It's annoying when the number of participants on your video conference isn't a number that forms a tidy grid on the screen e.g. 3x4 or 4x5 little pictures. The more 'prime' a number is the worse this problem becomes. So, a prime number like 17 can be represented as a 4x4 grid, plus one sad, lonely individual
stuck onto the side of this grid, but even a number like 22 which is not prime but has few factors is hard to represent as a 2x11 grid on a normal screen - you'd probably have to have a 4x5 grid with two extra people stuck on the side of the grid.

So this idea is for the configuration options for your conference call to have an option where you can specify how prime you will allow the number of participants to be, and what will happen if the number of participants deviates from this - e.g. random people could be booted off the call until a satisfactory grid is re-formed.

Stuff the feeds into a Voronoi diagram, and let the participants drive their focus around.
If someone's saying something important, everyone can back off and give their video the space to expand. If someone wants to blend into the background, they can sidle up behind someone else near a corner.

Be gracious and generous - if there is a non-griddable number of callers, have your system automatically disconnect you from the call and post the reason and apology into the chat.

The more inclusive solution is to volunteer, perhaps in a less than entirely voluntary manner, people to be dragged into the call to make up the numbers required to develop a pleasing grid.

Notional croissant for the Voronoi diagram suggestion also!

[calum] the tele-screen in your house flickers into life and says "welcome to the meeting" before all the other participants stare at you doing whatever you happen to be doing at that moment.

Somewhere there's a conjecture that talks about how far from prime a given number might be, I mean, you're only ever 1 away from a not-prime, but maybe what we're talking about is how far a given number is away from some number that meets an ideal aspect ratio.

When I've had to generate multi-plot visualisations that don't know until time of generation how many plots to place on the page, I determine my ideal aspect ratio, which I can then apply to the total number of required plots, and then round up to the nearest integer. That should give you an r x c grid with the final row only partially populated. More recently, I've just been plumbing for a square, in which case, I can use rows,cols = int(sqrt(N))+1, which guarantees a layout that will fit. 22 (4.69041 ^ 2) generates a 5x5 layout grid which after populating with 22 elements, leaves the last row (or column) only partly filled.

Now in some cases, there should be some way to double-up some of the callers, so that they take up 4 (assuming square) "places" in your layout grid, that means that you can rejig your number by +3 to see if you can get to something more amenable. In the 22 scenario, you could boost person 1 to occupy 4 squares, and fit everyone into a 5x5 grid quite nicely. Not sure how many times you can do that without it falling over, but there might be opportunities to go from 1x1, to 2x2 (+3), to 3x3 (+8) and so on, to try and find a pleasing tessellation that fills the space. If you're so inclined, you could choose some criteria that decides which of your meeting participants will be displayed twice as prominently as everyone else.

I love the Voronoi idea - and have been thinking for a while that more UI should be Voronoi-based. Limiting ourselves to rectangles carries with it so many hang-ups, while a cell-based representation is a lot more natural and I suspect would be quite friendly to use. There's a whole idea wrapped up in that somewhere. I was thinking about that in the context of strategy games where the square (or sometimes the hexagon) are the chosen tiles employed to make the gameworld, but think a Voronoi layout might be more natural.

I have just woken from a dream about the 1980s board game version of Civilisation, where the map was organised a bit like this (not in squares or hexes).