How do I typeset a tenfold powering (a tower-of-powers) with LaTeX?

Question

How do I write x^x^x^x^x^x^x^x^x^x?

My LaTeX distribution doesn't wanna eat it, it stops swallowing stuff at the double power x^x^x. What I'm aiming at is essentially building a telescope out of X's stacked on top of each other. Don't ask, that's my sense of humour.

I'm surprised noone has said $x\uparrow\uparrow10$ just because. — You, Nov 11 '13 at 23:53
@You It would in fact only be $x \uparrow 10$ ; two \uparrows and you would have x \uparrow 10 exponentiation steps. (But this is almost entirely beside the point. :)) — Sean Allred, Nov 12 '13 at 00:59
According to http://mathworld.wolfram.com/PowerTower.html, you would need two arrows. — Cephalopod, Nov 12 '13 at 08:44
@Arian as a CS major, it's an off-by-one error. As a math major, I'm ashamed. — Sean Allred, Nov 12 '13 at 17:21

score 69 · Answer 1 · answered Nov 11 '13 at 23:40

69

\documentclass{article}

\def\powertower#1#2{#1\ifnum#2>1 ^{\powertower{#1}{\numexpr#2-1\relax}}\fi}

\begin{document}

$\powertower{x}{100}$

\end{document}

enter image description here

answered Nov 11 '13 at 23:40

David Carlisle

757,742

32

This is the TeX version of Stairway to heaven. :) Amazing! – Paulo Cereda Nov 11 '13 at 23:45
5

@PauloCereda: maybe so, but that x at the bottom left is obviously under a lot of stress, and ready to fall over at any moment. – wasteofspace Nov 11 '13 at 23:50
@wasteofspace: I think David is inspired because we were singing The London bridge is falling down in the chatroom yesterday. :) – Paulo Cereda Nov 11 '13 at 23:53
Disclaimer: I apologize for inadvertently adding The before London bridge. :) – Paulo Cereda Nov 12 '13 at 00:09
2

@PauloCereda a minor sin compared to your earlier one of referencing a picture of tower bridge to accompany the lyrics – David Carlisle Nov 12 '13 at 00:11
@David: Hey, blame the YouTube video! :) And what if Tower bridge is falling down too? :) – Paulo Cereda Nov 12 '13 at 00:13
Hey it has a limit of 254! What if I need 256? – NictraSavios Nov 12 '13 at 04:16
@NictraSavios see http://chat.stackexchange.com/transcript/message/12126469#12126469 – David Carlisle Nov 12 '13 at 08:28
1

You could have called this macro \towerofpower in honor of a certain soul/funk band. :-) – Mico Nov 12 '13 at 14:23
You're aggregating \numexpr...\relax pairs in your \powertower, you could as well use \def\powertower#1#2{#1\ifnum\numexpr#2>1 ^{\powertower{#1}{#2-1}}\fi}, or you evaluate \numexpr before passing it on to the next recursion. – Skillmon Oct 06 '23 at 07:32

score 35 · Answer 2 · answered Nov 11 '13 at 21:28

35

\documentclass[12pt]{article}
\begin{document}

\[ x^{x^{x^{x^{x^{x^{x^{x^{x^x}}}}}}}} \]

\end{document}

enter image description here

answered Nov 11 '13 at 21:28

I'm curious—how hard would it be to generalize this as a \PowerTower{x}{10} (power tower reference) – Sean Allred Nov 11 '13 at 21:54
1

@Sean: done. :) – Paulo Cereda Nov 11 '13 at 23:25
@PauloCereda Yay!! – Sean Allred Nov 12 '13 at 01:00

Paulo Cereda · Answer 3 · 2013-11-12T00:11:41.207

29

_{I dedicate this code to Sean. :)}

Long live expl3! :)

\documentclass{article}

\usepackage{expl3}
\usepackage{xparse}

\ExplSyntaxOn

\cs_new:Npn \paulo_epicrecursion:nn #1 #2 {
   #1^{ \int_compare:nTF { #2 > 1 } { 
      \paulo_epicrecursion:nn { #1 } { \int_eval:n { #2 - 1 } }
   } {
     #1
   } }
}

