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Math::PlanePath -- points on a path through the 2-D plane


 use Math::PlanePath;
 # only a base class, see the subclasses for actual operation


This is the base class for some mathematical paths which map an integer position $n into coordinates $x,$y in the plane. The current classes include

    SquareSpiral           four-sided spiral
    PyramidSpiral          square based pyramid
    TriangleSpiral         equilateral triangle spiral
    TriangleSpiralSkewed   equilateral skewed for compactness
    DiamondSpiral          four-sided spiral, looping faster
    PentSpiral             five-sided spiral
    PentSpiralSkewed       five-sided spiral, compact
    HexSpiral              six-sided spiral
    HexSpiralSkewed        six-sided spiral skewed for compactness
    HeptSpiralSkewed       seven-sided spiral, compact
    OctagramSpiral         eight pointed star
    KnightSpiral           an infinite knight's tour

    SquareArms             four-arm square spiral
    DiamondArms            four-arm diamond spiral
    HexArms                six-arm hexagonal spiral
    AztecDiamondRings      four-sided rings
    GreekKeySpiral         spiral with Greek key motif

    SacksSpiral            quadratic on an Archimedean spiral
    VogelFloret            seeds in a sunflower
    TheodorusSpiral        unit steps at right angles
    ArchimedeanChords      chords on an Archimedean spiral
    MultipleRings          concentric circles
    PixelRings             concentric circles of pixels
    Hypot                  points by distance
    HypotOctant            first octant points by distance
    TriangularHypot        points by triangular lattice distance
    PythagoreanTree        primitive triples by tree
    RationalsTree          rationals X/Y by tree

    PeanoCurve             self-similar base-3 quadrant traversal
    HilbertCurve           self-similar base-2 quadrant traversal
    ZOrderCurve            replicating Z shapes
    BetaOmega              self-similar half-plane traversal
    ImaginaryBase          replicating in four directions
    SquareReplicate        replicating squares 3x3
    CornerReplicate        replicating squares 2x2
    DigitGroups            digit groups with high zero
    FibonacciWordFractal   turns by Fibonacci word bits

    Flowsnake              self-similar hexagonal tile traversal
    FlowsnakeCentres         likewise, but centres of hexagons
    GosperReplicate        self-similar hexagonal tiling
    GosperIslands          concentric island rings
    GosperSide             single side/radial

    QuintetCurve           self-similar "+" shape
    QuintetCentres           likewise, but centres of squares
    QuintetReplicate       self-similar "+" tiling

    DragonCurve            paper folding
    DragonRounded            same but rounding-off vertices
    DragonMidpoint         paper folding midpoints
    ComplexMinus           twindragon and other base i-r

    SierpinskiCurve        self-similar pattern by right-triangles
    HIndexing              self-similar right-triangle pairs

    KochCurve              replicating triangular notches
    KochPeaks              two replicating notches
    KochSnowflakes         concentric notched snowflake rings
    KochSquareflakes       concentric notched 4-sided rings
    QuadricCurve           eight segment zig-zag
    QuadricIslands         rings of those zig-zags
    SierpinskiTriangle     self-similar triangle by rows
    SierpinskiArrowhead    self-similar triangle connectedly
    SierpinskiArrowheadCentres  likewise, but centres of triangles

    Rows                   fixed-width rows
    Columns                fixed-height columns
    Diagonals              diagonals down from the Y to X axes
    Staircase              stairs down from the Y to X axes
    Corner                 expanding stripes around a corner
    PyramidRows            expanding stacked rows pyramid
    PyramidSides           along the sides of a 45-degree pyramid
    CellularRule54         cellular automaton rows pattern
    CellularRule190        cellular automaton rows pattern
    UlamWarburton          cellular automaton diamonds
    UlamWarburtonQuarter   cellular automaton quarter-plane

    CoprimeColumns         coprime X,Y
    DivisibleColumns       X divisible by Y
    File                   points from a disk file

The paths are object oriented to allow parameters, though many have none as yet. See examples/numbers.pl in the Math-PlanePath sources for a cute sample printout of selected paths or all paths.

Number Types

The $n and $x,$y parameters can be either integers or floating point. The paths are meant to do something sensible with floating point fractions. Expect rounding-off for big exponents.

Floating point infinities (when available system) are meant to give nan or infinite returns of some kind (some unspecified kind as yet). n_to_xy() on negative infinity $n is generally an empty return, the same as other negative $n. Calculations which break an input into digits of some base are meant not to loop infinitely on infinities.

