Before you start this project, look up both the pH and EC requirements for the several items you want to grow together. Many plants have very different nutrient and growing requirements, so you won't want to mix together plants that won't do well growing side by side.

I'm a hydroponic grower and grow most of my plants vertically, but I grow my hydroponic potatoes in individual tubs, sitting on the floor of my greenhouse. I don't mix cultivars; mostly because of vastly different feeding and nutrient uptake requirements. I'm assuming growing in soil would be a similar situation.

Personally, I would be concerned about how harvesting lower layers would impact higher levels. As you pull carrots, beets, radish, etc from the lower levels, that will cause a void, which could lead to a collapse of soil from the upper levels.

You would have to put the potatoes at the top, as you would want to continue to add soil to the potatoes as it grows, but they have a much longer growing season than the other items mentioned, so even doing a single harvest of the tower at once is not an option.

how would you plant carrots, radishes & beets in the side of a tube or container? i would like to know how it turns out, see some pics as they grow.