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Sadly there is precious little info on these machines, which appear in the novel Rainbows End.

A character who works on a transport tray for shop class comments on "solid-state robots" while looking through a parts catalog- I'm guessing these are machines that somehow do mechanical work without moving parts. In chapter 8 they apparently 'sound like ceramics when under stress' and mention something about microgrooves being the part that interacts with other objects.

How do they work? Are there any real-life equivalents?

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Al that we know for certain is that it's got an apparently smooth surface that can normally move objects, but not very quickly, and that initially Gu didn't trust them because he couldn't see how they worked. I don't think it indicates exactly how the mass is driven, but a few possible sources could be nanotechnology "cilia" (basically the surface has a lot of tiny cylinders that can be moved independently) or it's got something to do with accelerating the mass through magnetics or another force. The former seems more likely with how the description is generally of things moving pretty slowly with precision, but does not fit with Gu overclocking it to create a weapon.

So, in short, they were initially unknown technology to the kids, but Gu, with his intuitive abilities, is able to comprehend them enough to modify them.

As for contemporary non-fictional equivalents, there has been some study of creating sheets of cilia such as these to move objects, but it's still fairly primitive.

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As I recall not too long before the book came out the notion of using surfaces covered with dense collections of MEMS (in particular little actuater lever in groups that collectively act in the same direction but different patches point in different direction) to move macroscopic objects around in fairly arbitrary ways was hot.

The trays are presumably a extrapolation from that idea. Their working surfaces are covered with vast number of little actuators and they can slide things along, turn them around on the surface and so on.

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