Formica is a brand of thin melamine sheet - the traditional stuff of kitchen worktops.
Corian is much thicker, usually 12 mm, and is a resin sheet made up with powdered mineral. Used in much the same way - but the results more closely resemble solid stone worktops.
But its also used for things like shower enclosures - and particularly in hotel bathrooms, often with washbasins integrated into a seamless continuous surface.
Officially it's impossible. Du Pont are even pissy about where the waste offcuts go, and making sure that none of them end up in the hands of evil woodturners to make pens out of.
OTOH, Corian hasn't been the only game in town for some years now and the other brands are much less fussy. Most of it's The Other Brand (sheets are in the shed, but I've forgotten the name). I bet the eBay stuff isn't Corian, but you'll be hard-pressed to tell them apart. Watch out for signage grade though, as there are grades for making high-end outdoor signs that don't have anything like as much mineral content as the countertop grades and are far softer and risky for heat damage. The top of my waterproof sharpening bench has our old office address carved in the underside, following an office move and some skip-diving.
I don't know any Corian installers, but I do know two separate ex- Corian installers who got fed up of Du Pont's attitude and switched suppliers.
It's not hard to work it and you don't need to go on the course, but you do need to study the instructions, practice on scrap first and most of all, use the right tools and materials. These do make a difference. If you go to a posh restaurant in Bristol there's a vastly pretentious washbasin rig in the toilets and great big scored rings around all the basins where someone used a bearing-guided cutter with a hard bearing, not a proper soft bearing (more likely, they used a favourite bodge of wrapped insulation tape around it - which doesn't work on tapered basins, as the edge cuts through).
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