The 4-D hypercube, seen as a shadow in 3-D. Drag to orbit · scroll or pinch to zoom · tap a cell to highlight it.
Each of the 8 cubes hinges about the square it shares with its neighbour — a real rotation in 4-D. Fully unfolded you get Dalí’s cross: a column of 4 cubes with 4 arms.
| shape | V | E | F | C | V−E+F−C |
|---|
V = 2n, E = n·2n−1, and in general the number of k-dimensional faces of an n-cube is C(n,k)·2n−k.
Add back the solid n-cell itself and the alternating sum is 1 for every n (16−32+24−8+1 = 1 for the tesseract).
Where’s the maths? A dimension is just one more number in a list of coordinates: a point on a line needs 1 number, on a page 2, in the room 3. Nothing stops us writing down 4, and every formula keeps working — distances by Pythagoras, angles, rotations — because geometry is really algebra on coordinates. The tesseract is the simplest 4-D shape to count: its corners are all the lists of four ±1’s, and binomial coefficients count its edges, faces and cells. What you see on screen is a projection, the same maths a computer uses to draw a 3-D world on a flat screen, applied one dimension higher.
Beyond the syllabus: four-dimensional geometry is enrichment.
Start with a point. Drag a copy of it 1 unit sideways and join the two — a segment. Drag a copy of the segment in a new perpendicular direction — a square. Again — a cube. Do it once more, in a fourth direction perpendicular to all three, and you get a tesseract.
Each step doubles the corners, so V = 2n: 1, 2, 4, 8, 16. Every corner of the tesseract is a list of four ±1’s, like (+1, −1, −1, +1), and two corners are joined by an edge exactly when they differ in one slot — so 16 × 4 ÷ 2 = 32 edges.
To count the 8 cells: freeze one of the four coordinates at +1 or −1 and let the other three roam. That is 4 choices of coordinate × 2 signs = 8 solid cubes. Freeze two coordinates instead and you get the square faces: C(4,2) × 2² = 6 × 4 = 24 faces.
The fourth direction is a direction in space, not time. We can’t see it, so we look at a shadow. A cube casts a flat 2-D shadow in which the near face looks bigger than the far one; in the same way a tesseract casts a 3-D shadow in which the near cube looks bigger than the far one — the famous “cube inside a cube”. Nothing is really bent: every one of the 8 cells is a perfect cube in 4-D.
Unfolding works the same way by analogy. A cube unfolds into 6 squares laid flat in 2-D; a tesseract unfolds into 8 cubes laid out in 3-D, and one of the 261 possible nets is the Latin cross Salvador Dalí painted in Corpus Hypercubus (1954).