"Infinite City" - hyperbolic geometry, or something else? Background: I'm doing worldbuilding for my D&D world, and I want one location to be an "infinite city": finite circumference from the outside, but as you move toward the "center" there's always more room and space to go.
A specific detail I'd like is for "ring roads" circumnavigating the space inside to exist, and have larger circumference as you go further "in".
I think this can be described by saying that the region of space has a highly-curved hyperbolic geometry. Is this correct? If not, is there another way of describing the space that would have the effect I want? It's okay if the resulting geometry wouldn't be suitable for physics as we know it, I'm fine with hand-waving "it's magic!" to handle that; I'd just like something I can think about and possibly obtain some non-obvious interesting details from, and get a consistent description out of.
 A: Such examples are easy to construct indeed using classical hyperbolic geometry; however, you need to know some differential geometry.  
Start with the upper half-plane ${\mathbb H}^2$ in ${\mathbb R}^2$ equipped with the hyperbolic metric 
$$
ds^2= y^{-2}( dx^2 + dy^2). $$
Next, consider the rectangular (in the Euclidean sense) region 
$$
Q=\{(x,y): 0\le x\le 2\pi, 0< y\le 1\} \subset {\mathbb H}^2.$$
Identify the vertical boundary intervals of $Q$ via horizontal translations:
$$
(0,y)\sim (2\pi,y), 0< y\le 1.
$$
The quotient space $A$ is diffeomorphic to the half-open annulus (a closed disk with the center removed),
$$
A\cong S^1\times (0,1]\cong D =\{w\in {\mathbb C}: 0< |w|\le \frac{1}{e}\}
$$
 The hyperbolic metric $ds^2$ projects to a Riemannian metric on $A$ (since horizontal translations are isometries of the hyperbolic metric). You can  realize this diffeomorphism  via the map 
$$
z=x+iy\mapsto w=\exp( i x - y^{-1}), z\in Q, w\in D. 
$$
Your concentric roads $C_r$ in $D$ are the Euclidean circles $|w|=r$. But the hyperbolic lengths $L_r$ of such roads are equal to 
$$
\int_{0}^{2\pi} \frac{dx}{y}= \frac{2\pi}{y},
$$
where 
$$
r= \exp(-y^{-1}), y= \frac{-1}{\ln(r)}.
$$
 Hence, as $r\to 0$,  $L_r\to\infty$.  The distance between $C_r$ and the exterior road $C_{1/e}$ is constant, equal 
$-\ln(y)$, where $y= \frac{-1}{\ln(r)}$. The distance from any point of $D$ to the center is infinite: If $w\in D$ corresponds to $z=x+iy$ then the distance from $w$ to $0$ equals
$$
\int_{0}^y \frac{dt}{t}= - \lim_{t\to 0} \ln(t)= + \infty. 
$$
The total area of $D$ is infinite as well (you compute it using the same integral). 
A: The (Poincaré model of the) hyperbolic plane is slightly different from what you want: it looks like a circle, but as you move outwards from the centre you become ''smaller'' from the god's perspective, and will never reach the edge.
Your description may be impossible to realise as a Riemannian manifold, but I can't prove that yet.
A: Hyperrogue is a game in hyperbolic geometry and it already has something like that in it. I downloded that game and played it. Most boundaries between lands re straight lines. In classic mode, you have to kill 100 enemies to access the graveyard. In the graveyard, there is a forest. Its edge truly is curved inwards and doesn't just appear that way as a result of the projection. If you go far enough in, you pass an invisible straight line that was equidistant from the edge of that forest and wasn't that close to the edge of the forest. That's quite interesting. It's like getting sucked into a black hole. A person's mind could find the real world choppy and unreliable and then naturally get sucked into a very specific environment like a black hole and be truly focused to see anything other than that as small and unimportant and cannot be made to draw a lot of attention to it. You can always play it again from the beginning and in fact it' very hard to accomplish so many thing in the game without dying and getting sent back to the beginning so you can try again in your next attempt to enter the forest and again and again. I'm not sure if it's possible to use Hyperrogue in Dungeons and Dragons. Maybe it is. There is a video of Ocarina of Time in Banjo-Kazooie hown in the video https://www.youtube.com/watch?v=7_WliWHzLZQ. It was probably a typo saying Banjo-Kazooie in Ocarina of Time. If you can't find a way to do it, you want to accept it and maybe you can decide it's good enough playing Hyperrogue itself to get the thing you want and it doesn't have to be Dungeons and Dragons.
