math-image -- display some mathematical images


 math-image [--options]


math-image displays some mathematical images, either

There's lots of options for what to display, in particular it includes Ulam's spiral of prime numbers, and several variations on the numbers in a path such as Sacks spiral and Vogel floret. Try --random or the Randomize button for interesting combinations.

Most of the code is plain Perl, so it's not blindingly fast, but the GUI or root window is drawn progressively so you can see what's happening. In the GUI you can change the controls while drawing to start again on something else.

Mouse button 1 in the GUI drags the image to see parts away from the origin and which otherwise wouldn't fit on screen. This can become quite slow when displaying things like prime numbers which must be calculated all the way up to the desired part.

The number sequences displayed come from Math::NumSeq, and the paths they're plotted on from Math::PlanePath.


Values Options

The following options control what set of values to display. The --values option described last is the most general.


The prime numbers.


The twin primes. --twin is both twins like 11,13. --twin1 is just the first of each like 11, or --twin2 is just the second like 13.


The semi-prime or bi-prime numbers, meaning integers which have two prime factors p*q. This includes p==q squares of primes. --semi-primes-odd is just the odd semiprimes, so 2 excluded from p and q.


The perfect squares 1, 4, 9, 16, 25, 36, etc.


The pronic numbers 2, 6, 12, 20, 30, 42, etc, k*(k+1). These are half way between successive perfect squares, and twice the triangular numbers.


The triangular numbers 1, 3, 6, 10, 15, 21, etc, k*(k+1)/2.


The K-sided polygon numbers. For example --polygonal=3 is the triangular numbers, --polygonal=4 is the squares.


The cubes 1, 8, 27, 64, 125, etc or tetrahedral numbers 1, 4, 10, 20, 35, 56, etc. These tend to grow too quickly to display much of a pattern, though the Vogel floret is close,

    math-image --cubes --vogel

The Fibonacci numbers 1,1,2,3,5,8,13,21, etc. On the Vogel floret these fall on an axis going to the right. For other spirals and paths they tend to grow too quickly to show much.


The Perrin numbers 3, 0, 2, 3, 2, 5, 5, 7, 10, etc. Or Padovan numbers 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, etc. These are cubic recurrences and tend to grow too quickly to display much in the way of patterns.


The digits in the decimal expansion of a fraction. For example the default in the GUI is 5/29. A decimal like 1.234 means 1234/1000.

A fraction is always a repeating pattern, with length no longer than the denominator, but it can give interesting patterns for various paths. For example

    math-image --corner \

is the fine structure constant 1/137 in binary on the Corner path and is a repeating pattern of an angry man with a beard and a skull wearing a hat. No doubt this has deep cosmic significance.


All integers, or just odd or even integers. For the paths which fill the plane --all will just fill the screen (slowly!), but for things like --sacks and --vogel it shows where all the points lie.


Aronson's sequence 1,4,9,... of "T is the first, fourth, ninth, ...". This requires the Math::Aronson module.

--expression='i**2 + 2*i + 1'

Draw values following a formula. It should have a single variable which will be evaluated at 0,1,2, etc. The default is Perl syntax on an "i". See Math::NumSeq::Expression for more information.


Draw lines along the path instead of a set of selected points. This shows where a path travels though you may have to increase the --scale to see it properly.

When the scale is big enough the usual figure is drawn at each point (default a square or circle). Use --figure=point for just the lines.


Draw values from the given Math::NumSeq module (including experimental MathImageWhatever ones). For example

    math-image --values=Emirps

Parameters can be passed as comma separated NAME=VALUE, for example

    math-image --values=TwinPrimes,pairs=both

The File module can read values from a text file

    math-image --values=File,filename=/my/dir/data.txt

The OEIS module takes an A-number per Sloane's Online Encyclopedia of Integer Sequences and uses either a code module implementing the sequence or downloaded files under ~/OEIS/. See Math::NumSeq::OEIS for details. For example tribonnaci numbers are A000073,

    math-image --values=OEIS,anum=A000073

Path Options

The following control the path in the plane where on which the values will be displayed. The --path option described last is the most general.


Ulam's primes in a square spiral (currently the default).


Vogel's floret design for the positions of seeds in a sunflower (see Math::PlanePath::VogelFloret). Try the following to see all the points in the pattern before applying various special sets of values.

    math-image --vogel --all --scale=10

Scaling up helps the circles draw properly. When the values displayed are less than all the integers a lower scale can be used.


An Archimedian spiral with the square root as angle of rotation, by Robert Sacks (see Math::PlanePath::SacksSpiral).


The spiral of Theodorus or square-root spiral (see Math::PlanePath::TheodorusSpiral).


A diamond shaped spiral (see Math::PlanePath::DiamondSpiral).


The sides of a pyramid shape (see Math::PlanePath::PyramidSides).


A pyramid made from horizontal rows (see Math::PlanePath::PyramidRows).


Corners or diagonals between the X and Y axes, per Math::PlanePath::Corner and Math::PlanePath::Diagonals.


Points drawn in successive rows or columns.


Draw with the given Math::PlanePath module. For example

    math-image --path=HeptSpiralSkewed

This includes experimental paths "MathImageFoo", but expect them to change when finished.

Parameters to the path can be supplied as comma separated NAME=VALUE. For example,

    math-image --path=SquareSpiral,wider=3

Output Options

The default is to run the Gtk GUI.


Start the GUI in full screen mode. The Tools/Fullscreen menu entry can toggle between full screen and a normal window. In full screen mode the menus still work, just press Alt-F, Alt-T, etc as normal to pop up.


