A telescope with a bunch of small mirrors Together with colleagues we got this question. Imagine to take small mirrors, the size for example of a dentist mirror, and stick them to a wooden frame with a parabolic shape. Each mirror is flat, and once set of the frame it can be aligned properly with screws so that it redirects its reflection towards the focus of the parabola. All the parabolic surface is covered with these small mirrors.
Now, if you build a Newtonian telescope out of this system, would you get a decent telescope? I expect it to be affected by various aberrations, but would it work ? 
 A: The telescope you describe would work in some sense, but it depends strongly on how you define what it means for a telescope to work.
Combining many mirrors to behave as a single large mirror is possible, but those individual mirror segments must still be appropriately curved to achieve good optical performance. When building telescope mirrors for modern astronomy, we are concerned with deviations of the mirror shape from the ideal curve by distances on the order of nanometers. If the whole mirror surface were composed of small flat disks, that would be like using a mirror that deviates from the ideal shape by many millimeters, over it's whole surface!
This sort of thing would be fine for collecting light, and in fact similar systems are used in solar power generation, where one only needs to collect light at some central location, rather than produce an image. In a telescope however, you need very high optical quality to do much of anything.
A: Visit http://www.tmt.org/
You'll find an amazing telescope being build ....
 
From http://www.vikdhillon.staff.shef.ac.uk/teaching/phy217/telescopes/phy217_tel_active.html:

Segmented mirrors are the only feasible way of constructing telescopes with apertures significantly in excess of 8 m, as monolithic mirrors would become extremely expensive and ultimately impossible to manufacture, transport, install and maintain. The disadvantage of segmented mirrors is that they often require asymmetric profiles, making them difficult to manufacture. The active optics systems required to support them is also complex, and the gaps between the segments (typically a few mm) can cause diffraction effects and increased infrared background in the final image. 

A: (I have some experience with telescope mirror making.)
It would not work as described. It's not even close.
In a reflector telescope, the primary mirror needs to satisfy the Rayleigh criterion, which states that the surface error must not exceed λ/4. For visible light, that means 100 nanometers.
The ideal shape of the primary mirror in regular newtonian scopes is a paraboloid, so no matter what technique you use to make the mirror, the resulting surface needs to follow the ideal parabola with an error no greater than 100nm (due to Rayleigh).
For a mirror of reasonable aperture and focal length, your flat segments would diverge from the parabolic surface by an error far greater than 100nm, more like fractions of millimeter or so, which is about 1000x bigger than the acceptable value.
There's just no way around it. Each part of the mirror needs to track the ideal parabola with no more than 100nm error. Otherwise the mirror is not usable for visible light.
The technique you describe would be usable for microwaves. Since the wavelength in that case is on the order of 1cm or more, λ/4 would be a few millimeters. In that case, a wide shallow parabolic bowl, having multiple flat metal disks glued on the inside might work well enough. If the size of each metal disk is comparable to the wavelength, then it would definitely work well. Just make sure the error in each point is no greater than λ/4.
(Okay, you would then get some resonance effects in each disk itself, which would change the overall result, but let's not go there. Bottom line: it's feasible with microwaves, not feasible with visible light.)
If you don't need to create an image at focus, but just collect the energy of a microwave beam, then the error could be bigger than λ/4; the only ill effect would be that your reflector would become less and less efficient as the error increases.
One more thing: In theory, a mirror with an overall surface error of exactly λ/4 could have a total efficiency of exactly zero, but that's a pretty pathologic (unrealistic) case. In most real life situations, a few λ/4 zones here and there simply reduce the overall energy-gathering power. So keep the error down and it will work well.
A: Somehow all answers so far failed to mention that a segmented mirror consisting of flat segments could in principle yield a high optical quality (diffraction-limited) telescope provided the mirror segments are small enough. 
Spherical (or hexagonal or any other compact shape) flat mirrors would work perfectly provided: 
1) the flat segments are positioned so as to deviate no more than a fraction of the wavelength from a perfect parabolic shape, and
2) the surface area of the segments doesn't exceed (by more than a few times) the product of the focal length of the mirror times the wavelength of the light.
Practically, you will need many flat segments (a large telescope with 30 m focal length would need centimeter-size flat segments) and it will be a nightmare to position all these flat segments more accurately than a wavelength (0.5 micrometer), and to keep them in these positions and orientations despite mirror orientation and temperature changes.
