Kevin Ryde > math-image-110 > math-image

# NAME

math-image -- display some mathematical images

# SYNOPSIS

` math-image [--options]`

# DESCRIPTION

`math-image` displays some mathematical images, either

• in a Gtk2 or Wx GUI,
• as an image file output,
• or setting the root window.

There's many options for what to display, in particular the display 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 \

gives the fine structure constant 1/137 in binary on the Corner path. It's 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.

--oeis=A000217

Values selected by their A-number per Sloane's Online Encyclopedia of Integer Sequences. Some A-numbers are implemented by code modules, others can be read from files in a ~/OEIS/ directory. See Math::NumSeq::OEIS for details. For example the triangular numbers are A000217,

`    math-image --oeis=A000217`
--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::NumSeq::File)

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

## 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. `X11::Protocol` `--root` uses the server's database, usually /etc/X11/rgb.txt. Gtk2 uses a hard-coded copy of that /etc/X11/rgb.txt. `--png` with GD has the `GD::Simple` names. `--xpm` with `Image::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.

--figure=F

Draw a given shape figure at each point. The default is either a square or circle depending on the path. The choices are

```    point             single pixel
square            solid
box               unfilled square
circle            solid
ring              unfilled circle
diamond           solid
diamunf           unfilled diamond
plus              "+" shape
X                 "X" shape
L                 "L" shape
N                 "N" shape
V                 "V" shape
Z                 "Z" shape
arrow             arrow in direction of path```
--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. 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 chosen and displayed (output from ~/.xsession normally goes to the ~/.xsession-errors file). Sometimes `--random` may use a lot of memory, so consider `limit` or `timeout` or both, and perhaps low priority (see sh(1), timeout(1) and nice(1)).

The root window is set with `Gtk2`, or under X with `X11::Protocol::XSetRoot` if available. The `XSetRoot` method uses `Esetroot` style and so supports pseudo-transparency such as `Eterm --trans`. `XSetRoot` also preserves colormap entries on a `PseudoColor` visual and can act on an `__SWM_VROOT` style window manager virtual root.

--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, since it shows the completed image briefly where it might be hidden underneath 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 not moved so you don't lose any typing, but the flash 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 it can also use GD, PNGwriter, Imager, ImageMagick, Prima, Tk or Wx with the right libraries and `Image::Base` supporting module.

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

XPM output requires either `Image::Xpm`, ImageMagick, Prima, Tk or Wx.

Combinations `--prima --png`, or `--tk --xpm`, etc, force the respective output module rather than an automatic choice among available possibilities. Prima, Tk and Wx under X use the X server even when writing to a file and may give obscure error messages if no display.

--text

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

For images which would be colours in the GUI the text output is a digit which is the 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. This option is slightly experimental but works as far as it goes.

To make `math-image` available in `xscreensaver` add to the "programs:" section of your ~/.xscreensaver file,

`    math-image --xscreensaver \n\`

File xscreensaver/math-image.xml in the Math-Image sources can be used give a description in the `xscreensaver-demo` program. Currently "make install" doesn't try to install this so it must be copied manually to the /usr/share/xscreensaver/config/ directory..

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

For reference, under `xscreensaver` a saver program draws to a target window given either by `__SWM_VROOT` root window property from the `xscreensaver` daemon, or by a `-window-id` command line option under `xscreensaver-demo`. `__SWM_VROOT` is recognised 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`
`Tk::Pod`
`Wx::Perl::PodBrowser`

The "Help/POD Documentation" menu items to display this documentation and the various path and values classes, under Gtk, Tk or Wx respectively.

`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 VARIABLES

`DISPLAY`

The X display to use.

`TMPDIR`

A temporary directory to use, as per File::Temp and File::Spec (and on MS-DOS `File::Spec` may look at `TEMP` or `TMP` too, see File::Spec::Win32).

# BUGS

Some of the values plotted can be a 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 (eg. the `DragonCurve`), the colour shown is sometimes the smallest N or sometimes the biggest N due to overwriting. Not sure whether to try some colour mixing, or force the smallest among overlaps.

Colours for counts etc have some hard-coded scaling to show a range of colours for the typical range of values. There ought to be a user control for this. Perhaps relevant `NumSeq` modules should indicate their approximate growth rate to make a sensible initial scale.

xscreensaver(1), and in particular its ccurve(6x) which is a saver displaying various C curve, dragon curve and Koch curves.

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