\NewDocumentCommand{ \powertower } { m m } {
    \paulo_epicrecursion:nn { #1 } { #2 }
}

\ExplSyntaxOff

\begin{document}

$\powertower{x}{10}$

\end{document}

The output:

Quack

Update: egreg brilliantly noted that the above code generates a long list of nested \int_eval:n calls. The optimization is presented as follows:

\documentclass{article}

\usepackage{xparse}

\ExplSyntaxOn

\cs_new:Npn \paulo_epicrecursion:nn #1 #2
  {
   #1^{
     \int_compare:nTF { #2 > 1 }
      { \paulo_epicrecursion:nx { #1 } { \int_eval:n { #2 - 1 } } }
      { #1 }
   }
 }

\NewDocumentCommand{ \powertower } { m m }
 {
  \paulo_epicrecursion:nn { #1 } { #2 }
 }
\cs_generate_variant:Nn \paulo_epicrecursion:nn { nx }

\ExplSyntaxOff

\begin{document}

$\powertower{x}{10}$

\end{document}

The trick here is the generated variant. As egreg explained, the first call doesn't need the full expansion, but the recursive ones do.

edited Nov 12 '13 at 00:11

answered Nov 11 '13 at 23:24

Paulo Cereda

44,220

4

+1 for \powertower and epicrecursion. (Edit: Ah, I saw Sean's comment only after writing this one.) – Torbjørn T. Nov 11 '13 at 23:26
@TorbjørnT.: :) – Paulo Cereda Nov 11 '13 at 23:39
superb, absolutely superb :) – cmhughes Nov 11 '13 at 23:40
The command name is wrong: it should be \paulo_epicrecursion:nn because it has two arguments. – egreg Nov 11 '13 at 23:49
@egreg: oopsie, I forgot to update it when I added the second argument. Thanks, fixed. :) – Paulo Cereda Nov 11 '13 at 23:50
Wouldn't \paulo_epicrecursion:nn { #1 } { #2 - 1 } suffice? \int_compare:nTF { #2 > 1 } should evaluate #2 by itself. (Maybe then \int_compare:nTF { (#2) > 1 } just to be safe. :) – cgnieder Nov 12 '13 at 00:24
I'm touched :') The recursion is strong with this one. – Sean Allred Nov 12 '13 at 00:57
@cgnieder: I actually have absolutely no idea. :) – Paulo Cereda Nov 13 '13 at 02:51

egreg · Answer 4 · 2013-11-12T00:39:26.603

28

Here's a shorter version with LaTeX3 functions:

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn

\NewDocumentCommand{ \powertower } { m m }
 {
  #1
  \prg_replicate:nn { #2 - 1 } { ^\c_group_begin_token #1 }
  \prg_replicate:nn { #2 - 1 } { \c_group_end_token }
 }

\ExplSyntaxOff

\begin{document}

$\powertower{x}{10}$

\end{document}

enter image description here

Limitation: the tower maximum size is 254 because of the maximum nested group level is 255. This limitation on the nested groups is also in the other solutions, of course. Here's a picture of $\powertower{x}{253}$ (I was trying with standalone that adds a level of grouping and forgot to change).

With LuaLaTeX the limit is pushed to 499; at 500 the semantic nest size is overflown.

enter image description here

edited Nov 12 '13 at 00:39

answered Nov 12 '13 at 00:18

egreg

1,121,712

As always, brilliant! :) – Paulo Cereda Nov 12 '13 at 00:20
Very elegant; it's the simpler solutions that seem to be harder to think of, sometimes. :) – Sean Allred Nov 12 '13 at 00:56

score 21 · Answer 5 · 2016-01-22T07:51:54.643

What? afraid of going beyond 255? not me!

epictower1

This was obtained as $\epictower{x}{600}$ .

