What Is a Tesseract or Hypercube?


Tesseract or Hypercube
A tesseract or hypercube is the four-dimensional equivalent to a cube. In three dimensions, it is like a cube within a cube, except if all the vertices were connected by 90 degree angles.
Animated GIF of a tesseract
This animated GIF is a two-dimensional representation of a four-dimensional tesseract or hypercube. (Jason Hise)

A tesseract or hypercube is the four-dimensional equivalent to a cube, much like a cube is a three-dimensional equivalent to a square. While a cube has six square faces, a tesseract consists of eight cells.

It isn’t possible to represent a four-dimensional object in three-dimensional space, much less on a two-dimensional screen. But, you can consider a tesseract what you get if you have a cube-within-a-cube. Except, all of the vertices form right angles to one another. Rotating such an object appears very different from what you get if rotate a three-dimensional object.

Tesseracts are popular in art and science fiction. Salvador Dali painted a hypercube in his 1954 Crucifixion. Robert Heinlein described a tesseract buildng in his 1940 short story “And He Built a Crooked House.” Madeleine L’Engle describes a tesseract as a shortcut between three-dimensional places in her 1962 book “A Wrinkle in Time.” The Marvel Cinematic Universe includes a glowing blue crystalline tesseract.

But, the concept of a tesseract and other higher dimensional objects has practical applications, too. For example, virologists construct four-dimensional maps of DNA sequences, where each component of a three-dimensional DNA molecule has one of four possible attributes (A, T, G, or C). Spreadsheets and databases commonly form four-dimensional (or higher) shapes. The nested commands within computer programs also extend beyond three dimensions. For example, consider a spreadsheet consisting of three pages (which could be printed to form a three-dimensional object), where the elements in each layer link to new pages. The new pages add another dimension, yet you can’t print them in the normal 3D world to see the way the parts of the spreadsheet link together.

More Tesseract and Hypercube Names

The most common names for this four-dimensional shape are tesseract or hypercube, but the shape also goes by names tetracube, eight-cell, C8, cubic prism, octahedroid, and octachoron.

Tesseract Properties

Here is a quick summary of the properties of a tesseract or hypercube:

  • A tesseract is built from 8 cubes.
  • All of the lines that form the faces of the cubes are equal in length.
  • Al of the lines meet at right angles to each other.
  • A tesseract has 16 vertices.
  • A tesseract has 24 edges.
  • The shape has 36 edges.

From Zero Dimensions to Four Dimensions

A good way to grasp the concept of a tesseract is to consider the properties of objects as you move from one dimensions up to four dimensions.

  • A point has zero dimensions. It lacks length, width, or height.
  • A line has one dimension, which is length. A line is bounded by two zero-dimensional points.
  • A square has two dimensions, which are length and width. A square is bounded by four one-dimensional lines.
  • A cube has three dimensions, which are length, width, and height. A cube is bounded by six two-dimensional sides.
  • A tesseract or hypercube has four dimensions. A tesseract is bounded by eight three-dimensional cubes.

Note that moving up each dimensional step involves adding two more boundaries.

This video illustrates and explains the tesseract using math. (If math isn’t your strong suit, then skip to the video below it for a basic explanation.)

Still confused? Here’s an excellent explanation of how higher dimensions work and what they look like in our 3D world. In particular, check out the discussion of the shadow of a 4D cube (timestamp 3:40):

References

  • Coxeter, H.S.M. (1969). Introduction to Geometry (2nd ed.). Wiley. ISBN 0-471-50458-0.
  • Hall, T. Proctor (1893) “The Projection of Fourfold Figures on a Three-Flat“. American Journal of Mathematics 15:179–89. doi:10.2307/2369565
  • Johnson, Norman W. (2018). “§ 11.5 Spherical Coxeter groups“. Geometries and Transformations. Cambridge University Press. ISBN 978-1-107-10340-5.
  • Sommerville, D.M.Y. (2020) [1930]. “X. The Regular Polytopes“. Introduction to the Geometry of N Dimensions. Courier Dover. pp. 159–192. ISBN 978-0-486-84248-6.

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