Question 1: say you have an area for antennas that is a square with dimensions L by L feet and the frequency is f. You are allowed to have as many 1/4 wave verticals as you want, but the total power input is a fixed amount, so you are splitting power to all the verticals. You can place any number of verticals in the L by L square. For simplicity, say you can feed each vertical with any percentage of the total power and any phase you want. Feedline loss is zero and ground is perfectly conducting to the horizon to simplify the problem. (Hopefully not so idealized that the answer is useless!) We ignore SWR issues, assuming we can somehow feed the verticals efficiently. What array of 1/4 wave verticals would you choose, how would you split power between them and what phases would they get in order to achieve "a really useful gain pattern for DX"? And what does the far field plot look like? For “a really useful gain pattern for DX”, this is debatable, but let’s say we want to maximize the total radiated power “from desired heading plus or minus 10 degrees, and at a take-off angle between zero and ten degrees.” Feel free to substitute a better definition! We could narrow down the parameters to be the 20m band and a plot of land 20m by 20m as a concrete numerical example. This might (once made practical) make for a nice compact antenna with gain for hams with antenna height restrictions (like in the UK where antennas require planning permission and your neighbors can easily block you, potentially taking delight in said activity...asking for a friend...). Question 2: what if you are allowed to have some vertical elements parasitic instead of driven, and with slightly varying lengths (as in a Yagi)? In the second example the answer might look a bit like “horizontally stacked” Yagis? A recent YouTube video by Callum M0MCX (of DX Commander fame) discussed a partially parasitic vertical array for 40m - this is the inspiration for this question.