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Tower of powers

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  • Tower of powers

    This was something I had fun uncovering with a friend of mine. Others had already discovered it, but sometimes its fun to work out examples like this because they're actually fairly surprising at time.

    It started with me exploring iterated functions on a calculator, in this case, simple applying the power recursively.



    At first I'd though there would only be one trivial case z where this had a solution.



    However I discovered to my surprise that there was another,



    And so I started thinking why this works on computers. My intuition seemed to tell me that the number should be growing finitely for all z larger than 1. So I went back to studying what happens as you make this tower of powers progressively higher for various values of z.



    There were two problems to be solved. The first one was within what range of z was this limit defined, and when what was the limit when it was defined. It turned out that the latter problem was easier to solve than the first, and surprisingly that an analytical solution actually existed, using Lambert's W function!

    This function is defined as



    This function is fast and easy to compute using various algorithms. It has two principal branches in the real space, but if we follow its principal branch its defined from 1/e to infinity. It can be used kinda like an expanded logarithm. Like so.



    Using that its straightforward to derive a closed expression for the analytical function we're interested in.



    Now that assumes of course that the limit does in fact exist, but when it does it has this neat symmetric form.
    Last edited by Leonhard; 01-31-2015, 05:20 PM.

  • #2
    The main purpose of this thread is to illustrate to people how much better equations can look using LaTeX and (beyond the fun of the math) is a plea to the mods to include LaTeX support.

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    • #3
      LaTeX is awesome.
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      • #4
        Your symbols certainly do look good and are easy (!) to follow.

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        • #5
          Originally posted by Leonhard View Post

          Shouldn't it actually be?



          Neat stuff.
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          . . . that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures: . . . -- 1 Corinthians 15:3, 4.

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          • #6
            Originally posted by 37818 View Post
            Shouldn't it actually be?



            Neat stuff.
            Yes, you are right that the answer is 2. Here is a proof. It's just one big image, sorry.

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            • #7
              Yeah woops its clearly 2, must have made an error when I wrote the latex. I've fixed it now.

              If anyone is curious, the website I used was http://www.codecogs.com/latex/eqneditor.php in the lower half you can set it to URL ENCODED and simple paste link directly into [img ] [img/ ] tags.

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              • #8
                I am confused. The square root of 2 is 1.414214. The square root of two, to the power of the square root of 2 is 1.632527. The square root of two, to the power of the square root of 2, to the power of the square root of 2 is 2. There is no need to go to the limits at infinity. Am I missing something here?
                My Blog: http://oncreationism.blogspot.co.uk/

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                • #9
                  Sorry, no fancy formatting, but...

                  a^b^c = a^(bc)
                  So:
                  sqrt(2) ^ sqrt(2) ^ sqrt(2) = sqrt(2) ^ (sqrt(2) x sqrt(2)) = sqrt(2) ^ 2 = 2

                  In fact, more generally it looks as though (but not sure I can prove it!):

                  n = (nth root of n) ↑↑ n
                  My Blog: http://oncreationism.blogspot.co.uk/

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                  • #10
                    Originally posted by The Pixie View Post
                    I am confused. The square root of 2 is 1.414214. The square root of two, to the power of the square root of 2 is 1.632527. The square root of two, to the power of the square root of 2, to the power of the square root of 2 is 2. There is no need to go to the limits at infinity. Am I missing something here?
                    You do, indeed, seem to be missing something, since sqrt(2)^(sqrt(2)^sqrt(2)) is not equal to 2. It's approximately 1.7608395558800, and the exact value can be found here:

                    http://www.wolframalpha.com/input/?i...%5Esqrt%282%29
                    "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
                    --Thomas Bradwardine, De Continuo (c. 1325)

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                    • #11
                      Originally posted by The Pixie View Post
                      Sorry, no fancy formatting, but...

                      a^b^c = a^(bc)
                      Here's your mistake. You're confusing (a^b)^c = a^(bc) with a^b^c = a^(b^c)

                      EXAMPLE: (3^3)^3 = 3^9; however, 3^3^3 = 3^27
                      "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
                      --Thomas Bradwardine, De Continuo (c. 1325)

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                      • #12
                        Originally posted by The Pixie View Post
                        Sorry, no fancy formatting, but...

                        a^b^c = a^(bc)
                        So:
                        sqrt(2) ^ sqrt(2) ^ sqrt(2) = sqrt(2) ^ (sqrt(2) x sqrt(2)) = sqrt(2) ^ 2 = 2

                        In fact, more generally it looks as though (but not sure I can prove it!):

                        n = (nth root of n) ↑↑ n
                        Its actually fairly easy to prove that there's no convergence towards any number larger than e, which occurs for z = e^(1/e), any larger than that and the tower power will always diverge towards infinity.

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                        • #13
                          And both you and Pixie are confusing this for merely.

                          y = z^z^z

                          Its rather

                          y = z^z^z^z^z^z^z^z^...

                          Meaning the process goes on forever. If look close at the power tower, you'll notice a small ellipsis at the top. I should have specified this verbally though, but I thought it was obvious when I wrote it in up-arrow notation.

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                          • #14
                            Originally posted by The Pixie View Post
                            I am confused. The square root of 2 is 1.414214. The square root of two, to the power of the square root of 2 is 1.632527. The square root of two, to the power of the square root of 2, to the power of the square root of 2 is 2. There is no need to go to the limits at infinity. Am I missing something here?
                            Originally posted by Boxing Pythagoras View Post
                            Here's your mistake. You're confusing (a^b)^c = a^(bc) with a^b^c = a^(b^c)

                            EXAMPLE: (3^3)^3 = 3^9; however, 3^3^3 = 3^27
                            It's a natural error. Everything else in PEMDAS evaluates "ties" left to right, except exponents, which evaluate right to left.

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                            • #15
                              Originally posted by Leonhard View Post
                              And both you and Pixie are confusing this for merely.

                              y = z^z^z

                              Its rather

                              y = z^z^z^z^z^z^z^z^...

                              Meaning the process goes on forever. If look close at the power tower, you'll notice a small ellipsis at the top. I should have specified this verbally though, but I thought it was obvious when I wrote it in up-arrow notation.
                              I understood the original power tower. I was simply commenting on the mistake Pixie was making in his thought that sqrt(2)^sqrt(2)^sqrt(2)=2.
                              "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
                              --Thomas Bradwardine, De Continuo (c. 1325)

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