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
We can't picture 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.
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.
Rotate a tesseract through the XW, YW, and ZW planes and move a slice along the W axis to watch the 3D cross-section change shape. A live readout counts its vertices, edges, and faces, and a challenge asks you to find a tetrahedral slice.
Videos, papers, and websites
Work by others that has shaped how I think about higher dimensions.