 Experimental tetration 2D plot - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Experimental tetration 2D plot (/showthread.php?tid=884) Experimental tetration 2D plot - Lode - 06/11/2014 Hello, I made an experimental plot of tetration of x^^y, using complex color wheel (hue) coloring to indicate phase and magnitude. The upper right quadrant is for positive x and y, so it gets whiter and whiter there because tetration of positive numbers gets larger and larger. In the negative parts, pure experimental chaos, probably very wrong. I'm posting it because I don't think I've seen it rendered this way before, and it might be interesting to discuss what could be correct, and what could be totally wrong. I implemented what is called "Linear approximation for the extension to real heights" on Wikipedia, so it is of course not true tetration. Also, there may be programming and numerical errors. The source code is in JavaScript and can be found in the following file under "Jmat.Real.Tetration" and "Jmat.Complex.Tetration": https://github.com/lvandeve/jmat/blob/master/jmat.js So the coloring works as follows: -white = very high or infinity -black = zero -red = positive real number (darker = lower, brighter = higher) -cyan = negative real number (darker = closer to zero, brighter = more negative) -all other colors than cyan or red are complex or imaginary numbers, again, brighter is higher magnitude -grey = NaN (not a number, calculation error...) And then the features... So in the upper right it's obvious: it gets whiter and whiter because it's just so extremely high valued there. E.g. the most upperright pixel is 10^^10, it displays simply as white because it's so huge. The top left is a very strange chaos. This is where x is negative, but y is positive, in x ^^ y. Then the whole bottom has those white stripes of infinity, especially the slanted ones between 0 and 1 are weird. These may be a programming or numerical errors: given the colors around it, it seems as if in the limit from that region they would have a similar finite color. Does anyone recognise anything? Are there other plots like this, did anyone make a plot of x^^y? I'd love to see them. Do you think the plot is just totally wrong? Thanks. RE: Experimental tetration 2D plot - nuninho1980 - 06/12/2014 How do I open "jmat.js"? Tried to open in JavaScript but I got error: syntax error (line 6; car 1) with code "800A03EA". Can you calculate (10^10)^^0.5? RE: Experimental tetration 2D plot - Lode - 06/13/2014 Hi, You need to include the jmat.js file in an HTML file. Then you can use the javascript console to calculate things. I'm quite sure every browser has a JS console, but I can only describe how to use it in Chrome: load that HTML file with jmat.js in it, then press F12, and type the following to calculate (10^10)^^0.5: Jmat.tetration(Math.pow(10, 10), 0.5).toString() This answers: 100000 Does that seem plausible? The function has problems with all kinds of extreme values. The plot is in the region of -10 to 10 for x and y. Please do not use this for numerically very precise answers. I was mostly interested in comments about the plot however. Also, if you have software to calculate an approximation of tetration of real x and real y, could you please make such a plot as well? It's a "complex domain plot" using hue to represent phase (Wikipedia has lots of this type of plot). Thanks a lot! RE: Experimental tetration 2D plot - nuninho1980 - 06/13/2014 (06/13/2014, 03:42 PM)Lode Wrote: Hi, You need to include the jmat.js file in an HTML file. Then you can use the javascript console to calculate things. I'm quite sure every browser has a JS console, but I can only describe how to use it in Chrome: load that HTML file with jmat.js in it, then press F12, and type the following to calculate (10^10)^^0.5: Jmat.tetration(Math.pow(10, 10), 0.5).toString() This answers: 100000Ok. I have loaded "jmat.js" in Chrome, pressed F12, clicked "Console" tab,... I tried to run "Jmat.tetration(Math.pow(10, 10), 0.5).toString()" but I got error: ReferenceError: Jmat is not defined Seems 10,000 be not correct but yes I think ~41. 10^^0.5 ~= 2.4770 ("Kneser" code in Pari/GP). RE: Experimental tetration 2D plot - tommy1729 - 06/13/2014 I would love to see the plot by using a higher polynomial approximation of tetration. I expect to see fractal patterns. regards tommy1729 RE: Experimental tetration 2D plot - sheldonison - 06/14/2014 (06/13/2014, 10:50 PM)tommy1729 Wrote: I would love to see the plot by using a higher polynomial approximation of tetration. I expect to see fractal patterns. regards tommy1729Has anyone worked with the Schroeder equation for negative bases, or bases<1? The kneser program works for real bases > exp(1/e). So that part of the graph x^^y, for x>exp(1/e) wouldn't be too difficult. For 1