What you Need
A sharp knife
A set square can be useful
What to do
Making an arch is simple, you basically want to produce blocks of potato that are slightly wider at the top than the bottom.
So cut a few large chips out of the potato which are about 12-15mm square and as straight as possible.
Although the blocks you make are tapered in one direction all the other corners should be right angles or your arch will be wonky, so using a set square, or something similar with a good right angle is useful to make the blocks the right shape.
When you have made enough blocks to make a semi-circle try putting them together to make an arch. Does it stay up?
See how the fails if you abuse it, poke it and push it.
You often see arches on the top of pillars or walls. Have a go at modeling this by building your arch on the top of a couple of chips. Does it work?
Try making the blocks really really thin, does the arch still work?
What may happen
If you cut the blocks right the arch should be quite stable, unless you make it too thin when it will collapse.
If you build it on top of tall thin chips, you should find the chips are pushed outwards and it falls over.