Rotman Lens

If I understand correctly, is the application of this to quickly switch the direction of sending/receiving? Are there any other applications?

I was expecting some sort of fungal network with measurable compute on initial glance. Then the article image couldn't help but look quite like the outline of an Earthbound entity.

That's kind of it, yes.

You can aim an aerial by physically rotating it. You've probably done this, gone up on the roof to adjust the aim of your TV aerial or satellite dish. It makes sense, right? A Yagi aerial - a reflector, a driven element, and a bunch of directors - focuses the beam in a kind of aubergine-shaped blob in the direction of the pointy end.

But you can also aim aerials by having two of them, and varying the phase that you send a signal into them. This sounds a bit mental but consider how direction finding equipment like LoJack works - you have a transmitter in an unknown location and you have a cluster of aerials connected to one receiver. By comparing the phase of the incoming signal between two aerials you can work out which one it's nearer to! This trick works well enough if you make two dipoles spaced a half wavelength apart that you can easily homebrew something where by switching in a 180° phase shift at an audio rate, the difference in phase can be heard as a tone.

In this case you've got a bunch of aerials attached to the ports along the bottom and the phase of the signal reaching them depends on how long it's taken to cross the microstrip. If you fire it in at the top in the centre it'll be equal (notice the middle "legs" have kinks in them to keep the path length the same?), if you fire it in at the side then one of the ports at the side will get the signal sooner and its phase will appear advanced compared to the other one - and the beam will bend that way.

The type of old school black magick engineering Claude could never.

Imagine a power splitter + phase shifter that produces a correct phase shift for each element in a phased array to produce a directional beam from one radio transceiver.

Now this clever arrangement, instead of having only one radio transceiver port, has multiple. And each of those ports corresponds to a different set of phase shifts, producing a directional beam at a different azimuth angle.

And because this is an entirely passive device, it's linear, and all ports can be active at the same time (principle of superposition essentially). So you can use a single phased antenna array to serve multiple directional beams at the same time.