I was tempted to say something last night upon reading comments like this from you and Hari Seldon, but I forebore, so it’s been interesting to me to see how this debate has deveoped.
I’d say that statements like this are true in a sense, but potentially misleading. After all, people knew that the Earth’s surface was curved, and that therefore you could have “lines and patterns” on a curved surface, historically long before they accepted non-Euclidean geometry.
Is it okay to say the following?: If you treat lines, etc. on a curved surface as if you were working in a flat plane, they follow the rules of a non-Euclidean geometry. (But, as Apidastra pointed out, they’re still part of Euclidean 3-space.)
So, there are different ways to interpret the OP’s question. According to one interpretation, the answer is No, because any architecture that you actually build in the real world will have to follow the real world’s geometric structure, which is at least so close to Euclidean that you can’t tell the difference. And according to another interpretation, the answer is Yes, if you can somehow exploit the fact that plane geometry “looks” non-Euclidean on curved surfaces.
I’ve got a pretty good background in Math, even took a grad course in Transformational Geometry*. When someone mentioned “non-Euclidean” by itself it always meant planar geometry with an alternate parallel postulate.
Taking “non-Euclidean” to mean something other than what Euclid ever wrote about is …, well, not exactly a common interpretation.
If you want “curved” lines (such as on a spherical or hyperbolic plane) to look straight, you need the kind of space warping that is inadvisable in civil architecture.
You can find on YouTube computer renderings of what flying about such spaces looks like.
I’m sure you could design a respectable Cthulhu-cult temple that could still be built on a real city street without sucking the entire block into a black hole.
I picture the “what’s a straight line?” architecture of Frank Gehry and the like. I imagine something like this or this or this would look pretty damned trippy to someone of Lovecraft’s era, like the scenery in Cabinet of Dr. Caligari brought to life.