Square net pots?
 

Are there square net pots? I have seen larger ones - about 1 gallon size, or half gallon size. The reason I ask is that if I used square net pots I could put more plants in the same space. I know that root size might be an issue...is there some way to train roots to grow downwards rather than fanning out? Maybe by causing more flow in the nutrient - having deeper pots and more current?

Why could you fit more plants in the same space by using square vs round net pots?

My thinking is that all of the space could be utilized. A square that has the same area as a circle has a smaller side than the diameter of a circle. Unless the round pots were all pushed together with minimal space between them. I will try to draw a diagram of what I mean.

All your going to do by trying to push them as close together as you can is crowd them. The foliage needs space to grow.

What about root crops like radishes, beets and carrots, that have little foilage?

Root crops have just as much foliage as any other plant. They need the foliage (leaves) to do photosynthesis. Without enough foliage they can't do the photosynthesis needed to produce the the chemical changes needed to develop the root crops. Regardless if it's a root crop, or above ground fruit crops, they both need enough foliage to do the photosynthesis needed to produce the fruit (above ground or below it).

There are well known and published spacing figures for plants to grow optimally. Crowding them in will not give you the best yield, or even not fruit/produce at all.

Hydroponics isnt about super-density, its about being more efficient with the space you have - which sometimes means a tomato plant being 18" away from its nearest neighbor, etc.

Plants also need plenty of air flow to prevent disease and mold in the foliage. If you plan on planting close you better get a fan on the plants.