Floating point nans (when available) are meant to give nan, infinite, or empty/undef returns, but again of some unspecified kind as yet but in any case again not going into infinite loops.

A few of the classes can operate on Math::BigInt, Math::BigRat and Math::BigFloat inputs and give corresponding outputs, but this is experimental and many classes might truncate a bignum to a float as yet. In general the intention is to make the code generic enough that it can act on overloaded number types. Note that new enough versions of the bignum modules might be required, perhaps Perl 5.8 and up so for instance the ** exponentiation operator is available.

Also, for reference, an undef input $n, $x,$y, etc, is meant to provoke an uninitialized value warning (when warnings are enabled), but doesn't croak etc. Perhaps that will change, but the warning at least prevents bad inputs going unnoticed.


$path = Math::PlanePath::Foo->new (key=>value, ...)

Create and return a new path object. Optional key/value parameters may control aspects of the object.

Foo here is one of the various subclasses, see the list above and under "SEE ALSO".

($x,$y) = $path->n_to_xy ($n)

Return x,y coordinates of point $n on the path. If there's no point $n then the return is an empty list, so for example

    my ($x,$y) = $path->n_to_xy (-123)
      or next;   # usually no negatives in $path

Paths start from $path->n_start below, though some will give a position for N=0 or N=-0.5 too.

$n = $path->xy_to_n ($x,$y)

Return the point number for coordinates $x,$y. If there's nothing at $x,$y then return undef.

    my $n = $path->xy_to_n(20,20);
    if (! defined $n) {
      next;   # nothing at this x,y

$x and $y can be fractional and the path classes will give an integer $n which contains $x,$y within a unit square, circle, or intended figure centred on the integer $n.

For paths which completely tile the plane there's always an $n to return, but for the spread-out paths an $x,$y position may fall in between (no $n close enough).

($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)

Return a range of N values which occur in a rectangle with corners at $x1,$y1 and $x2,$y2. The range is inclusive. For example,

     my ($n_lo, $n_hi) = $path->rect_to_n_range (-5,-5, 5,5);
     foreach my $n ($n_lo .. $n_hi) {
       my ($x, $y) = $path->n_to_xy ($n) or next;
       print "$n  $x,$y";

The return may be an over-estimate of the range, and many of the points between $n_lo and $n_hi might be outside the rectangle, but the range at least bounds N in the rectangle. Classes which can guarantee an exact range say so in their docs.

$n_hi is usually no more than an extra partial row, revolution, or self-similar level. $n_lo is often merely the starting point $path->n_start() below, which is fine if the origin is in the rectangle, but something away from the origin might actually start higher.

$x1,$y1 and $x2,$y2 can be fractional and if they partly overlap some N figures then those N's are included in the return. If there's no points in the rectangle then the return may be a "crossed" range like $n_lo=1, $n_hi=0 (and which makes a foreach do no loops). But rect_to_n_range() might not notice there's no points in the rectangle and instead over-estimate the range.

$bool = $path->x_negative()
$bool = $path->y_negative()

Return true if the path extends into negative X coordinates and/or negative Y coordinates respectively.

$n = $path->n_start()

Return the first N in the path. In the current classes this is either 0 or 1.

Some classes have secret dubious undocumented support for N values below this (zero or negative), but n_start is the intended starting point.

$arms = $path->arms_count()

Return the number of arms in a "multi-arm" path.

For example in SquareArms this is 4 and each arm increments in turn, so the first arm is N=1,5,9,13, etc, incrementing by 4 each time.

$str = $path->figure()

Return a string name of the figure (shape) intended to be drawn at each $n position. This is currently either

    "square"     side 1 centred on $x,$y
    "circle"     diameter 1 centred on $x,$y

Of course this is only a suggestion since PlanePath doesn't draw anything itself. A figure like a diamond for instance can look good too.

$aref = Math::PlanePath::Foo->parameter_info_array()
@list = Math::PlanePath::Foo->parameter_info_list()

Return an arrayref of list describing the parameters taken by a given class. This meant to help making widgets etc for user interaction in a GUI. Each element is a hashref

      name        =>    parameter key arg for new()
      description =>    human readable string
      type        =>    string "integer","boolean","enum" etc
      default     =>    value
      minimum     =>    number, or undef
      maximum     =>    number, or undef
      width       =>    integer, suggested display size
      choices     =>    for enum, an arrayref

type is a string, one of


"filename" is separate from "string" since it might require subtly different handling to ensure it reaches Perl as a byte string, whereas a "string" type might in principle take Perl wide chars.

