Path: utzoo!utgpu!watserv1!watmath!att!dptg!ulysses!andante!princeton!udel!wuarchive!usc!ucsd!ucbvax!cck.coventry.ac.uk!esx038 From: esx038@cck.coventry.ac.uk ("W. J. G. Overington") Newsgroups: comp.sys.transputer Subject: (none) Message-ID: <21020.9007160837@cck.cov.ac.uk> Date: 16 Jul 90 10:37:13 GMT Sender: usenet@ucbvax.BERKELEY.EDU Organization: The Internet Lines: 128 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.