Set the root window background to the requested image and exit. For example draw a random image from your ~/.xsession startup,

    math-image --root --random &

Add --verbose to print what was in fact chosen and displayed. Output from ~/.xsession normally goes to the ~/.xsession-errors file. Sometimes --random may use a lot of memory, so consider a limit (see sh(1)) or timeout (see timeout(1)) or both, and perhaps low priority (see nice(1)).

Under X the root window is set with X11::Protocol if available, otherwise Gtk2. X11::Protocol is preferred as it allows --foreground and --background colours to be preserved on a PseudoColor visual. Gtk2 is fine on a TrueColor, or for black and white only (which are always permanent), but otherwise colours may not be preserved.


Flash the requested image on the screen instead of starting the GUI. Together with --root the image is drawn to the root then flashed as well. This is good if updating the background randomly every so often, as it shows the image when otherwise perhaps obscured by lots of windows.

    math-image --root --random --flash

The flash is done with a temporary full-screen window, either some X11 native or a Gtk2 per Gtk2::Ex::Splash. In both cases the keyboard focus is unchanged, so you don't lose any typing, though it does eat mouse clicks.


Select the X server for X11 or Gtk output. The default is from the DISPLAY environment variable (normally set at X startup).

    math-image --display=:3

Write a PNG or XPM image file to standard output and exit. PNG is always possible with GdkPixbuf but it can also use GD, PNGwriter, Imager, ImageMagick, Prima or Tk with the right libraries and Image::Base supporting module.

    math-image --png >/tmp/my-file.png

XPM output requires either Image::Xpm, ImageMagick, Prima, or Tk. Note that Prima and Tk for X11 require an X server even for file output (and will give obscure errors if no display).


Write a text-only image to standard output and exit. The default size follows the terminal with Term::Size. A typical tty size like 80x25 is usually too small to see much, but a bigger image might be cute to send to a line printer or similar.

    math-image --text --size=130x49 | lpr

Run the Prima GUI. This requires Prima and the separate Image::Base::Prima::Drawable modules. It doesn't yet have the full set of options the Gtk GUI does, but works as far as it goes.

A combination --prima --png means write a PNG image to standard output using Prima. This requires a graphics screen (which other PNG output modules don't).


Run the Tk GUI. This requires Perl-Tk. It doesn't yet have the full set of options the Gtk GUI does, but works as far as it goes.

A combination --tk --png or --tk --xpm means write a PNG or XPM image to standard output using Tk::Photo. This requires a graphics screen (which other output modules don't).


Run the Curses interactive text interface. This requires the Curses::UI modules. This is very experimental and the control options are minimal.

The Gtk and Prima GUIs have printer output through their usual printing mechanisms. In the current code the Gtk one is a screen dump but the Prima one is a PostScript re-run of the image drawing which might be a bit slow, but might be higher resolution for circle figures.

There's some very rudimentary support for other GUIs with --module=Wx for wxWidgets and --module=Gtk1 for the older Gtk 1.2 and corresponding Gtk-Perl. They're only meant to see how well those GUIs work as yet.

Other Options


Choose a path and values at random. For example in your ~/.xsession

    math-image --root --random

Set the foreground and background colours. The colours can be either names or hex style #RRGGBB or #RRRRGGGGBBBB. For example white on a shade of red,

    math-image --foreground=white --background=#A01010

The default is white foreground on black background. For a --root background a full white can be a bit hard on the eye when there's a lot of points shown. Try a shade of grey instead

    math-image --root --foreground=lightgrey

Available names depend on the output type. Gtk uses a hard-coded copy of the X /etc/X11/rgb.txt. The X11::Protocol --root uses the server's database. --png output with GD has the GD::Simple names. --xpm passes anything at all through to the file. For --text currently the colours can be single characters to show, though perhaps that will change.


Set the size of the image in pixels. A single value means that size square, otherwise WIDTHxHEIGHT. For --root this size is currently ignored and the full screen used.

For the GUI this is an initial size, though it might be widened to accommodate the menubar. Under --fullscreen the size is the unfullscreened window if you switch back to that (menu entry Tools/Fullscreen).

The default for the GUI is about 4/5 of the screen. The default for PNG etc image file output is an arbitrary 200x200, or for --text output the size of the terminal from Term::Size.


How many pixels for each value shown. The current default is 3 to show 3x3 pixel squares, or for --text output just 1 for a pixel per character

--help, -?

Print a summary of the options.


Print the program version number.


Standard Gtk options. See gtk-options(7) for the full list. The only one which does much for math-image is --display to set the X display (default from the DISPLAY environment variable).


In addition to the various modules noted above, the following are used in the Gtk GUI if available,


A "Help/POD Documentation" menu item to display this documentation and the Math::PlanePath classes.


Lines following the cursor, enabled from the Tools/Cross menu item.


Error messages in a dialog instead of to STDERR. Of course there shouldn't be any errors!


Scroll arrows in the bottom right corner.



The X display to use.


Some of the values plotted can be a bit slow to generate or use a lot of memory, or both. When the path goes out to large positions, or when scrolled out away from the origin the display might hang a little or a lot while generating values.

The paths which have big N values near the origin, such as RationalsTree or PythagoreanTree, are calculated with Math::BigInt for accuracy. This becomes very slow, and in some cases the values and/or path calculations might end up rounding off anyway.

Paths such as the dragon curve which duplicate points when plotting colours ends up sometimes showing the colour of the highest N and sometimes the lowest N.


gtk-options(7), xsetroot(1)

Math::PlanePath, Math::PlanePath::SquareSpiral

Math::NumSeq, Math::NumSeq::Aronson

Gtk2, X11::Protocol, Gtk2::Ex::PodViewer, Gtk2::Ex::CrossHair, Gtk2::Ex::ErrorTextDialog



Math-Image is Copyright 2010, 2011, 2012 Kevin Ryde

Math-Image is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-Image is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-Image. If not, see <>.