In this section particular constructions are presented.

As the first example let and . *
Cellular
* calculates and

`Martin`

Since is symmetric, . Now *
Growing
* chooses multiplication coefficient , extension vector and uses

`Colour`

*
Colour
* arranges copies into a blocks sized array receiving

*
Colour
* receives the indexing scheme and the colouring matrix transforming the elements of into digit length -ary numbers: .

Finally we colour the matrix using – that is multiply the elements of by and add the -th block of to both cells of the -th copy in :

Since , we use *
Colour
* again with and get the (8,1,2,64)-perfect sequence repeating 4 times, using the same indexing array and colouring array .

Another example is and . To guarantee the cellular property now we need a new alphabet size . Martin produces a (6,1,2,36)-perfect sequence , then *
Colour
* results a (12,1,2,144)-perfect sequence .

As the first example let and . Then . We construct the even sequence using *
Even
* and the symmetric perfect array in Figure 29.1.a using the meshing function (29.5). Since is symmetric, it can be used as . Now the greatest common divisor of and is 2, therefore indeed .

*
Growing
* chooses and

`Colour`

*
Colour
* uses the indexing scheme containing indices in the same arrangement as it was used in . Figure 29.1.b shows .

Transformation of the elements of into 4-digit -ary form results the colouring matrix represented in Figure 29.2.

Colouring of array using the colouring array results the (4,2,2,16)-square represented in Figure 29.3.

In the next iteration *
Colour
* constructs an 8-ary square repeating
times, using the same indexing scheme and colouring by . The result is , a -perfect square.

If , then the necessary condition (29.4) is for double cubes, implying is a cube number or is a multiple of 3. Therefore, either and then , or and so , that is, the smallest possible perfect double cube is the (8, 3, 2, 256)-cube.

As an example, let and . *
Cellular
* computes ,

`Mesh`

`Shift`

`Optimal-Martin`

`Cellular`

`Shift`

Let be a sized perfect, rectangular matrix, whose 0th layer is the matrix represented in Figure 29.1, and the -perfect array in Figure 29.5, where and .

*
Growing
* uses

`Colour`

has the size and . *
Colour
* gets the colouring matrix by transforming the elements of into 8-digit -ary numbers – and arrange the elements into sized cubes in lexicographic order – that is in order (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1). Finally colouring results a double cube .

contains elements therefore it is presented only in electronic form (on the home page of the first author).

If we repeat the colouring again with , then we get a 64-ary sized double cube .

In 4 dimensions the smallest 's satisfying (29.3) are and . But we do not know algorithm which can construct -perfect or -perfect hypercube. The third chance is the -perfect hypercube. Let and . *
Cellular
* calculates , then calls

`Optimal-Martin`

`Cellular`

`Mesh`

Now *
Shift
* calls

`Optimal-Martin`

`Cellular`

`Shift`

Up to this point the construction is the same as in [
115
], but now , therefore we use *
Shift
* again to get a -perfect prism, then we fill an empty cube with -sized prisms and finally colouring results the required 4-dimensional hypercube.

**Exercises**

29.6-1 Explain the construction of Figure 29.5.