Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

The Tesseract

March 2000

"Mathematics spans all dimensions" is the theme for the coming Math Awareness Month 2000. As in the past years, the Math Forum hosts a site devoted to the event that opens with a beautiful interactive poster. The poster highlights dimensions 0, 1, 2, 3, and 4. Probably in order to keep the work to a manageable amount, creators of the site have wisely skipped all the fractal dimensions of which we all are aware nowadays. This of course opens doors to a zenonean inquiry, how does one get, say, from 1 to 2 with infinitude of dimensions in-between? On the other hand, the site gives an inspiring coverage to the human dimension of mathematics.

Hypercube is a multidimensional analogue of a 3-dimensional cube in that each coordinate of a point in a hypercube is restricted to the same 1-dimensional (line) segment. Tesseract is a 4-dimensional hypercube. In anticipation of MAM 2000, a remark by A. K. Dewdney served an additional reason to write about the tesseract. Wrote he, "Dimensions seem to creep up in everywhere as HYPERCUBE is written." Dewdney was referring to matrices (2-dimensional objects) and vectors (1-dimensional objects) that are part of any modern computer language. They are also handy in describing and manipulating multidimensional objects. It's a tribute to these mathematical notations that they make a CUBE variant of the program virtually indistinguishable from its HYPERCUBE analogue.

The first applet below serves to demonstrate the inductive construction of the tesseract. (Links to other related sites are listed at the bottom of the page.)

Press the Start button to begin the demonstration. The label then converts to Continue. Keep pressing the Continue button to watch the successive steps of the construction. When finished, you'll be able to rotate the tesseract with sliders or by dragging the mouse. (You may also Skip the demonstration but remember that holding down the Shift or Control key changes the plane of rotation.)

The applet also shows the cross-section of the tesseract by a hyperplane given by the equation:

Ax + By + Cz + Dh = E.

(Throughout, the fourth coordinate is denoted by h or H as a reminder that we deal with a hyperspace.)

alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

The number of vertices doubles with every dimension: the segment has 2 of them, the square 4, the cube 8, and the tesseract has 16. In general, the n-dimensional hypercube has 2n vertices. Such a hypercube is built up of (n-1)-, (n-2)-, ..., and 0-dimensional elements. The inductive construction provides a clue to the formula [M. Gardner] used to calculate their number: for the hypercube these appear as the coefficients of the expanded polynomial (2x + 1)n. For example,

(2x + 1)4 = 16x4 + 32x3 + 24x2 + 8x + 1

which says that, in addition to 16 vertices, the tesseract has 32 edges, 24 squares, and 8 cubes - all in 1 tesseract.

What the applet shows is only a 2-dimensional projection of the tesseract. (A stereoscopic view is available on the Web.) The difference of 2 dimensions makes it difficult to depict a 4-dimensional object on a flat 2-dimensional screen. We try learning by analogy.

A segment, as a portion of a line (a 1-dimensional space), is bounded by two points, each a 0-dimensional object. A 2-dimensional square is bounded by 4 1-dimensional segments. A 3-dimensional cube is bounded by 6 2-dimensional squares. A 4-dimensional tesseract is bounded by 8 3-dimensional cubes.

In a horizontal plane, a square has an upside and a downside. Only one is visible when its rotation is confined to the plane. In the 3-dimensional space both sides are in principle visible. In 3D, a cube has an inside and an outside. However it is turned in the 3-dimensional space, only its outside is visible, the inside remains hidden. In 4D, a cube can be turned inside out by rotating around one of its 2-dimensional faces. That's right. In 2D, we can only rotate a shape around a point. In 3D, we can also rotate around a 1-dimensional axis - for example, an edge in the case of a cube. In 4D, a shape can be rotated around a plane. (In the above applet one can clearly observe the phenomenon by fixing the location of the origin.) It must be understood that in 4D a 3-dimensional cube has neither inside nor outside. All points of a cube are as much exposed in 4D as are the points of a square in 3D. (This is what makes a prospect of 4D-travel so unpleasant. It also follows from the above that 4D-travel is extremely dangerous. Back in 3D, a traveller may find himself in a state of excessive introversion.)

Vacuously, in a square there is only 1 square that contains a given edge. In a cube, every edge is shared by 2 squares. In a tesseract, 3 squares meet at every edge. Taken pairwise, squares through the same edge define three cubes. Detecting the three cubes seems akin to shifting a view point when observing the Necker cube.

I found this observation useful when playing with the applet below. What is it about? Travelling in 4D may have a milder effect on a 3D body than turning it inside out. It may only change its orientation. For example, a left-hand glove sucked into 4D may on return fit the right hand instead. (Future 4D travel guides are bound to offer an advice to the effect that gloves and shoes should always be carried in pairs.)

In the applet below, two repers - a pair of perpendicular segments - are randomly placed on one of 24 squares of the tesseract. One reper remains on that square for the duration of the experiment. The other reper can be moved to any of the 8 squares that have a common edge with the current one. (Obviously, there are 8 candidate squares, right?) The reper moves without rotation: if the two squares (the from-square and the to-square) were placed on the same plane, the reper would just glide from one to the other. The task is to take the moving reper on a ride at the end of which, back at the original square, the two repers will have different orientations.

The tesseract is the set of points

{(x, y, z, h): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1, 0 ≤ h ≤ 1}.

Its boundary cubes are defined by fixing value of one of the coordinates to either 0 or 1. This is why there are 8 of them. Each of the 24 squares is defined by fixing values of any two coordinates. There are 6 possible pairs and 4 possible values (00, 01, 10, 11) for each. Every square is assigned a 4 symbol name. X01H, for example, denotes the square for which y = 0 and z = 1.

alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

Note that any chain of squares that solves the problem forms a one-sided surface, like a Möbius strip. After many attempts I discovered the shortest one. To say how short it is is to give the solution away because it is so short it actually carries a proof that it is the shortest. I was very pleasantly surprised when I realized how simple it is. The solution is hinted to somewhere on this page. I can offer another hint: experience with toy engines and railways may prove handy.

Isn't it a good example of how mathematics adds a dimension to one's life?


  1. T.F. Banchoff, Beyond the Third dimension, Scientific American Library, 1996
  2. A.K. Dewdney, The Armchair Universe, W.H. Freeman & Co, 1988
  3. M. Gardner, Mathematical Carnival, Vintage Books, 1977
  4. I. Stewart, Concepts of Modern Mathematics, Dover, 1995

On the Web

  1. Tesseract, Eric's World of Mathematics
  2. The Tesseract (or Hypercube), A guided demonstration, Geometry Center, University of Minnesota
  3. Stereoscopic Animated Hypercube by Mark Newbold

Generating Functions

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Copyright © 1996-2017 Alexander Bogomolny


Choose any of the four edges of the given square and the other two squares that share that edge with the given one.

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