The 4D analog of the cube: 16 vertices, 32 edges, 24 faces, 8 cubic cells. Watch it rotate in 4D space — the XW or YW rotations don't exist in 3D, so the projection appears to turn inside out. Inner cubes become outer; cells get swallowed and ejected through their own boundaries.
what your eye is doing
A tesseract has 16 vertices — every point with each coordinate ∈ {−1, +1} in ℝ⁴. To see it, we apply two projections: first from 4D to 3D (perspective from the W direction), then 3D to 2D (perspective from Z). The result on your screen is a cube-within-a-cube connected by edges — the standard Schlegel diagram.
A 4D rotation in the XY plane looks like a normal 3D rotation around the Z-axis. But a rotation in the XW or YW or ZW planes — rotations involving the 4th dimension — has no 3D analog. Under such rotations, vertices that were "inside" the cube travel out through faces, and vertices that were "outside" come in. The cube-within-a-cube structure dissolves, reforms inverted, then dissolves again. You're watching a 4D solid pass through your 3D space.
The same construction works in any dimension. The n-cube has 2^n vertices, n · 2^(n-1) edges, and exactly n independent rotation planes (where 3D has 3 — XY, YZ, XZ — and 4D has 6 — XY, YZ, XZ, XW, YW, ZW). The mathematics is uniform; only our intuition runs out at 3. The tesseract has been a touchstone for 4D visualization since Hinton's 1888 essay; the rotation experience is itself the proof.