I study multi-dimensional geometry: what shapes look like in four or more spatial dimensions, and how we can learn to see them from down here in three.
It grew naturally out of my work in computer graphics. Lighting and rendering is already about projecting a 3D world onto a 2D image, so asking what a 4D world would look like projected into 3D felt like the next question to ask.
Seeing in higher dimensions
I'm fascinated by hidden structures: things human brains can't picture, yet mathematics can describe them exactly, and that math reaches into our three-dimensional world. We can't see a fourth spatial dimension directly, but we can reason about it by analogy, the way a flat creature in Edwin Abbott's Flatland might reason about our three.
A few of the ideas I keep coming back to:
A sphere passing through a plane appears as a circle that grows and shrinks. A 4D object passing through our space would do the same thing in 3D. Slice a tesseract corner-first and the cross-section begins as a tetrahedron and swells into an octahedron at the middle.
A cube's shadow on a wall is a 2D picture of a 3D object. The familiar "cube within a cube" is the same trick one dimension up: a 3D shadow of a 4D hypercube.
In 3D we rotate around an axis, but rotation really happens in a plane. Four dimensions have six of them (XY, XZ, XW, YZ, YW, ZW), and rotations that involve W turn a shape "inside out" in ways that have no 3D counterpart.
A tesseract has 16 vertices, 32 edges, 24 square faces, and 8 cubic cells, and 16 − 32 + 24 − 8 = 0, just as Euler's formula predicts. There are five Platonic solids in 3D, six regular polytopes in 4D, and only three in every dimension above that.
I wrote a guided journey through the regular polytopes, the most symmetrical shapes there are, from a single point up to shapes in eight dimensions.
It starts with a single point, stretches it into a line, builds polygons, folds them into the five Platonic solids, then climbs through the six regular shapes of 4D to the three that exist in every dimension beyond. Every figure is drawn live in the browser from exact vertex coordinates, so you can rotate them, fold them, and build them up one dimension at a time.
Reading about the fourth dimension only goes so far, so I built a tool to explore it. Hyperslice is an interactive geometry lab, and it's also listed with my other Development projects.
Hyperslice pairs a tesseract explorer with a primer on how slicing works. A live readout counts each slice's vertices, edges, and faces, and presets jump to shapes like a tetrahedron, octahedron, or hexagonal prism.
Videos, papers, and websites
Work by others that has shaped how I think about higher dimensions.