NAME

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

SYNOPSIS

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

DESCRIPTION

This is the base class for some mathematical paths which map an integer position $n to and from coordinates $x,$y in the plane.

The current classes include the following. The intention is that any Math::PlanePath::Something is a PlanePath, and supporting base classes or related things are further down like Math::PlanePath::Base::Xyzzy.

    SquareSpiral           four-sided spiral
    PyramidSpiral          square base 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
    AnvilSpiral            anvil shape
    OctagramSpiral         eight pointed star
    KnightSpiral           an infinite knight's tour

    SquareArms             four-arm square spiral
    DiamondArms            four-arm diamond spiral
    AztecDiamondRings      four-sided rings
    HexArms                six-arm hexagonal spiral
    GreekKeySpiral         spiral with Greek key motif
    MPeaks                 "M" shape layers

    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

    PeanoCurve             3x3 self-similar quadrant traversal
    HilbertCurve           2x2 self-similar quadrant traversal
    HilbertSpiral          2x2 self-similar whole-plane traversal
    ZOrderCurve            replicating Z shapes
    WunderlichMeander      3x3 "R" pattern quadrant traversal
    BetaOmega              2x2 self-similar half-plane traversal
    AR2W2Curve             2x2 self-similar of four shapes
    KochelCurve            3x3 self-similar two shapes
    CincoCurve             5x5 self-similar

    ImaginaryBase          replicating in four directions
    SquareReplicate        3x3 replicating squares
    CornerReplicate        2x2 replicating squares
    LTiling                self-simlar L shapes
    DigitGroups            digit groups of 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 or 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
    AlternatePaper         paper folding in alternating directions
    TerdragonCurve         ternary dragon
    TerdragonMidpoint      ternary dragon midpoints
    ComplexPlus            base i+r
    ComplexMinus           base i-r, including twindragon
    ComplexRevolving       revolving base i+1

    SierpinskiCurve        self-similar right-triangles
    HIndexing              self-similar right-triangles, squared up

    KochCurve              replicating triangular notches
    KochPeaks              two replicating notches
    KochSnowflakes         concentric notched 3-sided 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
    DiagonalsAlternating   diagonals Y to X and back again
    Staircase              stairs down from the Y to X axes
    StaircaseAlternating   stairs Y to X and back again
    Corner                 expanding stripes around a corner
    PyramidRows            expanding stacked rows pyramid
    PyramidSides           along the sides of a 45-degree pyramid
    CellularRule           cellular automaton by rule number
    CellularRule54         cellular automaton rows pattern
    CellularRule190        cellular automaton rows pattern
    UlamWarburton          cellular automaton diamonds
    UlamWarburtonQuarter   cellular automaton quarter-plane

    DiagonalRationals      rationals X/Y by diagonals
    FactorRationals        rationals X/Y by prime factorization
    GcdRationals           rationals X/Y by rows with GCD integer
    RationalsTree          rationals X/Y by tree
    FractionsTree          fractions 0<X/Y<1 by tree
    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. See examples/numbers.pl in the Math-PlanePath sources for a cute sample printout of the numbering for 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) are meant to give nan or infinite returns of some kind (some unspecified kind as yet). n_to_xy() on negative infinity $n is 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 not going into infinite loops.

Many of the classes can operate on overloaded number types as inputs and give corresponding outputs.

    Math::BigInt        maybe perl 5.8 up, for ** operator
    Math::BigRat
    Math::BigFloat
    Number::Fraction    1.14 or higher (for abs())

This is slightly experimental and some classes might truncate a bignum or a fraction to a float as yet. In general the intention is to make the code generic enough that it can act on sensible number types. Recent versions of the bignum modules might be required, perhaps Perl 5.8 and up for the ** exponentiation operator in particular.

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

FUNCTIONS

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

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

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

($x,$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 N 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 covering or exceeding 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 in all cases many of the points between $n_lo and $n_hi might be outside the rectangle. But the range at least bounds the N values which occur in the rectangle. Classes which guarantee an exact lo/hi 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 $path->n_start(), 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 can be a "crossed" range like $n_lo=1, $n_hi=0 (and which makes a foreach do no loops). Though rect_to_n_range() might not notice there's no points in the rectangle and instead over-estimate the range.

$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.

$f = $path->n_frac_discontinuity()

Return the fraction of N at which there's discontinuities in the path. For example if there's a jump in the coordinates between N=7.4999 and N=7.5 then the returned $f is 0.5. Or $f is 0 if there's a discontinuity between 6.999 and 7.0.

If there's no discontinuities in the path, so that for example fractions between N=7 to N=8 give smooth X,Y values (of some kind) then the return is undef.

This is mainly of interest for drawing line segments between successive N points. If there's discontinuities then the idea is to draw from say N=7.0 to N=7.499 and then another line from N=7.5 to N=8. The returned $f is whether there's discontinuities anywhere in $path.

$bool = $path->x_negative()

$bool = $path->y_negative()

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

$bool = Math::PlanePath::Foo->class_x_negative()

$bool = Math::PlanePath::Foo->class_y_negative()

$bool = $path->class_x_negative()

$bool = $path->class_y_negative()

Return true if any paths made by this class extends into negative X coordinates and/or negative Y coordinates, respectively.

For some classes the X or Y extent may depend on parameter values.

