🚀 go-pugleaf

The Infinite Pavement                               February 1989

I have been impressed by demonstrations of the computation of
parts of the Mandelbrot set using a many transputer network; the
more so as I have seen the time that such computation can take on
a personal computer.
I decided that I would like to design a two dimensional surface
myself, where the design is based upon integers, so that the
design was automatically at its maximum fineness of structure
right at the start. I decided to base my design upon whether any
given integer is or is not a prime number. I decided on the
layout of numbering pixels as in figure 1.
I devised the design primarily as an example that might well be a
good subject for a demonstration of transputer speed, thinking
that it might be possible to move from one part to another part
of the design at high speed and investigate just how fast a
network of transputers could compute a screenful when huge
numbers were encountered. I believe that screenfuls of the design
would take much longer to compute as the numbers get larger. I do
not, as yet, have access to a many transputer system with colour
graphics suitable for this work and I do not as yet know how to go
about programming a graphic output of the infinite pavement on a
many transputer system, but I have produced a few diagrams using
pascal on a unix system. A part of one of these diagrams, showing
the start of the infinite pavement design, is shown in figure 2.
In the event, I was amazed to find the various straight lines,
and straight lines with interestingly spaced gaps, that appear in
the design of the infinite pavement.
The design is quite simple to produce. All of the numbered pixels
are coloured as colour 0, except for those pixels where the
number is prime, which are coloured as colour 1. The numbers 0
and 1 in the design are coloured as colour 1.

   0
   1   2   3
   4   5   6   7   8
   9  10  11  12  13  14  15
  16  17  18  19  20  21  22  23  24
  25  26  27  28  29  30  31  32  33  34  35
  36  37  38  39  40  41  42  43  44  45  46  47  48
  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63
   .
   .
 etc
   .
   .

Figure 1. The numbering of pixels in the infinite pavement.


An interesting extension of the design is shown in figure 3.
Here, those pixels not in the infinite pavement itself, and thus
not already colourable as colour 0 or colour 1, are coloured in
colour 3, if and only if the number of the pixel is prime, with
the pixel numbered 1 in the top row being coloured as colour 3.
The remaining pixels are colourable as colour 2.


Sheet 1 of 2, The Infinite Pavement, by W. J. G. Overington. 







*
***
.*.*.
..*.*..
.*.*...*.
....*.*....
.*...*.*...*.
....*.....*.*..
...*...*.*.....*.
..*.....*.......*..
.*.*...*.*...*.......
......*...*.....*.*....
.....*.*.....*.....*...*.
....*.....*.*.........*.*..
.*.*...........*...........*.
..*.*...*.....*.*.........*....
.*.....*.....*.*.....*...*.*.....
....*.............*...*.*...*......
.......*.....*.........*.*...*.....*.
......*.....*.....*...*.....*.......*..
.*.......*.........*.*.........*.*.....*.
..*.....*.......*...*.*...*...........*....
...*...*.......*...*.....*...........*.*.....
............*.....*.........*.....*.....*.*....
.*.........*.....*.....*.*.....*.....*...*.*.....
......*.........*.*...*.....*.....*.*...........*..
.*.....*.......*.........*.......*.........*.......*.
....*.....*...*.......*.....*...*.......*...*..........
...*.........*...........*.*.........*.*...*.*.........*.
............*...*.*...*.............*...*.*...*............
.......*...*.......*.........*.......*...*.....*.....*.......
......*...*.....*.....*.......*.....*...........*...*.....*.*..
.......*.*.....*.........*.*.........*.*.....*.................*.

Figure 2. The start of the infinite pavement, with colour 0
represented by '.' and colour 1 represented by '*', showing lines
of primes, one with interestingly spaced gaps.

   0   1   2   3   4   5   6   7   8   9  10  11  12  13  ..  ..
   1   2   3   4   5   6   7   8   9  10  11  12  13  14  ..  ..
   4   5   6   7   8   9  10  11  12  13  14  15  16  17  ..  ..
   9  10  11  12  13  14  15  16  17  18  19  20  21  22  ..  ..
  16  17  18  19  20  21  22  23  24  25  26  27  28  29  ..  ..
  25  26  27  28  29  30  31  32  33  34  35  36  37  38  ..  ..
  36  37  38  39  40  41  42  43  44  45  46  47  48  49  ..  ..
  49  50  51  52  53  54  55  56  57  58  59  60  61  62  ..  ..
   .
   .
 etc
   .
   .

Figure 3. The numbering of pixels in the extended design.

The Infinite Pavement is written by W. J. G. Overington,
Department of Electrical, Electronic and Systems Engineering,
Coventry Polytechnic, Priory Street, Coventry CV1 5FB, England.

Sheet 2 of 2, The Infinite Pavement, by W. J. G. Overington.