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human
[Warhammer 40k] The Eldar & their Empire, and the current Eldar & the Tyranid.

Second question first:  How do the Eldar see the Tyranid?  How do they react to them (aside from defending their worlds, ofc :P).  I mean, do they try to manipulate the Hive fleets as they do with Ork WAAGH!s?  Have any Farseers had any Tyranid-relevant visions or anything like that?

As for the other question: Are the Eldar in a state similar to that of Mankind in terms of having lost a lot of knowledge (think of our search for STCs or even fragments/printouts) and the capability to make their best stuff (see: The origins of the Rhino, Terminator Armor, and the formerly lowly status of the Mighty Baneblade)?

On the one hand I'd imagine that would be the case, but on the other hand their stuff is based on their psychic powers and related stuff (see: Wraithbone) so maybe that isn't the case, and all they really lost was numbers + some old archived knowledge... ?
gpt
>How do the Eldar see the Tyranid?

They're referred to by their Farseers, usually, as the 'Great Devourer'. An all-consuming tide. A scourge to be paid close attention, especially after the de-population of Craftworld Iyanden. Something to be manipulated, as seen in Farseer Idranel. She masterminded the distraction of the Orks (to keep Imperial Forces busy) and the luring of the Tyranid Hivefleet during the first Aurelian Crusade to prevent it from striking her craftworld. The Tyranids are attracted to the galaxy, probably, by the Astronomican. The Zoanthropes are almost certainly the result of a Norn-Queen hooking into some tasty, delicious, nutritious Eldar flesh. So they see psychic beacons, they tend to get curious.

>As for the other question: Are the Eldar in a state similar to that of Mankind in terms of having lost a lot of knowledge (think of our search for STCs or even fragments/printouts) and the capability to make their best stuff (see: The origins of the Rhino, Terminator Armor, and the formerly lowly status of the Mighty Baneblade)?

Sort of. They've lost a lot of ability, but for entirely different reasons. The Eldar still understand their science. They know how to build things, why these things work and the like. They just don't have the infrastructure anymore. It helps that a lot of their technology is created through psychic manufacture, meaning that the capacity to build it is inherent to every member of their race, at least on the lower end.

For example, the Webway. Originally built by the Old Ones, added to by the Eldar at the height of their Empire. It's one thing to add new spars and new layers of protection, new nodes when you're sitting atop an Empire that's essentially one comprised of demigods. It's another when you're a hunted race on the verge of extinction, hiding in worldships in the dark of the stars, unable to use your full abilities for the fear of being eaten by Slaanesh. Given sufficient energy, materials and labour they could build new craftworlds, reconstruct the broken spars of the Webway, craft entire new worlds again.

Some things are irreplaceable, though. For different reasons. Some are cultural - the loss of the spirit stone of Irrillyth, for example, put paid to the Shadow Spectres aspect until it was recovered. Others are knowledge based: Spirit-stones themselves can't be manufactured (yet), as they're the crystallized energies of the Eye of Terror, filtered through the lingering Eldar resonance on the Crone Worlds.