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Journey 3 of 10 · 10 steps

Into the Fourth Dimension

Square → cube → tesseract → beyond

Learn to see four dimensions the way a flatlander would learn to see three: through shadows, slices and rotation.

Play it in SacredForm

In the app each step moves: figures draw themselves, rise from the page, turn and slice, and the narration can be read aloud. Here they are held still, one picture each.

  1. A square: Cube

    Step 1 · Cube

    A square

    Start with a square. Imagine you were a flat being who could only see the plane. How could we show you a cube?

  2. Pull it into depth: Cube

    Step 2 · Cube

    Pull it into depth

    Drag the square in a direction you cannot point to, perpendicular to the page, and join old corners to new. That is a cube: 8 corners, 12 edges, 6 square faces.

  3. Now do it again: Tesseract

    Step 3 · Tesseract

    Now do it again

    This looks like a cube. It is a tesseract seen with its fourth coordinate squashed flat, exactly as the square was the cube squashed flat.

  4. The tesseract: Tesseract

    Step 4 · Tesseract

    The tesseract

    Release the fourth direction, W. A cube inside a cube appears: 16 corners, 32 edges, 8 cubic cells. Colour shows each point’s W coordinate.

  5. Turning through W: Tesseract

    Step 5 · Tesseract

    Turning through W

    Rotating in the XW plane makes the inner cube swell and swallow the outer one. Nothing bends: every cell stays a perfect cube, only our shadow distorts.

  6. The 4D triangle: 5-Cell

    Step 6 · 5-Cell

    The 4D triangle

    The simplest 4D shape: five points, each joined to all the others. Point, segment, triangle, tetrahedron, 5-cell.

  7. The 16-cell: 16-Cell

    Step 7 · 16-Cell

    The 16-cell

    One point at each end of four perpendicular axes: the 4D octahedron, bounded by sixteen tetrahedra, the dual of the tesseract.

  8. A shape only 4D has: 24-Cell

    Step 8 · 24-Cell

    A shape only 4D has

    The 24-cell, built from 24 octahedra, has no equivalent in 2D, 3D, or any other dimension. It exists only here.

  9. The 120-cell: 120-Cell

    Step 9 · 120-Cell

    The 120-cell

    120 dodecahedra closing into a hypersphere. Stereographic projection maps that sphere into our space, turning straight edges into arcs, like a map of the globe.

  10. The Hopf fibration: Hopf Fibration

    Step 10 · Hopf Fibration

    The Hopf fibration

    The 3-sphere can be filled entirely with circles, no two touching, every pair linked like chain links. Projected here, they nest into tori of rings.