tesseract

tesseract, geometric shape that is the four-dimensional equivalent of the three-dimensional cube. Because a tesseract cannot be accurately pictured in two or three dimensions, it is often approximated as a cube within a cube.

British mathematician Charles Howard Hinton first introduced the tesseract in his books A New Era of Thought (1888) and The Fourth Dimension (1904). The etymology of the word tesseract has been the source of some confusion. Hinton first spelled the word tessaract. This spelling combines the Greek word tessara, meaning “four,” with the Greek word act meaning “rays.” Thus, tessaract could mean “four rays,” referring to the spatial axes of the hypercube. However, in The Fourth Dimension, Hinton spelled the word tesseract, possibly based on Latin word origins since the Latin word tessera means “cube.” This spelling is the most widely used today.

The geometry of a tesseract is not easily understood. It can be helpful to begin with lower geometric dimensions to grasp the concept. In geometry, the zeroth dimension is represented by a point that has no length, width, or height. In one-dimensional geometry, a point is extended in one direction to another point to create a line segment that has length but no width or height. In two-dimensional geometry, a line segment can be extended at right angles to itself to form a flat square with equal length and width but still no height. The flat square has four vertices and one surface. In three-dimensional geometry, the flat square is raised at a right angle to itself to form a cube with equal length, width, and height. The cube has six faces and eight vertices that define its three-dimensional volume.

However, in four-dimensional geometry, shapes become more complicated. In a four-dimensional universe, the actual tesseract consists of eight cubes. The cubes have a total of 16 vertices, and the four lines that meet at each vertex all form 90-degree angles to each other, which is impossible in three-dimensional geometry. As a result, a 4-D tesseract has 8 cubes, 16 vertices, 24 faces, and 32 edges—which cannot be represented in 3-D space.

Even though Hinton discussed the fourth dimension only as space and not time, his work anticipated the development of the union of the three spatial dimensions and one temporal dimension in the concept of space-time used in relativity. His vision of a tesseract also anticipated the concept of “world lines” or paths taken by objects in four-dimensional space-time.

The tesseract has been a popular theme in science fiction. Robert Heinlein used the tesseract in his short story “—And He Built a Crooked House—” (1941), in which an architect builds a three-dimensional unfolding of a tesseract that collapses into an actual four-dimensional tesseract after an earthquake. Perhaps the most famous fictional example is in Madeleine L’Engle’s A Wrinkle in Time (1962), in which L’Engle takes artistic license with the science of the tesseract, using it as a means of traveling through space-time.

The tesseract has even been featured in fine art. Artist Salvador Dalí, in his painting Crucifixion (Corpus Hypercubus) (1954), portrayed Christ floating with his arms outstretched in front of a three-dimensional cross composed of eight cubes.