Say I have twelve flat-screen TVs with HDMI ports that are physically arranged in a 3 by 4 matrix. What would I need to "mirror" an image from a laptop display (also with HDMI port) to that 3 by 4 matrix as if it was a single giant display? Can I do it only through software or do I need a special hardware?
It's practical to consider that you need special hardware, if only to have the physical capability to connect twelve screens to a source device. A quick search for "tiling computer image over multiple screens" provided too many links covering two monitor wallpaper, but a couple more useful resources.
One of them is an NVIDIA product line called Mosaic. I expect that the company would be able to provide substantially high performance video devices, considering their history in the gaming video world. I did not follow the pricing links!
Another link I discovered also appears to be a commercial resource, but seems to be a service rather than a solution. One of the links from that source sent me to Matrox which advertises "Dual, Quad and beyond," for capability.
AMD appears to offer support via graphics adapter cards up to six displays.
I popped off another search using "How many monitors will windows support," and got a few links to indicate "no limit," except for hardware limitations and performance consideration. The search also provided an article written in 2012 that might be of value https://www.pcworld.com/article/251521/when_two_monitors_arent_enough.html describing alternative devices, which again is a hardware consideration.
You'll be limited by the number of slots in your computer and the number of ports per card, but if you have sufficient room to plug in four cards with three ports, or the reverse, it is suggested in the articles noted above that you should use all the same device, to keep the drivers uniform.