Hyperplane Explorer

You stand in a 3D slice of a 4D world. Everything you see is where a 4D solid crosses your slice. Step ana or kata to slide the slice along the fourth axis, or rotate it to cut the same solids at a new angle.

Play 4D chess → 4D spinning cat →

View keys 1 · 2 · 3

Move hold to keep going

within slice

height

4th axis

Rotate the slice

TurnR–F plane

Pitch forwardF–A plane

Swing sidewaysR–A plane

Volume view V

Lays your slice level: the view turns 90° in the up–ana plane, so your ana direction shows as up (that’s w until you tilt) and R/F step through heights. Your own frame and the other controls don’t change.

Shapes

kata −6your slice+6 ana

Each bar is a shape's extent along your ana direction (kata to the left). Bars crossing the line intersect your slice and are drawn in the view. Tap a name to center your slice on that shape; use the eye to show or hide it.

4D Rubik’s cube solved

Layer axis

Layer

Aim

4D tricks

Puzzle

A 3×3×3×3 with 80 movable pieces. A twist turns one layer (a whole 3×3×3 block) 90° in a plane that excludes the layer’s own axis. The selected layer is outlined in violet.

Earth's layers in view

Crust0–35 km*
Upper mantle35–660 km
Lower mantle660–2,890 km
Outer core, liquid2,890–5,150 km
Inner core, solid5,150–6,371 km

Depths are measured from the surface. The globe is an ordinary 3D ball with a thin extent of 0.6 along w, so with your slice level you see the whole planet. Tilt into ana (I/K or U/O) and the slice cuts it open: the flat faces show the interior. *Crust drawn about 5× thicker than real so it stays visible.

Klein bottles

Klein bottle. A ring whose tube turns half a turn through w as it goes around, so it comes back flipped. No inside, no outside, and in 4D it never passes through itself. It is a surface, so slices are curves. In Shadow, the bright line starts on top of the tube and returns underneath; it takes two laps to close.

3D Klein bottle (S² × S¹). The same idea one dimension up: a sphere carried around a loop that comes back mirrored, inside out. It can’t fit in 4D without passing through itself, just as the glass Klein bottle can’t in 3D. Its slices are surfaces: a level slice through its middle is exactly the glass bottle, and stepping ana pinches off the thin neck first. In Shadow, the bright lines are the glass bottle’s two outlines; after one lap each becomes the other.

4D Möbius strip Möbius strip

A Möbius strip is a segment carried around a loop that comes back flipped. Carry a disk instead and bring it back mirrored, and the flip has to go through w: a solid that only fits in 4D. Your slice cuts each disk in a segment, so you see a ribbon. Where it crosses every disk, the ribbon is an ordinary Möbius strip, with one edge (drawn bright) that runs around twice; elsewhere it breaks into open ribbons. Tap its name to go to a Möbius slice. Its whole surface is a Klein bottle, the way a Möbius strip’s edge is a circle.

Your frame basis in world x y z w

The compass in the corner of the view shows the world x, y, z and w axes from your camera. An arrow is shorter the more its axis leaves your slice, and dashed once it is mostly out of it. Axes too short to draw are listed under it: ⊥ means mostly out of your slice, · means pointing along your line of sight.

Spin solids in 4D