06/11/2014, 09:30 PM

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.

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.