Non-Euclidean Space in Dungeons and Dragons In Dungeons and Dragons, the world is mapped out into five-foot squares. Spheres are represented as cubes, and cones look really weird. However, straight lines remain straight, and a rectangular room has 90-degree angle corners. Most troublesome is that you move diagonally at the same rate as you move laterally: it seems that moving $5\sqrt{2}$ ft occurs at precisely the same rate as moving a mere $5$ ft.
Assuming that these aren't merely simplifications for bookkeeping (which they are), what might this environment look like in normal, Euclidean space? I'm not a mathematician, so I'm not even sure it's possible. I'm not even sure what to tag this with.
 A: You might be surprised to know that such spaces do exist; in fact they are an example of one of the most well-studied types of the mathematical formalizations of space!
I don't know what you mean by "I'm not a mathematician" so I'll give the answer in somewhat technical terms for now and if you'd like some clarification, I'll come back when I'm less tired to give a more thorough explanation.
We want a Euclidean space to be one in which we may perform traditional Euclidean geometry, under this interpretation a Euclidean space contains (at least) five important and mutually coherent notions:


*

*Angle

*Length

*Distance

*Area

*Lines


In fact, in the usual model for a Euclidean space ($\mathbb R^n$) these are all essential features the same thing, the notion of the so-called inner product. With this we define in steps the norm, the metric, the topology, and the measure. By "in steps" I mean that each definition only depends on the previous one. These account for the first four:


*

*Inner product $\rightarrow$ Angle

*Norm $\rightarrow$ Length

*Metric $\rightarrow$ Distance

*Measure $\rightarrow$ Area


(The topology doesn't do much of anything geometrically; intuitively it induces a notion of "nearness" but we don't really think of that as a geometric concept.)
An inner product comes with some prerequisites; the technical name is that the elements of the space must form a real vector space, which means roughly there is a notion of direction. Not every real vector space admits an inner product, but an inner product cannot exist without a real vector space [technically, a complex vector space would work too]
Directions, of course, induce a notion of lines, which rounds out the list. (Directions also induce a notion of planes and hyperplanes which are important geometrically, but we won't need this.) Cones come about via a more complex process, but the real vector space structure is all they need; not even the inner product is required for general cones.
The upshot of this is the following: in order to have an inner product, we cannot live by D&D rules. There are theorems to this effect; I don't know all the details myself.
However! If we are willing to leave behind inner products, and take only norms, then we can construct an explicit model of D&D rules (at least as you've described them). Sanity check: all of the things you've described make sense as long as we have a norm, because it also requires a real vector space.
The model is actually quite simple; it is $\mathbb R^n$ equipped with the so-called infinity norm; so that the norm of a vector is the maximum of its components.
$$ || (a_1, a_2, a_3, \dots, a_n ) ||_\infty = \max(a_1, a_2, a_3, \dots, a_n)$$
In the infinity norm, the set $\{ x\in\mathbb R^n : ||x||_\infty=1\}$, which describes a hypersphere, is in fact shaped like a hypercube (you can check this with $n=2$). However, since lines come solely from the real vector space, and not from the gadgets the norm provides. Therefore, the lines in this space look exactly like lines in the normal Euclidean $\mathbb R^n$ (with the inner product).
Other things are preserved as well; distance depends pretty crucially on the notion of norm, but the definition of area is actually so far-removed that it is identical for this norm as in the Euclidean norm with respect to the underlying set. So for example the area of the unit circle in this new space is the same as the area of the (symmetric) unit square in the Euclidean space, which is just 4.
Cones are, as described, tricky. But my guess is that the notions may diverge here. In particular, a cone over a circle is the set of lines connecting each point in that circle to the vertex point. My guess is that in D&D cones look "blockier" because they are trying to project the usual cone onto a square grid. These objects can almost surely be defined; they just are not what a mathematician would usually call a cone. But this is simply a practical concern: we've discovered that a different generalization of classical cones can be used to great effect in the subjects we study :)
