How to see a cube in the fourth dimension: its shadow in our space, its slices, and counting its edges and corners with help from Flatland, a two-dimensional world.
Step 1 of 4 · A tesseract's shadow
Step 1 of 4
A tesseract’s shadow
We cannot see four dimensions, but we can check how a flat creature sees three. In Flatland, a world on a sheet of paper, a lamp above a cube casts its shadow on the sheet: a square inside a square. The same works one dimension up: the tesseract, the cube of the fourth dimension, casts a shadow into our space, and the fourth direction is called .
Step 2 of 4
Counting
Each shape in the series is two copies of the one before, joined corner to corner: a segment from two points, a square from two segments, a cube from two squares, and a tesseract from two cubes along . So the number of corners doubles at each step: .
Step 3 of 4
Slices
Now not a shadow but a slice: our space cuts through the tesseract as Flatland’s sheet cuts through a cube. When the cube passes through the sheet Corner first, Flatland sees a triangle near the tip. The tesseract’s slice in the same pose is a tetrahedron. The Pose is fixed, and you move the Cut height yourself.
Step 4 of 4
Turning in the fourth dimension
Turn in 4D turns the tesseract in a plane that includes . With no turn, a whole cell faces the lamp, and its eight corners are nearer to it than all the rest. After a turn the blue cell slides out, and a square face faces the lamp.
Loading the interactive version…
Loading the interactive version…
Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.
Practice · Choose
In the tesseract’s shadow, Square on to the lamp, you see a small cube inside a big one. What is the small cube?
Practice · Choose
In the tesseract’s shadow, Square on to the lamp, you see a small cube inside a big one. What is the small cube?
Practice · Calculate
A cube is two squares, plus an edge joining each corner of one to the matching corner of the other: edges. The tesseract is two cubes joined the same way along . How many edges does it have?
Practice · Calculate
A cube is two squares, plus an edge joining each corner of one to the matching corner of the other: edges. The tesseract is two cubes joined the same way along . How many edges does it have?
Practice · Predict, then look
The tesseract passes through our space Corner first, and the slice at this height is a tetrahedron. What will the slice be when you move the Cut height to the middle?
Practice · Predict, then look
The tesseract passes through our space Corner first, and the slice at this height is a tetrahedron. What will the slice be when you move the Cut height to the middle?
Practice · Calculate
After a turn in the fourth dimension, a square face of the tesseract faces the lamp. How many of the tesseract’s corners are nearer the lamp than all the others?
Practice · Calculate
After a turn in the fourth dimension, a square face of the tesseract faces the lamp. How many of the tesseract’s corners are nearer the lamp than all the others?
Lesson complete. Every practice question here is done.