https://player.vimeo.com/video/434244025

3-dimensional∘space

Exploring the Thurston Geometries

Raymarching 3-dimensional Geometries

Space around us has a special kind of symmetry: it behaves the same at every point. (Of course this isn't strictly true as Einstein taught us, but it is a fantastically accurate idealization.) Generalizing this, mathematicians call a space homogeneous if there is a symmetry taking any point to any other point. The first thing one might ask following this definition is classification: how many possible types of geometry have rich symmetry like the euclidean space we live in? Given some reasonable constraints, it turns out there are eight homogeneous geometries in dimension three. These eight geometries are together called the Thurston Geometries in honor of William Thurston, who first realized their fundamental importance in three dimensional topology.

Here you can explore these worlds through real-time simulations which run live in your browser. These are computed via raymarching, a computer graphics technique which produces an image by shooting out rays of light from an imagined camera and seeing what they hit. Much as light travels on the shortest path, or geodesic, in the real world, these simulations were produced by tracing light along geodesics in each of these eight spaces.

To visit each of these worlds, click the 'fly' buttons below. Use the keys Q,E to rotate the screen about its center; A,D and W,S to rotate your field of view. To move around, use the arrow keys to go into/out of the screen as well as left/right, and the keys '/ to move up and down. If the simulation is running slowly on your screen, make your browser window smaller. Remember these images are being produced by simulating light rays in curved spaces for every pixel, so a smaller screen = less pixels = faster performance!

This project is joint work by Remi Coulon, Sabetta Matsumoto, Henry Segerman, and Steve Trettel.

$$\mathbb{E}^3$$

Euclidean space is constant curvature, isotropic and flat (to an excellent approximation, the space around us is euclidean). Geodesics diverge linearly and the surface area of spheres is quadratic in their radius.

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$$\mathbb{S}^3$$

The three dimensional 'surface' of the four dimensional ball is an isotropic geometry of constant curvature. Geodesics initially diverge sublinearly, before reconverging at the antipode to their source. The area of geodesic spheres grows with $\cos(\mathrm{radius})^2$.

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$$\mathbb{H}^3$$

Hyperbolic space is isotropic, with constant curvature $-1$. Geodesics diverge exponentially, and the surface area of geodesic spheres is exponential in their radius. There is an extremely rich collection of hyperbolic manifolds, which are a topic of contemporary research.

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$$\mathbb{S}^2\times\mathbb{E}$$

The product of a sphere and a line is not isotropic, and contains totally geodesic spheres and euclidean cylinders. The rate of geodesic divergence depends on their initial starting direction. Geodesic spheres grow sub-quadratically in area (and are immersed for large radii).

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$$\mathbb{H}^2\times\mathbb{E}$$

A product of the hyperbolic plane and a line, this geometry is not isotropic, and contains both totally geodesic hyperbolic and euclidean planes. The rate of geodesic divergence depends on their initial starting direction, and spheres grow (coarsely) exponentially with radius.

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$$\mathsf{Nil}$$

Nil is a homogeneous geometry built from the 3-dimensional Heisenberg group. It is anisotropic, with a single axis of rotational symmetry. Generic geodesics in Nil spiral around this symmetry axis.

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$$\widetilde{\mathsf{SL}}_2\mathbb{R}$$

This geometry is built from the unit tangent bundle to the hyperbolic plane. Due to the negative curvature of $\mathbb{H}^2$, traversing a loop in this plane causes additional vertical motion- corresponding to holonomy around the loop.

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$$\mathsf{Sol}$$

The homogeneous geometry Sol is built from a three dimensional (solvable) Lie group. It is the least symmetric three dimensional geometry, with no continuous symmetries fixing a point. Sol contains two hyperbolic planes through each point. Away from these, geodesics come in two families of spirals about orthogonal axes.

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