A four-dimensional cube turning through a direction we cannot point to. Its far cube swells, passes through the near one and turns inside out, edges never bending.
Ludwig Schläfli, 1852 · Charles Howard Hinton, 1888
A square drawn in perspective is a small square inside a big one, joined at the corners; that is a cube seen down one axis. Do the same one dimension up and you get the tesseract: a small cube inside a big one, joined corner to corner, eight cubes in all, the same as a cube has six squares. Charles Howard Hinton coined the name in 1888 and spent years teaching people, with coloured blocks, to picture it.
Here it turns in a plane that contains the fourth direction, so the cube that was far away comes round to the front: it swells, sweeps out through the cube that was near, and the whole thing seems to turn inside out. Nothing is really bending, which is the clue: every edge stays straight and every edge stays the same length in four dimensions, as a spinning cube's square faces turn into thin diamonds and back without changing at all.
Ludwig Schläfli found in 1852 that four dimensions have six regular solids, one more than three dimensions; four of them are here: the simplex, with five cells, the tesseract, with eight, the 16-cell, its dual, and the 24-cell, which has no counterpart in any other dimension. Echoes draw the same shape a moment earlier, fainter, as a long exposure would.
Choose Double to turn in two planes at once, the way only four dimensions allow: a rotation with no fixed axis at all.
Behind the mathematics
Named the tesseract, A New Era of Thought, 1888
British mathematician and writer who spent years teaching people to picture the fourth dimension, with coloured cubes, and who coined the word tesseract in A New Era of Thought, 1888.
Found the six regular polytopes of four dimensions, 1852
Swiss mathematician who worked out the geometry of many dimensions decades before others took it up, and found that four dimensions have six regular polytopes.