This reproduces exactly what TeX would have done (were it not for the limitation to 255 group levels) inclusive of a quite odd feature of TeX regarding extra horizontal space on the right.

update 2015. I forgot to say in 2013 what I was referring too: the horizontal whitespace extending to the the right of the power towers, on the side of the character glyphs. The image with the 600 x's does not show the resulting (big) extra horizontal space because the figure was clipped to the contained ink. The images below with shorter staircases display using fboxes the extra whitespace, look on right of the rightmost characters the distance to the fbox frame. You can also experiment with David Carlisle's macro inside an fbox (with \fboxsep=0pt or rather -0.4pt).

In the next picture, on top left what is produced by \epictower, on the right and bottom the original thing, to demonstrate that horizontal and vertical placement is correct (the font size is set to 60pt and then standard size).

And here is the code:

\documentclass{article}
% use this for 600 x's!
% \usepackage [paperheight=65cm,paperwidth=100cm]{geometry}
\usepackage {geometry}
\pagestyle{empty}

\makeatletter
\newdimen \epic@x
\newdimen \epic@y
\newdimen \epic@extrax
\newdimen \epic@extray
\newbox \epic@one
\newbox \epic@two
\newcommand{\epictower}[2]{\ifcase #2\relax
         1\or #1\or {#1}^{#1}\or {#1}^{{#1}^{#1}}\else \epic@tower {#1}{#2}\fi }
\def\epic@tower #1#2{%
    \sbox\epic@one{\m@th $\scriptscriptstyle #1$}%
    \sbox\epic@two{\m@th $\scriptscriptstyle {#1}^{#1}$}%
    \epic@x=\wd\epic@one 
    \epic@extrax=\wd\epic@two
    \advance\epic@extrax by -\epic@x
    \advance\epic@extrax by -\epic@x
    \epic@y=\ht\epic@two
    \advance\epic@y by -\ht\epic@one
    \epic@extray=\ht\epic@one
    \advance\epic@extray by -\epic@y
    \toks@ \expandafter{\the\numexpr #2-3}%
    {#1}^{{#1}^{%
    \setlength{\unitlength}{1sp}%
    \begin{picture}({\numexpr (#2-2)*\epic@x+(#2-3)*\epic@extrax\relax},%
                    {\numexpr (#2-2)*\epic@y+\epic@extray\relax})
    \count@\z@
    \loop
        \put (\numexpr \count@ * \epic@x\relax,%
              \numexpr \count@ * \epic@y\relax){\copy\epic@one}%
    \ifnum\the\toks@>\count@
    \advance\count@ \@ne
    \repeat
    \end{picture}}}}
\makeatother


\begin{document}\thispagestyle{empty}

\fboxsep-.4pt
\fontsize{60}{60}

\newcommand{\testtexpowers}[1]{%
\setbox 1 \hbox{$\scriptscriptstyle #1$}%
\setbox 2 \hbox{$\scriptscriptstyle #1^{\fbox{\copy1}}$}%
\setbox 3 \hbox{$\scriptscriptstyle #1^{#1^{\fbox{\copy1}}}$}%
\setbox 4 \hbox{$\scriptscriptstyle #1^{#1^{#1^{\fbox{\copy1}}}}$}%
\setbox 5 \hbox{$\scriptscriptstyle #1^{#1^{#1^{#1^{\fbox{\copy1}}}}}$}%
\setbox 6 \hbox{$\scriptscriptstyle #1^{#1^{#1^{#1^{#1^{\fbox{\copy1}}}}}}$}%
\setbox 7 \hbox{$\scriptscriptstyle #1^{#1^{#1^{#1^{#1^{#1^{\fbox{\copy1}}}}}}}$}%
\setbox 8 \hbox{$\scriptscriptstyle #1^{#1^{#1^{#1^{#1^{#1^{#1^{\fbox{\copy1}}}}}}}}$}%
\setbox 9 \hbox{$\scriptscriptstyle #1^{#1^{#1^{#1^{#1^{#1^{#1^{#1^{\fbox{\copy1}}}}}}}}}$}%
\leavevmode\rlap{\rlap{\rlap{\rlap{\rlap{\rlap{\rlap{\rlap{\fbox{\box1}}\fbox{\box2}}\fbox{\box3}}\fbox{\box4}}\fbox{\box5}}\fbox{\box6}}\fbox{\box7}}\fbox{\box8}}\fbox{\box9}}