$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()
      share_key   =>    string, or undef
      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

    "integer"
    "enum"
    "boolean"
    "string"
    "filename"

"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/or maximum are omitted if there's no hard limit on the parameter.

share_key is designed to indicate when parameters from different NumSeq classes can done by a single control widget in a GUI etc. Normally the name is enough, but when the same name has slightly different meanings in different classes a share_key allows the same meanings to be matched up.

$hashref = Math::PlanePath::Foo->parameter_info_hash()

Return a hashref mapping parameter names $info->{'name'} to their $info records.

GENERAL CHARACTERISTICS

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().

    foreach my $n ($path->n_start .. 100) {
      my ($x,$y) = $path->n_to_xy($n);
      print "$n  $x,$y\n";
    }

The separate n_to_xy() calls were motivated by plotting just some points of a path, such as just the primes or the perfect squares. Successive positions in 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 N into digits of some radix might increment instead of recalculate.

A disadvantage of an iterator is that if you're only interested in a particular rectangular or similar region then the iteration may stray outside for a long time, making it much less useful than it seems. For wild paths it can be better to apply xy_to_n() by rows or similar, on the square-grid paths at least.

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, for example 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  (clockwise)
    -X,-Y       rotate 180 degrees

Flip vertically makes the spirals go clockwise instead of anti-clockwise, or a flip horizontally the same but starting on the left at the negative X axis. See "Triangular Lattice" below for 60 degree rotations of the triangular grid paths.

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

For scaling and shifting see Transform::Canvas or to rotate 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, DiagonalsAlternating (2 rows for +4)
        5       PentSpiral, PentSpiralSkewed
       5.65     PixelRings (average about 4*sqrt(2))
        6       HexSpiral, HexSpiralSkewed, MPeaks,
                  MultipleRings (default)
       6/2      CellularRule190 (2 rows for +6)
       6.28     ArchimedeanChords (approaching 2*pi)
        7       HeptSpiralSkewed
        8       SquareSpiral, PyramidSpiral
      16/2      StaircaseAlternating (up and back for +16)
        9       TriangleSpiral, TriangleSpiralSkewed
       12       AnvilSpiral
       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, DiagonalRationals
    divcount    DivisibleColumns
    various     CellularRule

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 stepped paths are quadratics

   N = a*k^2 + b*k + c    where 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. For example see "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 4*step path 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). In that bigger step there'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 relationship to such a power for things like KochPeaks and GosperIslands.

    Base          Path
    ----          ----
      2         HilbertCurve, HilbertSpiral, ZOrderCurve (default),
                  BetaOmega, AR2W2Curve, SierpinskiCurve, HIndexing
                  ImaginaryBase (default), CornerReplicate,
                  ComplexMinus (default), ComplexPlus (default),
                  ComplexRevolving,
                  DragonCurve, DragonRounded, DragonMidpoint,
                  AlternatePaper, DigitGroups (default)
      3         PeanoCurve (default), GosperIslands, GosperSide
                  WunderlichMeander, KochelCurve,
                  SierpinskiTriangle, SierpinskiArrowhead,
                  SierpinskiArrowheadCentres,
                  TerdragonCurve, TerdragonMidpoint,
                  UlamWarburton, UlamWarburtonQuarter (each level)
      4         KochCurve, KochPeaks, KochSnowflakes, KochSquareflakes,
                  LTiling
      5         QuintetCurve, QuintetCentres, QuintetReplicate,
                  CincoCurve
      7         Flowsnake, FlowsnakeCentres, GosperReplicate
      8         QuadricCurve, QuadricIslands
      9         SquareReplicate
    Fibonacci   FibonacciWordFractal
    parameter   PeanoCurve, ZOrderCurve, ImaginaryBase, DigitGroups
                  ComplexPlus, ComplexMinus

Many number sequences plotted on these paths tend to be fairly random, or merely show the tiling or path layout rather than much about the number sequence. 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. See for example "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 and offset on alternate rows so 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 triangle base is width=2 and top is 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 on the usual kind of raster computer 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 will be wrong. 60 degree rotations can be made within the integer X,Y coordinates directly as follows (all giving integer results),

    (X-3Y)/2, (X+Y)/2       rotate +60   (anti-clockwise)
    (X+3Y)/2, (Y-X)/2       rotate -60   (clockwise)
    -(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 which is triangular points ordered by this radial distance.

FORMULAS

Triangular Calculations

For a triangular lattice the rotation formulas above allow calculations to be done in the rectangular X,Y coordinates which are the inputs and outputs of the PlanePath functions. An alternative is to number vertically on a 60 degree angle with coordinates i,j,

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

Such coordinates are sometimes used for hexagonal grid board games etc, and using this internally can simplify rotations a little,

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

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 degrees angle can be used too,

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

This is redundant since it doesn't number anything i,j alone can't already, but it the advantage of turning rotations into 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,k and the rectangular X,Y are similar to the i,j 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

HOME PAGE

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

http://user42.tuxfamily.org/math-planepath/gallery.html

LICENSE

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/>.

To install Math::PlanePath, copy and paste the appropriate command in to your terminal.

cpanm

cpanm Math::PlanePath

CPAN shell

perl -MCPAN -e shell
install Math::PlanePath

For more information on module installation, please visit the detailed CPAN module installation guide.

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