How to build a four-dimensional cube, one sliding step at a time
Slide a line segment sideways and it sweeps out a square. Push that square at right angles to itself and you get a cube. Now shove the cube along a fourth direction we cannot picture, and the result is a tesseract, a shape bounded by eight cubes that Salvador Dalí unfolded into a crucifixion painting.
Each step doubles the corners. A square has four vertices and four edges; a cube has eight vertices, twelve edges and six square faces. The tesseract, also called a 4-cube or 8-cell, has sixteen vertices, thirty-two edges, twenty-four squares and eight cubical cells meeting at right angles. Three cubes gather around every edge, and at each corner four edges meet, so the shape formed around a vertex is a regular tetrahedron. It is one of just six convex regular polytopes in four dimensions.
You can unfold a cube into a flat cross of six squares, and likewise a tesseract into three-dimensional nets of eight cubes. There are 261 distinct ones, and every one can fill ordinary space without gaps. The best known is the Dalí cross, a column of four cubes with four more attached around the second from the top, depicted in the 1954 painting Corpus Hypercubus.
The name comes from Charles Howard Hinton, who coined it in his 1888 book A New Era of Thought, first spelling it tessaract before settling on the current form in 1904. It blends Greek words for four and ray, referring to the four edges leaving each corner. Science fiction has since borrowed the term freely, often with little link to geometry.
The shape has practical uses. Because all edges are equal, linking parallel processors in a tesseract pattern keeps any two nodes at most 4 steps apart, with many alternative routes for balancing traffic. Mathematicians have also counted 92487256 ways to carve it into four-dimensional simplices sharing its corners; the fewest pieces any such division needs is 16.
Source: Tesseract
