Everywhere I look, it is recited as a quasi-mantra that "segmented mirrors are much lighter than monolithic mirrors", and I can't manage to find an explanation anywhere no matter how hard I look. To me, getting a monolithic mirror and cutting it up should provide strictly no weight savings.
I get that it's not exactly "cutting up a monolithic mirror", and I understand that there are many other advantages to segmented mirrors, but where does this weight economy come from?
A telescope's precision mirror needs to be very large, to collect lots of light, and needs to be very precisely shaped. Holding that proper shape is based on the mirror's ability to resist stress on its physical structure. A paper-thin mirror would be wobbly, every bump and movement would make it ripple and blur the image. But the biggest challenge a mirror has to deal with on earth is trying not to bend under its own weight. Even with a cradle to hold the mirror, a small deviation can affect the whole telescope's performance.
And that's where smaller mirrors are useful, because the square-cube law comes into play. A mirror's mass (which creates most of the forces it needs to deal with) is based on the cube of its size, but its structural strength (what resists those forces) is related to the cross-section, which increases by only the square of the size. Or to put that another way, a small mirror needs less thickness to have the same resistance to deformation, which means less overall weight. The composite mirror can be thinner relative to its overall width.
In theory, a single large mirror could be made thinner by cutting away some of the "back" of the mirror and then supporting it in a cradle or other support structure instead of depending on its internal strength (and indeed single large mirrors usually do need a special support cradle). But if you're going to make your mirror thin enough that it can't really support itself, you might as well just slice the big thin mirror into easy-to-handle pieces and mount them in the support structure independently, and that's just what a composite mirror is.
The down-side to this is that you have to mechanically support the mirror segments and hold them in a very precise alignment, but even that is often a feature rather than a bug. For space telescopes like the JWST, a segmented mirror can stow for launch in a much smaller space and then deploy on-orbit, and any Hubble-like mirror aberrations can potentially be managed by adjusting the support structure instead of needing corrective lenses or a whole new mirror. For earthly telescopes, a segmented mirror with a clever control system can functionally "flex" during observation to counteract the effects of atmospheric distortions and other sources of error.
When creating a mirror, you need to take a block of glass and grind and polish it until it has the desired shape. For telescopes this is typically a parabolic shape. So The thickness of the slab of glass depends on the size of the mirror as you still need a finite thickness at the center while you only remove very little at the edges.
If you now create the same diameter telescope from a total of 7 segments, the same reasoning applies - but to each piece separately. Thus you can save a lot of weight in the glass the mirror's surface is vapour-deposited on. Of course you still need to correctly align the 7 elements with respect to eachother, but the overall weight including the support structure will be much less than a giant solid piece of glass.
A similar reasoning also applies if you do not manufacture your mirror from a slab of glass but other material.
The bottom line is that an (active) support mechanism to hold a segment in the right position will be lighter than the (passive) glass required if the mirror were not segmented.
There have been precursors of this for a long time, in the form of local support mechanisms with counterweights that compensated for sag as the mirror was repositioned.