\testtexpowers{x}

\testtexpowers{X}

\testtexpowers{xX}

\clearpage

\fbox{$\epictower {x}{1}$}\fbox{$x$}\hrule\fbox{$x$}
\hrule
\fbox{$\epictower {x}{2}$}\fbox{$x^x$}\hrule\fbox{$x^x$}
\hrule
\fbox{$\epictower {x}{3}$}\fbox{$x^{x^x}$}\hrule\fbox{$x^{x^x}$}
\hrule
\fbox{$\epictower {x}{4}$}\fbox{$x^{x^{x^x}}$}\hrule\fbox{$x^{x^{x^x}}$}
\hrule
\fbox{$\epictower {x}{5}$}\fbox{$x^{x^{x^{x^x}}}$}\hrule\fbox{$x^{x^{x^{x^x}}}$}
\hrule
\fbox{$\epictower {x}{6}$}\fbox{$x^{x^{x^{x^{x^x}}}}$}\hrule\fbox{$x^{x^{x^{x^{x^x}}}}$}
\hrule



\clearpage

\fbox{$\epictower {X}{1}$}\fbox{$X$}\hrule\fbox{$X$}
\hrule
\fbox{$\epictower {X}{2}$}\fbox{$X^X$}\hrule\fbox{$X^X$}
\hrule
\fbox{$\epictower {X}{3}$}\fbox{$X^{X^X}$}\hrule\fbox{$X^{X^X}$}
\hrule
\fbox{$\epictower {X}{4}$}\fbox{$X^{X^{X^X}}$}\hrule\fbox{$X^{X^{X^X}}$}
\hrule
\fbox{$\epictower {X}{5}$}\fbox{$X^{X^{X^{X^X}}}$}\hrule\fbox{$X^{X^{X^{X^X}}}$}
\hrule
\fbox{$\epictower {X}{6}$}\fbox{$X^{X^{X^{X^{X^X}}}}$}\hrule\fbox{$X^{X^{X^{X^{X^X}}}}$}
\hrule

\end{document}

% comment-out the preceding, sets page dimensions in geometry
% only limited by TeX's \maxdimen !

$\epictower{x}{600}$

\end{document}

it is easy to modify \epictower to print the exponents along an arbitrary algebraically parametrized curve, for nice special effects. Size of the bounding box will need extra coding if the curve zig-zags; or one may just set the bounding box to the space taken by x^{x^x} for simplicity. — , Nov 12 '13 at 14:23

score 11 · Answer 6 · answered Nov 13 '13 at 23:13

As more than three powers get very unpleasant to read, I would really recommend using another notation. As already proposed in the comments (hence a community wiki answer), you can represent power towers by the arrow notation:

(x\uparrow\uparrow10)

enter image description here

You might want to watch Numberphiles YouTube video about Grahams number to understand this notation.

This notation is much easier to read. In fact, you can read it at all in contrast to the big tower of powers notation.

The recursion: this notation was introduced by Someone that is already a legend...=). — TheVal, Nov 14 '13 at 12:50

score 7 · Answer 7 · answered Sep 09 '14 at 20:50

This is a version of David's answer, but without fancy etex for those being stuck with Knuth TeX.

\newcount\powercount
\def\powertower#1#2{
  #1\powercount=#2
  \advance\powercount by -1
  \ifnum\powercount>1
  ^{\powertower{#1}{\the\powercount}}
  \fi
}
$\powertower{x}{100}$
\bye

How do I typeset a tenfold powering (a tower-of-powers) with LaTeX?

7 Answers7

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