My climbing partner and I are at the top of a pitch and we drop a belay plate to the ground. There is no safe descent, we were planning on rappelling down off safe anchors.
What do we do?
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If you drop a belay plate you can use a Munter Hitch to descend down the rope.
It works like a belay plate so if you hold your hand close to your leg it will lock off. Moving it forward releases the slip knot allowing you to rappel. You can use a munter with two ropes (or one rope in half) or a single rope.
It's a good idea to attach a Prusik as a back up if you can.
Here's a video describing the whole process in detail.
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One method is to build a brake out of carabiners. The minimum equipment for this is three oval biners plus a locking biner, and the diagrams below show how to construct the system with only this many biners. However, this setup doesn't give much friction, especially with a thin rope, and normally people use at least one more oval, as described in the text below.
To construct the brake start with the usual large locking biner clipped to your belay loop, as you would for any rappel. The diagrams show a double-strand rappel.
As noted previously, the setup shown in figure 4 doesn't really give much friction. If you have a fourth oval biner D, it's a good idea to add it next to C, so that they form a sort of double-thickness crossbar. Both C and D should have their spines touching the rope, not their gates. (They can be reversed, but not opposed.) Only A and B should be opposite and opposed.
Don't try to do without the locking biner. If you simply clip A and B to your belay loop, the rope will rub against your belay loop and possibly destroy it through intense frictional heating. It is OK, however, to replace the locking biner with a pair of nonlocking biners, opposite and opposed.
The standard braking position for this standard setup is to hold the brake strand down at your hip, just as you would with an ATC. If you raise the brake strand to a higher angle, it could press against the gates of C and D, possibly causing them to open.
There are various ways of increasing the friction. If you have a huge load to lower and lots of oval biners, you can build two of these brakes in series. You can rearrange the brake bars to send the rope through in an S shape. In comments, Steed suggests using a locking biner for C, in which case it becomes safe to brake with the brake strand raised.
A possible advantage of the carabiner brake over a Munter is that the Munter will tend to twist up your rope. A possible disadvantage is that you may not have enough oval biners.
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This topic will be incomplete without mentioning the good old dulfersitz method used by our fathers when there were no belay plates and no carabiners.
This method doesn't require any equipment other than the rope itself. And, well, sturdy clothes.
The method is to pass the rope around your body in a special way shown in the picture 1 below. Picture 2 shows a variation with a carabiner.
This is a "last resort" type method, because it is quite hard on your body on inclined slopes and simply painful on verticals. But it works and you can even use a prusik to back it up.
Because the rope presses hard on your leg and eats up your clothes by friction, people started to add more fabric to their pants at the main friction point. Then they started to sew metallic parts to their pants. Then they realized that they didn't need pants if they had metallic parts. This way the first belay devices were born.
DISCLAIMER: this is definitely not a proven advice from the book and may be suitable only for experienced climbers, who do it on their own risk!
Wonder what to do if you have no spare carabiners for the carabiner brake and need to descend many rope lengths so that Munter Hitch would twist and damage it like hell?
Use your ascender (I mean Petzl Ascension or something similar). If you look at it from the bottom (handle) side and add some imagination, you will see a figure eight there:
So you can use this instead of a figure eight.
PS. This is mostly an example of how you can think out of the box.
PPS. Thinking out of the box is your own risk. I personally would use this method only as the last resort if other methods fail (e.g. in the middle of an expedition, when you can't throw away your ropes after the descent).
PPPS. Yes, this was used in practice.
PPPPS. No, there are even weirder methods, e.g.: http://video.yandex.ru/users/rus-alp/view/1/?ncrnd=739692