Comments on: Rectangular Oscillators in the 2×2 (B36/S125) Cellular Automaton http://www.njohnston.ca/2009/05/rectangular-oscillators-in-the-2x2-b36s125-cellular-automaton/ A blog of recreational math and quantum information theory Mon, 06 Feb 2012 21:25:22 +0000 hourly 1 http://wordpress.org/?v=3.3.1 By: Nathaniel http://www.njohnston.ca/2009/05/rectangular-oscillators-in-the-2x2-b36s125-cellular-automaton/#comment-62 Nathaniel Tue, 18 May 2010 12:48:24 +0000 http://www.nathanieljohnston.com/?p=267#comment-62 Whoops, you're of course right. If you "double" one of these rectangular oscillators, you get a pattern that evolves the same way but takes twice as long to do it. So of course there are the 4x8 oscillator with period 4 and things like that. This is explained more rigorously (and correctly) in Section 5 of <a href="http://www.nathanieljohnston.com/wp-content/uploads/2010/01/2x2.pdf" rel="nofollow">this paper</a>, if you're interested. All the best, and thanks for the correction! Whoops, you’re of course right. If you “double” one of these rectangular oscillators, you get a pattern that evolves the same way but takes twice as long to do it. So of course there are the 4×8 oscillator with period 4 and things like that.

This is explained more rigorously (and correctly) in Section 5 of this paper, if you’re interested.

All the best, and thanks for the correction!

]]> By: Nathan http://www.njohnston.ca/2009/05/rectangular-oscillators-in-the-2x2-b36s125-cellular-automaton/#comment-61 Nathan Tue, 18 May 2010 03:32:26 +0000 http://www.nathanieljohnston.com/?p=267#comment-61 Oops. Wrong again. A (2^k-1)x(2^(k+1)n) rectangle has 2^(k-1) times the period of a 2x4n one. This time it's right; I checked. :-) Oops. Wrong again. A (2^k-1)x(2^(k+1)n) rectangle has 2^(k-1) times the period of a 2x4n one. This time it’s right; I checked. :-)

]]> By: Nathan http://www.njohnston.ca/2009/05/rectangular-oscillators-in-the-2x2-b36s125-cellular-automaton/#comment-60 Nathan Tue, 18 May 2010 03:30:34 +0000 http://www.nathanieljohnston.com/?p=267#comment-60 I intended for those to be superscripts. It should say a 2^(k-1)x(2^(k+1)n) rectangle has 2^(k-1) times the period of a 2x4n one. I intended for those to be superscripts. It should say a 2^(k-1)x(2^(k+1)n) rectangle has 2^(k-1) times the period of a 2x4n one.

]]> By: Nathan http://www.njohnston.ca/2009/05/rectangular-oscillators-in-the-2x2-b36s125-cellular-automaton/#comment-59 Nathan Tue, 18 May 2010 03:29:24 +0000 http://www.nathanieljohnston.com/?p=267#comment-59 Indeed, a (2k-1)x2k+1n rectangle appears to have 2k-1 times the period of a 2x4n one. Indeed, a (2k-1)x2k+1n rectangle appears to have 2k-1 times the period of a 2x4n one.

]]> By: Nathan http://www.njohnston.ca/2009/05/rectangular-oscillators-in-the-2x2-b36s125-cellular-automaton/#comment-58 Nathan Tue, 18 May 2010 03:04:40 +0000 http://www.nathanieljohnston.com/?p=267#comment-58 It looks like a 3x8n rectangle always has twice the period of a 2x4n one. It looks like a 3x8n rectangle always has twice the period of a 2x4n one.

]]> By: Nathan http://www.njohnston.ca/2009/05/rectangular-oscillators-in-the-2x2-b36s125-cellular-automaton/#comment-57 Nathan Mon, 17 May 2010 21:48:21 +0000 http://www.nathanieljohnston.com/?p=267#comment-57 Thanks for this; I had noticed the pattern but hadn't worked out the math yet. But when you say "the reason for these restrictions on the sizes of the rectangles is simply that they aren’t oscillators otherwise" that seems to be implying something false. These are definitely not the only rectangular oscillators in this CA. There are also 3x8, 3x16 (=8x11), 4x16 (=8x12), 7x16, 8x16, 3x24 (=8x19=11x16), 3x32 (=4x31=7x28=8x27=16x19), 4x32 (=8x28=16x20), and likely infinitely many others. Thanks for this; I had noticed the pattern but hadn’t worked out the math yet. But when you say “the reason for these restrictions on the sizes of the rectangles is simply that they aren’t oscillators otherwise” that seems to be implying something false. These are definitely not the only rectangular oscillators in this CA. There are also 3×8, 3×16 (=8×11), 4×16 (=8×12), 7×16, 8×16, 3×24 (=8×19=11×16), 3×32 (=4×31=7×28=8×27=16×19), 4×32 (=8×28=16×20), and likely infinitely many others.

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