Why do telescopes use hexagonal mirror pieces instead of pie slice shaped ones? Some new large telescopes in the near future, on Earth and in space, will use multiple mirrors. The individual mirror pieces will be hexagonal in shape. I wonder why this shape is prefered over the circle sector shape (like pie slices). I would think that corners and joints between mirrors cause practical problems. And a circle sector slice has half as many corners and two rather than three or six neighboring pieces, and shorter interfacing edges with neighboring pieces overall. So shy is the hexagonal shape prefered?
The ideal shape for the mirror is round. It's the easiest to make. It's the best-behaved while in use.

The hex tiles are already harder. The mirror is a revolution surface generated by a conic curve (circle, parabola, hyperbola, ellipse), which needs to be machined with a precision greater than 0.1 microns. That's extremely difficult already with a round mirror, where the surface can be easily distorted near the edge in the process of manufacturing.

With the hex tiles, maintaining that precision near the corners is very hard - corners are like the edge but exponentially harder. Unless you take special precautions and unless you perform extra work, you're guaranteed to end up with surface errors near the corners (the surface is either too raised, or too low, or has a more complex distortion). Based on my experience making round mirrors, I would guess the surface near the corners would tend to be turned down (too low relative to the rest), which is a defect that's extremely hard to correct - one of the hardest, in fact.

With a pie slice, all problems mentioned above increase greatly in magnitude. The blunt angle of the hex tile gives you some support in machining the surface, but the sharp angle of the pie slice gives you almost none. I can't even begin to imagine the difficulties one would encounter while making a high precision surface on a pie slice. The sharp end of the slice would be almost guaranteed to be worse quality than the rest.



What most people don't realize is that maintaining the precise shape of the mirror while in use is difficult too. You can't just lay it on a hard surface - the uneven support would distort the precise shape. Mirrors are supported on the back by a precise mechanism, with a number of support elements precisely sharing the weight in equal amounts (or, if not equal, then deduced from finite element analysis), and located in places carefully calculated.

With a round or hex tile this is, if not easy, then at least a well-understood and manageable procedure. A pie slice would be very difficult to support in the same even manner near the sharp tip. You can't rely on the symmetry of the shape to provide the even load on the support points. The support force would have to be specially adjusted there. It would probably be doable, but harder.

This is only an issue with terrestrial telescopes, which operate under gravity. Space telescopes don't suffer from this issue.



Alternatives to hex tiles could also be square tiles, or triangle tiles. These are worse than hex tiles for all the reasons shown above - the corners have sharper angles which make everything more difficult. The blunt angles of the hex tiles alleviate these issues somewhat.

TLDR: Non-round mirrors are hard to make anyway, but hex tiles are less hard compared to the alternatives.