Sketch · 2026-09-02
Tesseract, six planes of rotation.
A cube has three axes and three planes of rotation. A tesseract has four axes, so the number of planes jumps to six: xy, xz, xw, yz, yw, zw. The three planes that touch the fourth axis are the ones you cannot picture without help. What follows is the help: sixteen vertices at every combination of ±1, projected first through w to three-space, then through z to the screen.
How the projection works
Each vertex is a length-four vector. A rotation in the plane spanned by axes a and b leaves the other two coordinates alone and turns the (a, b) pair by an angle: the same 2×2 matrix you know from planar geometry, embedded in a 4×4. Six independent planes give the group its dimension, and any orientation of the hypercube can be built by composing them.
Reducing four dimensions to two takes two perspective steps. First w to 3D: put a camera at w = 2.5 and divide the remaining coordinates by (2.5 − w), so vertices with high w shrink like a face receding down a hallway. Then z to 2D: another camera at z = 3.5 and another divide. Under an xw rotation, a vertex whose w was small is now large, and the whole inner cube swaps places with the outer one. That swap is the fourth axis making itself visible.
Reading the wireframe
The 4-cube has eight cubical faces, called cells. In the default pose two of them read as an inner cube and an outer cube, joined by eight edges that connect matching corners. The other six cells are the frustum-shaped shells wrapping the sides. Every cell is a plain cube; only the projection makes some of them look bent.
Edges are shaded by the average w of their endpoints. Faint means the edge is deep in the fourth axis; solid means it is close. Watch a slow xw rotation: the faint edges walk forward while the solid ones fall back, and the inner and outer cubes trade roles once per half turn.
Sources
- Hinton, C. H. (1888). A New Era of Thought. Swan Sonnenschein. The book that coined the word tesseract and worked out the sections a viewer would see.
- Boole Stott, A. (1900). On certain series of sections of the regular four-dimensional hypersolids. Verhandelingen der Koninklijke Akademie van Wetenschappen, 7(3). The systematic drawings that pinned down what the projections look like.
- Coxeter, H. S. M. (1973). Regular Polytopes, 3rd ed. Dover. Chapter VII covers the tesseract with the counts used above: 16 vertices, 32 edges, 24 square faces, 8 cubical cells.