For "enum" the choices field is the possible values, such as

    { name => "flavour",
      type => "enum",
      choices => ["strawberry","chocolate"],

minimum and maximum are omitted if there's no hard limit on the parameter.


The classes are mostly based on integer $n positions and those designed for a square grid turn an integer $n into integer $x,$y. Usually they give in-between positions for fractional $n too. Classes not on a square grid but instead giving fractional X,Y such as SacksSpiral and VogelFloret are designed for a unit circle at each $n but they too can give in-between positions on request.

All X,Y positions are calculated by separate n_to_xy() calls. To follow a path use successive $n values starting from $path->n_start.

The separate n_to_xy() calls were motivated by plotting just some points on a path, such as just the primes or the perfect squares. Perhaps successive positions in some paths could be done in an iterator style more efficiently. The paths with a quadratic "step" are not much worse than a sqrt() to break N into a segment and offset, but the self-similar paths which chop into digits of some radix might increment instead of recalculate.

But when interested in a particular region an iterator will often stray outside for a long time, and going by X,Y in rows or similar may be better, for the square-grid paths.

Scaling and Orientation

The paths generally make a first move horizontally to the right, or from the X axis anti-clockwise, unless there's some more natural orientation. There's no parameters for scaling, offset or reflection as those things are thought better left to a general coordinate transformer to expand or invert for display. But some easy transformations can be had just from the X,Y with

    -X,Y        flip horizontally (mirror image)
    X,-Y        flip vertically (across the X axis)

    -Y,X        rotate +90 degrees  (anti-clockwise)
    Y,-X        rotate -90 degrees
    -X,-Y       rotate 180 degrees

A vertical flip makes the spirals go clockwise instead of anti-clockwise, or a horizontal flip the same but starting on the left at the negative X axis.

The Rows and Columns paths are slight exceptions to the rule of not having rotated versions of paths. They started as ways to pass in width and height as generic parameters, and have the path use the one or the other.

For scaling and shifting see for example Transform::Canvas, or for rotating as well see Geometry::AffineTransform.

Loop Step

The paths can be characterized by how much longer each loop or repetition is than the preceding one. For example each cycle around the SquareSpiral is 8 more N points than the preceding.

      Step        Path
      ----        ----
        0       Rows, Columns (fixed widths)
        1       Diagonals
        2       SacksSpiral, PyramidSides, Corner, PyramidRows (default)
        4       DiamondSpiral, AztecDiamondRings, Staircase
        4/2     CellularRule54 (2 rows for +4)
        5       PentSpiral, PentSpiralSkewed
        5.65    PixelRings (average about 4*sqrt(2))
        6       HexSpiral, HexSpiralSkewed, MultipleRings (default)
        6/2     CellularRule190 (2 rows for +6)
        6.28    ArchimedeanChords (approaching 2*pi)
        7       HeptSpiralSkewed
        8       SquareSpiral, PyramidSpiral
        9       TriangleSpiral, TriangleSpiralSkewed
       16       OctagramSpiral
       19.74    TheodorusSpiral (approaching 2*pi^2)
       32/4     KnightSpiral (4 loops 2-wide for +32)
       64       DiamondArms (each arm)
       72       GreekKeySpiral
      128       SquareArms (each arm)
      216       HexArms (each arm)
    parameter   MultipleRings, PyramidRows

     totient    CoprimeColumns
     divcount   DivisibleColumns

The step determines which quadratic number sequences make straight lines. For example the gap between successive perfect squares increases by 2 each time (4 to 9 is +5, 9 to 16 is +7, 16 to 25 is +9, etc), so the perfect squares make a straight line in the paths of step 2.

In general straight lines on the stepped paths are quadratics a*k^2+b*k+c with a=step/2. The polygonal numbers are like this, with the (step+2)-gonal numbers making a straight line on a "step" path. For example the 7-gonals (heptagonals) are 5/2*k^2-3/2*k and make a straight line on the step=5 PentSpiral. Or the 8-gonal octagonal numbers 6/2*k^2-4/2*k on the step=6 HexSpiral.

There are various interesting properties of primes in quadratic progressions. Some quadratics seem to have more primes than others, see for example "Lucky Numbers of Euler" in Math::PlanePath::PyramidSides. Many quadratics have no primes at all, or none above a certain point, either trivially if always a multiple of 2 etc, or by a more sophisticated reasoning. See "Step 3 Pentagonals" in Math::PlanePath::PyramidRows for a factorization on the roots making a no-primes gap.

