I've got bots in a rectangular formation with rows and columns. A problem arises when a bot is added or removed from the formation. When this happens, the bots have to rearrange themselves so that the rectangular formation is still roughly the same aspect ratio, and is as rectangular as possible. How to do this?
When a bot is added or removed, use the new total number of bots and a desired constant aspect ratio to calculate the new width and height of the formation that most closely fits that aspect ratio. Then somehow reshuffle the bots to fit the new dimensions.
When a bot is removed, move the bot that was behind it into it's place, and continue until you reach the end of the formation. Then even out the back rank as much as possible by somehow shuffling the bots in the back rank.
Another idea that's completely different is to mimic the way molecule structures stay together. Make every bot want to be surrounded by four other bots by attracting the four closest bots, and repelling the rest. Repel all bots (including the four) that are too close to ensure separation using inverse square law. You'd also need an additional force to shape of the entire structure. But, this sounds very computationally expensive.
UPDATE: So looking into sarahm's answer, I came up with a good general function that gives good dimensions.
First I solved the below simultaneous equation for width and height, and then rounded the answers.
width/height=aspect ratio of your choice width*height=number of bots
This gives you the closest integer rectangle to that aspect ratio for your number of bots. The closest rectangle will half the time be a too large, and half the time be a too small (of course sometimes it will be just right but who cares about those). In the cases where the rectangle is a little too large, nothing needs to be done. The back rank will just end up being almost full, which is ideal. In the cases where the rectangle is a little too small, you got problems because that teeny tiny overflow will have to go to its own rank created a rank with only a few bots on it, which doesn't look pretty. There are also cases where the difference is large (larger than half the width), in which case add or subtract one rank to make the difference small. Then, when the rectangle is too small, add one column to make it just a little bit larger. After doing that it looks like the back rank will always have at least half as many bots as the other ranks.
Once you got the dimensions, compare them to the current dimensions. If the frontage of the new dimension is bigger, for every rank, pop bots from the rank below, and push them onto the current rank until that the number of bots on that rank is equal to the frontage. Continue that algorithm until you get to the back rank. Using this algorithm, bots will move to fit into the new dimension efficiently. After that, I simply push the new old onto the back rank. The algorithm is slightly different for the cases where the new frontage is smaller, but you can figure it out!
There's two more problems next. Deletion, and a more flexible addition method where new bots are not necessarily assigned to the back rank but whichever position is closest to them at the moment they are added.