real 4d geometry · the tesseract · an accurate picture of the space

The Tesseract

real 4D geometry — the only unreal step is flattening four dimensions onto a screen
Here is the 4-cube, drawn honestly. Every vertex, every edge is real; what you see is its true shadow in 3D. The outer cube and the inner cube are two of its eight cells — the inner one looks small only because it sits farther along the fourth axis. Turn it through any of its six rotation planes and read the space directly.

The 4-cube

16 vertices · 32 edges · projected 4D → 3D → screen
w = +1 cell (near — outer) w = −1 cell (far — the shadow) edges along w (connect the two)
4D depth

Reading the picture

what the space actually contains
16 vertices32 edges24 faces8 cells

The six rotation planes

a cube turns in 3 — a tesseract in 6
plane speed

Why the inner cube is a shadow

To draw 4D on a flat screen you project twice: 4D → 3D (perspective along the w-axis), then the usual 3D → 2D. In the first step, things farther out in w shrink — exactly like 3D perspective makes distant things small.

So the inner cube isn't really inside the outer one — it's the cell at w = −1, drawn smaller because it's farther away in the fourth direction. Slide 4D depth down toward 1 and watch the whole figure swell and turn through itself as the projection point nears.

What the six planes mean

In 3D you rotate about an axis — really, in a plane: xy, xz, yz. Three planes. In 4D there are six: add xw, yw, zw.

The three pure-3D planes (xy, xz, yz) spin it like an ordinary cube. The three w-planes (xw, yw, zw) do the 4D-only move — rotating space into the fourth dimension, which is what turns the cube inside-out. Toggle them one at a time to feel the difference.

Honest footing. This is genuine 4-dimensional geometry — the 16 vertices, 32 edges, six rotation planes, and the perspective projection are all real and standard. The single "unreal" step is the unavoidable one: four dimensions can't sit on a screen, so we view a shadow. Nothing here is metaphor — the inner cube is a real cell seen at depth, and the inside-out tumble is what an actual 4D rotation looks like from where we stand.