A step factor 4 splits a straight line in two, so for example the perfect squares are a straight line on the step=2 "Corner" path, and then on the step=8 SquareSpiral they instead fall on two lines (lower left and upper right). Effectively in that bigger step it's one line of the even squares (2k)^2 == 4*k^2 and another of the odd squares (2k+1)^2. The gap between successive even squares increases by 8 each time and likewise between odd squares.

Self-Similar Powers

The self-similar patterns such as PeanoCurve generally have a base pattern which repeats at powers N=base^level, or some multiple or relation to such a power for things like KochPeaks and GosperIslands.

    Base          Path
    ----          ----
      2         HilbertCurve, ZOrderCurve (default),
                  BetaOmega, SierpinskiCurve, HIndexing
                  ImaginaryBase (default), ComplexMinus (default)
                  DragonCurve, DragonRounded, DragonMidpoint,
                  DigitGroups (default), CornerReplicate
      3         PeanoCurve (default), GosperIslands, GosperSide
                  SierpinskiTriangle, SierpinskiArrowhead,
                  UlamWarburton, UlamWarburtonQuarter (per level)
      4         KochCurve, KochPeaks, KochSnowflakes, KochSquareflakes
      5         QuintetCurve, QuintetCentres, QuintetReplicate
      7         Flowsnake, FlowsnakeCentres, GosperReplicate
      8         QuadricCurve, QuadricIslands
      9         SquareReplicate
    Fibonacci   FibonacciWordFractal
    parameter   PeanoCurve, ZOrderCurve, ImaginaryBase, DigitGroups

Many number sequences on these paths tend to come out fairly random, or merely show the tiling or nature of the path layout rather than much about the number sequence. Number sequences related to the base can make holes or patterns picking out parts of the path. For example numbers without a particular digit (or digits) in the relevant base show up as holes, eg. "Power of 2 Values" in Math::PlanePath::ZOrderCurve.

Triangular Lattice

Some paths are on triangular or "A2" lattice points like

      *   *   *   *   *   *
    *   *   *   *   *   *
      *   *   *   *   *   *
    *   *   *   *   *   *
      *   *   *   *   *   *
    *   *   *   *   *   *

These are done in integer X,Y on a square grid using every second square,

    . * . * . * . * . * . *
    * . * . * . * . * . * .
    . * . * . * . * . * . *
    * . * . * . * . * . * .
    . * . * . * . * . * . *
    * . * . * . * . * . * .

In these coordinates X and Y are either both even or both odd. The X axis and the diagonals X=Y and X=-Y divide the plane into six parts.

       X=-Y     X=Y
         \     /
          \   /
           \ /
    ----------------- X=0
           / \
          /   \
         /     \

The diagonal X=3*Y is the middle of the first sixth, representing a twelfth of the plane.

The resulting triangles are a little flatter than they should be. The base is width=2 and top height=1, whereas height=sqrt(3) would be equilateral triangles. That sqrt(3) factor can be applied if desired,

    X, Y*sqrt(3)          side length 2
    X/2, Y*sqrt(3)/2      side length 1

Integer Y values have the advantage of fitting pixels of the usual kind of raster screen, and not losing precision in floating point results.

If doing a general-purpose coordinate rotation then be sure to apply the sqrt(3) scale factor first, or the rotation is wrong. Rotations can be made within the integer X,Y coordinates directly as follows (all resulting in integers),

    (X-3Y)/2, (X+Y)/2       rotate +60   (anti-clockwise)
    (X+3Y)/2, (Y-X)/2       rotate -60
    -(X+3Y)/2, (X-Y)/2      rotate +120
    (3Y-X)/2, -(X+Y)/2      rotate -120
    -X,-Y                   rotate 180

    (X+3Y)/2, (X-Y)/2       mirror across the X=3*Y twelfth line

The sqrt(3) factor can be worked into a hypotenuse radial distance calculation as follows if comparing distances from the origin.

    hypot = sqrt(X*X + 3*Y*Y)

See for instance TriangularHypot taking triangular points in order of this radial distance.


