HYPERCUBES

  What is a hypercube?

We now look at a different sequence of objects. Again the figures we get will be skeletal, in the sense that the construction will give the vertices and edges of each figure.

In 0 dimensions we have a point. (That is still all there is.)

In 1 dimension, we take the point and move it unit distance, drawing in the trace path of the vertex, to obtain a line segment.

In 2 dimensions, we take the line segment and move it a unit distance perpendicular to itself, drawing in the trace paths of the vertices, to obtain a square.

In 3 dimensions, we take the square and move it a unit distance perpendicular to itself, drawing in the trace paths of the vertices, to obtain a cube.

The general figure defined in this way is called a hypercube. If you want to specify the dimension d, you can speak of a d - hypercube. For example a cube is a 3-hypercube.



•  Now, what statement would define the figure in 4 dimensions? (You may have problems understanding the statement!)

The first four members of the sequence of figures we obtain will look like this (we have added some shading for the faces):

How would we picture the 4-dimensional hypercube? Again there are several options: we could construct a 3-dimensional model, or draw a 2-dimensional picture.

First of all, look at the picture of the cube above (right). We can ‘see’ that this represents a cube because we know what a 3-dimensional cube looks like. But really, it is just a 2-dimensional picture. We can think of it as two congruent and similarly placed squares with corresponding vertices joined together.

• 1. This might suggest what a 3-dimensional picture of a 4-dimensional hypercube will look like. Can you describe it?
      
   2. What would a 2-dimensional picture of a 4-dimensioal hypercube look like? (How many vertices? How are they joined?) Try drawing a couple of pictures.

• 1. This might suggest what a 3-dimensional picture of a 4-dimensional hypercube will look like. Can you describe it?
      
   2. What would a 2-dimensional picture of a 4-dimensioal hypercube look like? (How many vertices? How are they joined?) Try drawing a couple of pictures.

To obtain the 3-dimensional representation, we take a two cubes which are similarly placed, and join each pair of corresponding vertices. The figure at left below gives a picture with congruent cubes. A neater picture (and one which produces a nice model when made of struts and joiners) has one cube inside the other, as in the central figure below. You can think of this model having some perspective! After all, you can represent a cube quite realistically in the same way with one square inside the other (see figure at right below).

               

Of course, each of the above pictures of the 4-hypercube is automatically a 2-dimensional representation. One of the prettiest such figures is given below.




  Properties of the hypercube

Let us now proceed for the hypercubes as we did with the simplexes, investigating whether there are nice relationships between the quantities V, E, F and C. We work with the extended table we introduced for the simplexes (deciding that the effort is worthwhile!).

 
Point
Line Segment
Square
Cube
4-hypercube
  5-hypercube
1
1
1
1
1
V
0
1
2
4
8
E
0
0
1
4
12
F
0
0
0
1
6
C
0
0
0
0
1
C4
0
0
0
0
0
C5
0
0
0
0
0

• See if you can fill in the missing values for the 4-hypercube and 5-hypercube. Several of the missing entries are obvious. But there is again a simple general rule which relates various entries in the table. Can you see what it is? (Unfortunately our artificial top row does not fit the construction rule this time, but we shall keep it for a different reason – to be revealed later!)

Check your answers ...




































Your completed table should look like this:

 
Point
Line Segment
Square
Cube
4-hypercube
  5-hypercube
1
1
1
1
1
1
1
V
0
1
2
4
8
16
32
E
0
0
1
4
12
32
80
F
0
0
0
1
6
24
80
C
0
0
0
0
1
8
40
C4
0
0
0
0
0
1
10
C5
0
0
0
0
0
0
1

It is easy to fill in the top two rows, the diagonal of 1s, and the zeros. After filling in the top two rows, each non-zero entry is the sum of twice the entry to its left and the entry above that. This means that we can fairly confidently give the values for a hypercube of any dimension.

Again this doesn’t prove that these results are the correct ones. But looking back on our method of construction, we can understand where the factor 2 comes from.


Let us check our extended Euler’s Formula.

For the line segment : V = 2.
For the square : V – E = 0.
For the cube : V – E + F = 2.
For the 4-hypercube : V – E + F + C = 16 – 32 + 24 – 8 = 0.   ...      As expected!


A nice sequence of 2-dimensional representations from the 2-hypercube (square) to the 7-hypercube is given in Wolfram Math World (reference and link below):


  Other sequence properties

In 1 dimension, a line segment of (edge-) length 2 can be represented algebraically by   –1 x 1.

In 2 dimensions, a square of edge-length 2 can be represented by   –1 x, y 1.

In 3 dimensions, a cube of edge-length 2 can be represented by   –1 x, y, z 1.

It is not hard to guess the representation for further members of the sequence!


In the case where our hypercubes are regular, (which has automatically occurred here with our definition), we can continue the labelling we used for the square and cube.

Thus in 2 dimensions (the plane), we denote the square by  {4}.

In 3 dimensions, we denote the cube by {4, 3}.

In 4 dimensions, we denote the hypercube by {4, 3, 3}.

In general d-dimension (d > 2), we denote the d-hypercube by {4, 3, ... 3} – a 4 followed by (d – 2) 3s.

We won’t try to justify this in general, but in the case of the cube the faces are square {4}s, with 3 meeting at each vertex. Or, we can say that each vertex figure is an equilateral triangle {3}. In 4 dimensions, the cells are cubes {4, 3}, with 3 cubes meeting at each vertex (see figure at right). This actually means that each ‘vertex figure’ is a tetrahedron {3, 3}. So we can think of the symbol {4, 3, 3} as being a contraction of {4, 3} and {3, 3}.



  The 4-hypercube

The hypercube in 4 dimensions is sometimes called a tesseract (the Greek tesseres means ‘four’).

You might recall that the model of the cube can be constructed from a net of six squares. The analogue for the 4-hypercube would be a ‘net’ of eight cubes, as shown below. You may have some trouble folding them! This does make a nice 3-dimensional model though.

It is interesting that this configuration of cubes is used by Salvador Dali in his 1954 painting Crucifixion or Corpus hypercubicus. A host of literary and film references to the 4-hypercube is given in http://en.wikipedia.org/wiki/Tesseract.




  References

Abbott, E. A., Flatland, a romance of many dimensions, Dover (Edition 6) 1953.

(The full text of this delightful classic is now available online: http://www.ibiblio.org/eldritch/eaa/FL.htm .)

Coxeter, H.S.M., Introduction to Geometry, Wiley 1969. (See Chapter 22.)

Coxeter, H.S.M., Regular Polytopes, 2nd Edition (Macmillan 1963). (This is more of a specialist book.)

Wikipedia: http://en.wikipedia.org/wiki/Tesseract

Wolfram Math World : http://mathworld.wolfram.com/Hypercube.html