Tesseract

A tesseract, also known as a 4-cube or an 8-cell, is a four-dimensional hypercube. In geometry, it is classified as a convex regular 4-polytope. Just as a square is a two-dimensional shape and a cube is a three-dimensional shape, the tesseract exists in the fourth dimension. It serves as the fundamental measure polytope for hypervolume. This means it is used as a basic unit to measure space in four dimensions.

To understand how a tesseract is built, we can follow a pattern of increasing dimensions. First, we start with two points connected by a specific length to form a line segment. If we move an identical line segment in a perpendicular direction, it sweeps out a square, or a 2-cube. A square consists of four vertices and four edges.

If we move that square in a direction perpendicular to its plane, it generates a cube, or a 3-cube. This cube has eight vertices, twelve edges, and six square faces. Finally, moving the cube into the fourth dimension generates the tesseract.

The structure of a tesseract is highly complex and precise. It is bounded by eight cubical cells, which act as its outer boundaries. Every cube in the tesseract shares its faces with other cubes. At every edge, three cubes and three squares meet. At every vertex, the structure is even more crowded, with four cubes, six squares, and four edges meeting at a single point. In total, a single tesseract consists of sixteen vertices, thirty-two edges, twenty-four squares, and eight cubes.

Mathematics provides different ways to describe these properties. For example, the tesseract can be viewed as a 4-4 duoprism, which is the Cartesian product of two squares. It also has a Schläfli symbol of {4,3,3}, representing its regular polytope nature. The tesseract possesses hyperoctahedral symmetry of order 384. Interestingly, the tesseract is the only hypercube, besides a zero-dimensional point, that is radially equilateral. This means its radius, the distance from the center to a vertex, is equal to its edge length.

The history of the term "tesseract" is tied to early explorations of higher dimensions. The word was first used by Charles Howard Hinton in his 1888 book, *A New Era of Thought*. He originally spelled it "tessaract," but changed it to "tesseract" in his 1904 book, *The Fourth Dimension*. The name comes from Greek roots meaning "four rays." This refers to the four edges that extend from each vertex to other vertices.

Because we live in a three-dimensional world, we cannot see a tesseract directly. Instead, we use projections or "nets" to study it. A net is an unfolding of a polytope into a lower dimension. There are 261 distinct nets of a tesseract, and each one can tile 3-space.

We can also create projections, which are like shadows. A vertex-first parallel projection of a tesseract into 3D space creates a shape called a rhombic dodecahedron.

Other projections can result in a cubical envelope or a hexagonal prism.

Artists and architects have used the concept of the tesseract for inspiration for decades. In 1954, the Spanish surrealist Salvador Dalí created the painting *Corpus Hypercubus*, which depicts a tesseract unfolding.

In architecture, the Grande Arche near Paris, France, was designed by engineer Erik Reitzel to resemble a projected tesseract. This massive monument was completed in 1989.

The tesseract also appears in science fiction, such as Robert Heinlein's 1940 story "And He Built a Crooked House." These examples show how a mathematical idea can influence art and design.