made, they simply float liquid glass on a molten lead bath)

pfdietz 6y ago
The float glass process uses molten tin, not molten lead.
jfengel 6y ago
I'd have expected that the convex-concave and concave-convex solutions would be unstable, and that they'd average out to flat-flat. If I understand you, you're saying that you get symmetry breaking instead, so that once it starts with either concave-convex or convex-concave, it continues going down that route. Is that right? Is there a way to predict whether you get convex or concave on top, or do you just look at it after a while and say, "Huh, today I got one of these. Guess that's what I'll be looking at for the next N weeks of this project".

gwern 6y ago

Worth noting is https://ericweinhoffer.com/blog/2017/7/30/the-whitworth-three-plates-method https://ericweinhoffer.com/blog/2017/7/30/the-whitworth-thre...