i'd second the person who said the den-realm is an extension of the bastet's personality. but i don't think it's that strong. with a den-realm score of only one, it would be only mildly like the bastet wanted it. to use the example of the khan in L.A.: let's say she first establishes the realm with a scoe of 2. of course, there is no change in the material world. but in the denrealm, it is fairly heavily vegetated. any trees or shrubs are healthier, lusher, more overgrown. window boxes are replaced with huge patches of mosses, epiphytes, creaper vines, and other tropical jungle plants. the scraggly grass in the sidewalk cracks is replaced with patches of ferns. there may be an occaisional small jungle rodent, or some tropical insects, but it still looks basically like L.A. there are apartment buildings, homes, parking lots, stores, and businesses (as appropriate to the part of the city). brings new meaning to the term "concrete jungle!" :-)
later, the Khan's player spends some experience points and raises the den-realm score to a 4. in game reality, the Khan has spent time there, and it gradually attunes to her beliefs. additionally, she spends some time chasing out an occasional invading spirit and patrolling the real-world area that corresponds to the den-realm, extending her influence so that the sleepers begin to feel the delerium. she then goes further: she offers a few genre-appropriate spirits the opportunity of a safe place, in return for aid in manipulating the pocket of the umbra, and added "atmosphere." she begins leading a grass-roots campaign to promote green-space and houseplants for their psychological boost. and she spreads the word of the therapeutic benefits of pet ownership. finally, she frees a few appropriate animals from the local zoo, and turns them loose in the den realm. [for that matter, i'd say that if the character went to this sort of trouble, the den-realm strength ought to go up without the expenditure of experience points.] now, the den-realm looks only vaguely like the actual city. it is more like a long-lost incan temple. the buildings are mostly gone, and those that aren't are in shambles. everything is covered in vines and ferns and epiphytes. all of the plant life is larger-than-life and terrifically healthy. perhaps a few scyscrapers are maintained only by the gargantuan strangler figs that have grown up around and through it. the atmosphere and lighting are like a deep tropical rainforest. there are the sounds and occasional rustlings of countless strange birds, monkeys, insects, and other wildlife. and roaming through it all are several actual jungle animals; perhaps a mated pair of tigers, a few monkeys, several families of birds, and a hoard of insects and small rodents (imported by the spirits and/or stolen from pet stores).
as to its relationship with the umbra, i'd say that it is separate from the umbra, but can be reached. the same guantlet has to be pierced as when travelling from the material world. and it's co-spacial with the material world (as is all of the umbra) *and* the umbra itself. so it doesn't take up any "space" in the umbra, and won't just be seen. perhaps a power that enables one to see into the umbra from the material world would let umbral denizens see into the den realm and therefore be aware of it's existence.
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