Space(time)

The Mathematics

The still life gathers objects from the study of geometry from antiquity through the 1800s: the Platonic solids in three and four dimensions, Archimedes’ sphere and cylinder, and on to fractals, hyperbolic space, knots, and the Klein bottle. The first panel shows this scene in flat space, lit as we would ordinarily see it.The second panel adds the mathematics of the late 19th and early 20th centuries — the geometry of curved space — by sprinkling six black holes into the scene and rendering an accurate image of what one would actually see, as light travels along geodesics through that spacetime: the same objects stretched, multiplied, and drawn toward the horizons.

Technique

Both panels come from a general-relativistic path tracer I wrote. It works for static spacetimes: objects living on a spacelike hypersurface are extended into world-tubes, and from the simulated camera a light cone is traced out into the scene in 3+1 dimensions. The standard path-tracing algorithm then runs on top of this, adapted to the relativistic setting. The first panel is rendered in Minkowski space; the second in a Majumdar–Papapetrou spacetime with six extremally charged black holes.