I have not heard of anyone doing this with aquariums, but ponds, yes. As a matter of fact, on an even larger scale, in swampy areas there is a 'new' system of sewer purification, that uses a planted bed to clean the leach field for raised bed septic systems. So I would say that if you wanted to do it, one thing to keep in mind is that if you have a tank that is as large as the aquarium you are filtering, have it lit with strong lights, and do not have even the norman fish/gallon, you might try it. I think the idea of having say a canaster filter to purify your tank is that the number of square inches of filter material, compacted into a small space is cost efficient. The cost of running a canaster filter would only be that of a pump, where the filter you are talking about would still require a pump, but also lights.
The other thing to consider is that there are three types of filtration, biological, mechanical and chemical. The planted tank would have less biological filtration or mechanical filtration than conventional systems. As for the chemical filtration, many chemicals would tend to disrupt a heavily planted system, so you would not have the possibility of chemical filtration with out risking problems that could end up causing your fish harm, before you noticed the problem.
I did read recently about a fellow who was using a conventional filter system, but wanted to have a planted tank to take only the nitrates out of the filtered water. This seems more logical to me, as the nitrogen cycle depends on the bacteria in the biological filtration, changing ammonia to nitrite, to nitrate. In a planted system, the ammonia build up would not only be lethal to your fish, it would not be good for the plants until it is converted to nitrate.
In reality it would be hard to use plants as a main filtration. If you have very few fish and one's that are small and produce small amounts of waste then it is more feasible. You can use plants as chemical filtration like sustina flower said but with lots of fish it helps to have some sort of mechanical filtration. Sometimes people may not use a large filter when they have a heavily planted tank but that's only when it's heavily planted.
The reason I asked is because I had read that, with a very lightly stocked tank and a lot of plants, you can run a tank without a filter, but the book I read it in didn't give an idea of gallons per fish, so I was hoping someone here had an idea.
No idea. The accepted standard for aquariums is to have a filter designed for the size tank you are running, heater, and lights. To do otherwise, you are treading in so many unknowns, you risk having a tank that is an unhealthy environment for fish. It would be your risk. :-X
What I said before was theory, the fact is that plants do use nitrogen and sunlight to produce chlorophyll, to grow. Nitrate, the end result of your filters bacteria colony, is part of what plants need, so they would extract a certain amount from your tank. They do not however convert ammonia to nitrite and then to nitrate. That is the biological part of filtration. The bacteria that do this job have to come in contact with the water, (flowing water) there has to be oxygen available to the bacteria for them to live. These things all are accomplished in a filter.
A filter has other applications in a tank however, oxygenation is one, and circulation is another. Fish to be healthy need water that resembles their natural environment. Very few fish can live in a stagnant pond. The movement and oxygen supply are important. Stagnant water stinks, which doesn't add to the enjoyment of a tank.
Of course the very next post I opened had some answers that tell me that I really should be more careful about spouting my ideas !!! :-[ Check out this article. It was posted by Butterfly on the idea of using soil in a planted aquarium, but also answere some of your questions. Hope this helps.
I had a 4ft goldfish tank that had no filtration (it was outside so no power point )...I had 2 huge faintails in there....they bred all the time and had been in there for 3 yrs happily (well until the neighbours kids put poison in the water!)
The tank was heavily planted with elodea