Australian philosopher Frank Jackson proposed the thought experiment of a scientist who is devoted to gathering all the knowledge she can about color, despite being stuck in a strictly black-and-white room.
After learning all there is know about color, save for directly witnessing it, Mary leaves the room. Does she learn anything new?
The thought experiment points to a notion of direct experience conferring a type of understanding that can’t be substituted, and I feel the same notion can apply to math.
A geometry on a smooth manifold $X$ is a maximal Lie group $G$ acting transitively on $X$ with compact point stabilizers, which implies the existence of a $G$-invariant Riemannian metric.
The uniformization theorem classifies simply connected Riemannian 2-manifolds into three classes up to conformal mapping: the sphere, the torus (the euclidean plane), and the open disk (the hyperbolic plane).
Since the metric on a Riemannian manifold induces a metric on its universal cover, the geometry on a Riemannian manifold is classified by the induced geometry on its universal cover.
By simulating light traveling along geodesics, we can render a scene inside of a 2-manifold like the sphere.
The rendered image is 1-dimensional, with each point corresponding to a unit vector in the tangent plane at the camera’s position (represented by the small white sphere).
Thurston’s geometrization conjecture is an analogue of the uniformization theorem for oriented and closed 3-manifolds.
Unlike 2-manifolds, not every closed 3-manifold is diffeomorphic to one with a single geometric structure. Instead, oriented and closed 3-manifolds decompose as connected sums of prime 3-manifolds, each of which has one of eight geometric structures.
We’ll render scenes inside of each, which will be 2-dimensional since the tangent spaces are now $\mathbb{R}^3$, and see how curved geodesics give rise to strange and surprising results.
Our model space is $\mathbb{R}^3$ with the usual Euclidean metric, and geodesics are straight lines.
If $M$ is any finite-volume manifold with a geometric structure and $\pi_1(M)$ is virtually abelian (i.e. contains an abelian subgroup of finite index) but not virtually cyclic, then $M$ has $\mathbb{E}^3$ geometry.
Our model space is the length-1 elements of $\mathbb{R}^4$ with the usual Euclidean metric, and geodesics are great circles.
If $M$ is any finite-volume manifold with a geometric structure and $\pi_1(M)$ is finite, then $M$ has $S^3$ geometry.
Our model space is the top sheet of the hyperboloid $x^2 + y^2 + z^2 - w^2 = -1$, and geodesics are hyperbolas.
If $M$ is any finite-volume manifold with a geometric structure and $\pi_1(M)$ has no infinite normal cyclic subgroup and is not virtually solvable, then $M$ has $\mathbb{H}^3$ geometry (effectively the catch-all case).
Our model space is the cylinder $S^2 \times \mathbb{R} \subseteq \mathbb{R}^4$; geodesics are formed from lower-dimensional geodesics in the factors.
If $M$ is any finite-volume manifold with a geometric structure and $\pi_1(M)$ is virtually cyclic but not finite, then $M$ has $S^2 \times \mathbb{E}$ geometry.
Our model space is exactly analogous to that of $S^2 \times \mathbb{E}$.
If $M$ is any finite-volume manifold with a geometric structure and $\pi_1(M)$ has an infinite normal cyclic subgroup but is not virtually solvable, then $M$ has either $\mathbb{H}^2 \times \mathbb{E}$ geometry or that of $\widetilde{\operatorname{SL}}(2, \mathbb{R})$.
Our model space is $\mathbb{R}^3$ with isometry group given by the Heisenberg group, and geodesics are helices about lines parallel to the $z$-axis.
If $M$ is any finite-volume manifold with a geometric structure and $\pi_1(M)$ is virtually nilpotent but not virtually abelian, then $M$ has Nil geometry.
Our model space is given by identifying $\operatorname{SL}(2, \mathbb{R})$ with $\mathbb{R}^4$ and passing to the universal cover; geodesics are helices parallel to the vertical axis.
Any finite-volume manifold with this geometry has a fundamental group that has an infinite normal cyclic subgroup but is not virtually solvable.
Our model space is $\mathbb{R}^3$ with a solvable group structure, and geodesics are variously helices, exponential curves, and straight lines.
If $M$ is any finite-volume manifold with a geometric structure and $\pi_1(M)$ is virtually solvable but not virtually nilpotent, then $M$ has Sol geometry.
In the future, I’d like to develop a framework to teleport not just between positions within a manifold, but between any two arbitrary manifolds, effectively rendering portals between geometries.
A much more difficult undertaking that I’d like to work on eventually is to render the 18 maximal 4-dimensional geometries by taking 3D slices of the tangent spaces.