Triangular Calculations

For a triangular lattice the rotation formulas above allow calculations to be done in the rectangular X,Y which are the inputs and outputs of the PlanePath functions. But an alternative is to number vertically on an angle with coordinates i,j

          *   *   *      2
        *   *   *       1
      *   *   *      j=0
    i=0  1   2 

This is usual for hex grid games, and using this internally can simplify the rotations a little,

    -j, i+j         rotate +60   (anti-clockwise)
    i+j, -i         rotate -60
    -i-j, i         rotate +120
    j, -i-j         rotate -120
    -i, -j          rotate 180

The conversions between i,j and the rectangular X,Y are

    X = 2*i + j         i = (X-Y)/2
    Y = j               j = Y

A third coordinate k at a +120 angle can be used too,

     k=0  k=1 k=2   
        *   *   *   
          *   *   * 
            *   *   *   
             0   1   2 

This is redundant, but has the advantage of making rotations just sign changes and swaps,

    -k, i, j        rotate +60
    j, k, -i        rotate -60
    -j, -k, i       rotate +120
    k, -i, -j       rotate -120
    -i, -j, -k      rotate 180

The conversions between i,j and the rectangular X,Y are as above with k worked into the X,Y.

    X = 2i + j - k        i = (X-Y)/2        i = (X+Y)/2
    Y = j + k             j = Y         or   j = 0
                          k = 0              k = Y


Math::PlanePath::SquareSpiral, Math::PlanePath::PyramidSpiral, Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangleSpiralSkewed, Math::PlanePath::DiamondSpiral, Math::PlanePath::PentSpiral, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::HexSpiral, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::HeptSpiralSkewed, Math::PlanePath::OctagramSpiral, Math::PlanePath::KnightSpiral

Math::PlanePath::HexArms, Math::PlanePath::SquareArms, Math::PlanePath::DiamondArms, Math::PlanePath::AztecDiamondRings, Math::PlanePath::GreekKeySpiral

Math::PlanePath::SacksSpiral, Math::PlanePath::VogelFloret, Math::PlanePath::TheodorusSpiral, Math::PlanePath::ArchimedeanChords, Math::PlanePath::MultipleRings, Math::PlanePath::PixelRings, Math::PlanePath::Hypot, Math::PlanePath::HypotOctant, Math::PlanePath::TriangularHypot

Math::PlanePath::PeanoCurve, Math::PlanePath::HilbertCurve, Math::PlanePath::ZOrderCurve, Math::PlanePath::BetaOmega, Math::PlanePath::ImaginaryBase, Math::PlanePath::SquareReplicate, Math::PlanePath::CornerReplicate, Math::PlanePath::DigitGroups, Math::PlanePath::FibonacciWordFractal

Math::PlanePath::Flowsnake, Math::PlanePath::FlowsnakeCentres, Math::PlanePath::GosperReplicate, Math::PlanePath::GosperIslands, Math::PlanePath::GosperSide

Math::PlanePath::QuintetCurve, Math::PlanePath::QuintetCentres, Math::PlanePath::QuintetReplicate

Math::PlanePath::KochCurve, Math::PlanePath::KochPeaks, Math::PlanePath::KochSnowflakes, Math::PlanePath::KochSquareflakes

Math::PlanePath::QuadricCurve, Math::PlanePath::QuadricIslands

Math::PlanePath::SierpinskiCurve, Math::PlanePath::HIndexing

Math::PlanePath::SierpinskiTriangle, Math::PlanePath::SierpinskiArrowhead, Math::PlanePath::SierpinskiArrowheadCentres

Math::PlanePath::DragonCurve, Math::PlanePath::DragonRounded, Math::PlanePath::DragonMidpoint, Math::PlanePath::ComplexMinus

Math::PlanePath::Rows, Math::PlanePath::Columns, Math::PlanePath::Diagonals, Math::PlanePath::Staircase, Math::PlanePath::Corner

Math::PlanePath::PyramidRows, Math::PlanePath::PyramidSides, Math::PlanePath::CellularRule54, Math::PlanePath::CellularRule190, Math::PlanePath::UlamWarburton, Math::PlanePath::UlamWarburtonQuarter

Math::PlanePath::PythagoreanTree, Math::PlanePath::RationalsTree, Math::PlanePath::CoprimeColumns, Math::PlanePath::DivisibleColumns, Math::PlanePath::File

math-image, displaying various sequences on these paths.

examples/numbers.pl in the Math-PlanePath source code, to print all the paths.

Math::Fractal::Curve, Math::Curve::Hilbert, Algorithm::SpatialIndex::Strategy::QuadTree

PerlMagick (Image::Magick) demo scripts lsys.pl and tree.pl





Copyright 2010, 2011 Kevin Ryde

This file is part of Math-PlanePath.

Math-PlanePath 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-PlanePath 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-PlanePath. If not, see <http://www.gnu.org/licenses/>.