/ 
math-image-101
River stage zero No dependents
/ math-image

NAME

math-image -- display some mathematical images

SYNOPSIS

math-image [--options]

DESCRIPTION

math-image displays some mathematical images, either

  • in a Gtk2 GUI,

  • as an image file output,

  • or setting the root window.

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.

OPTIONS

Values Options

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

--primes

The prime numbers.

--twin
--twin1
--twin2

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.

--semi-primes
--semi-primes-odd

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.

--squares

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

--pronic

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.

--triangular

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

--polygonal=K

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

--cubes
--tetrahedral

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
--fibonacci

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.

--perrin

The Perrin numbers 3, 0, 2, 3, 2, 5, 5, 7, 10, etc. These are a cubic recurrence and tend to grow too quickly to display much of a pattern.

--fraction=5/29
--fraction=1.234

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 \
           --values=FractionDigits,radix=2,fraction=1/137

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
--odd
--even

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

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

--lines

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.

--values=MODULE
--values=MODULE,NAME=VALUE,NAME=VALUE,...

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

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

--vogel

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.

--sacks

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

--theodorus

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

--diamond

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

--pyramid

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

--pyramid-rows

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

--corner
--diagonals

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

--rows
--columns

Points drawn in successive rows or columns.

--path=MODULE
--path=MODULE,NAME=VALUE,NAME=VALUE,...

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

Other Options

--random

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

math-image --root --random
--foreground=COLOUR
--background=COLOUR

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 module. Gtk2 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.

--size=PIXELS
--size=WIDTHxHEIGHT

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 the menu bar might make the window wider than requested. 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.

--scale=PIXELS

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 single character per point.

--help, -?

Print a summary of the options.

--version

Print the program version number.

GUI Options

The default is to run the Gtk GUI.

--display=DPY

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
--fullscreen

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.

--wx

Run the wxWidgets GUI. This requires wxPerl (see Wx), probably for wxWidgets 2.8 or higher. It might have a few less features than the Gtk2 GUI.

--prima

Run the Prima GUI. This requires the Prima module (see Prima) and the separate Image::Base::Prima::Drawable module. It doesn't yet have the full set of options the Gtk2 GUI does, but works as far as it goes.

--tk

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

--<gtk-options>

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).

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=Curses for Curses::UI 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.

Output Types

--root

Set the root window background to the requested image and exit. For example to 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 limit (see sh(1)) or timeout (see timeout(1)) or both, and perhaps low priority (see nice(1)).

The root window is set with Gtk2, or under X with X11::Protocol is preferred if available because it allows --foreground and --background colours to be preserved on a PseudoColor visual. Gtk2 is fine on a TrueColor and for black and white (being permanent pixels), but on a PseudoColor allocated colours may not be preserved.

--flash

Flash the requested image on the screen instead of starting the GUI. A combination --root --flash means draw to the root and then flash. This is good if updating the background randomly every so often, as it shows the completed image briefly when it might be hidden 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 (see Gtk2::Ex::Splash). In both cases the keyboard focus is unchanged so you don't lose any typing, but it does eat mouse clicks.

--png
--xpm

Write a PNG or XPM image file to standard output and exit. PNG is always possible with Gtk2::Gdk::Pixbuf but 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).

Combinations --prima --png, or --tk --xpm, etc, force the respective output module rather than an automatic choice among available possibilities.

--text

Write a text-only image to standard output and exit. 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. The default size follows the terminal with Term::Size or can be set with --size=WIDTH,HEIGHT,

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

For images which would be colours in the GUI the text output is a digit which is the sequence value at that point. This is slightly experimental, especially for big sequence values, but currently for example

math-image --values=PrimeFactorCount --text --size=5x5

14221
31213
12011
31322
22142
--xscreensaver

Run under the xscreensaver(1) program. This requires the X11::Protocol::XSetRoot module and is slightly experimental, but works as far as it goes.

To make math-image available as a screensaver add it to the "programs:" section of your ~/.xscreensaver file,

math-image --xscreensaver \n\

xscreensaver/math-image.xml can be used give a description in the xscreensaver-demo program, but currently the package "make install" doesn't try to install this.

There's no options for screensaver mode yet. The intention would be a control for the redraw rate, unless there's a common xscreensaver option for that, and to limit each image drawing to the redraw time so slow things aren't continued indefinitely.

For reference under xscreensaver the target X window is either a -window-id command line option from xscreensaver-demo, or __SWM_VROOT from the daemon when it activates. The latter is recognised automatically by X11::Protocol::XSetRoot version 18 and up.

MODULES

In addition to the various modules noted above, the following are used in the Gtk2 GUI if available,

Gtk2::Ex::PodViewer

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

Gtk2::Ex::CrossHair

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

Gtk2::Ex::ErrorTextDialog

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

Gtk2::Ex::QuadButton

Scroll arrows in the bottom right corner.

ENVIRONMENT

DISPLAY

The X display to use.

BUGS

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. In some cases the values and/or path calculations might end up rounding off anyway.

When plotting colours on paths which duplicate points, such as the dragon curve, the colour shown is sometimes the biggest N and sometimes the smallest N.

Colours for counts etc have some hard-coded scaling to try to show a range of colours for the typical range of values. There ought to be a user control for this, and/or perhaps relevant NumSeq modules could indicate their approximate growth rate to make a sensible initial scale.

SEE ALSO

gtk-options(7), xsetroot(1)

Math::PlanePath, Math::NumSeq

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

HOME PAGE

http://user42.tuxfamily.org/math-image/index.html

LICENSE

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 <http://www.gnu.org/licenses/>.