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Ax + By + Cz + Dh = E.
(Throughout, the fourth coordinate is denoted by h or H as a reminder that we deal with a hyperspace.)
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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,
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(2x + 1)4 = 16x4 + 32x3 + 24x2 + 8x + 1
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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.
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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
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{(x, y, z, h): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1, 0 ≤ h ≤ 1}.
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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.
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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?
References
- T.F. Banchoff, Beyond the Third dimension, Scientific American Library, 1996
- A.K. Dewdney, The Armchair Universe, W.H. Freeman & Co, 1988
- M. Gardner, Mathematical Carnival, Vintage Books, 1977
- I. Stewart, Concepts of Modern Mathematics, Dover, 1995
On the Web
- Tesseract, Eric's World of Mathematics
- The Tesseract (or Hypercube), A guided demonstration, Geometry Center, University of Minnesota
- The Tesseract, A look into 4-dimensional space, Harry J. Smith
- Math Expands, The Math Forum
- Math Expands: Madeleine L'Engle, The Math Forum
- Stereoscopic Animated Hypercube by Mark Newbold
- A Property of the Powers of 2
- An USAMTS problem with light switches
- Examples with series of figurate numbers
- Euler's derivation of the binary representation
- Examples with finite sums with binomial coefficients
- Fast Power Indices and Coin Change
- Number of elements of various dimensions in a tesseract
- Straight Tromino on a Chessboard
- Ways To Count
Copyright © 1996-2008 Alexander Bogomolny
Solution
Choose any of the four edges of the given square and the other two squares that share that edge with the given one.
Copyright © 1996-2008 Alexander Bogomolny
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