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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 a base class for some mathematical paths which map an integer position $n to and from coordinates $x,$y in the 2D 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
    CretanLabyrinth        7-circuit extended infinitely

    SquareArms             four-arm square spiral
    DiamondArms            four-arm diamond spiral
    AztecDiamondRings      four-sided rings
    HexArms                six-arm hexagonal spiral
    GreekKeySpiral         square 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      unit chords on an Archimedean spiral
    MultipleRings          concentric circles
    PixelRings             concentric rings of midpoint pixels
    FilledRings            concentric rings of pixels
    Hypot                  points by distance
    HypotOctant            first octant points by distance
    TriangularHypot        points by triangular distance
    PythagoreanTree        X^2+Y^2=Z^2 by trees

    PeanoCurve             3x3 self-similar quadrant
    WunderlichSerpentine   transpose parts of PeanoCurve
    HilbertCurve           2x2 self-similar quadrant
    HilbertSpiral          2x2 self-similar whole-plane
    ZOrderCurve            replicating Z shapes
    GrayCode               Gray code splits
    WunderlichMeander      3x3 "R" pattern quadrant
    BetaOmega              2x2 self-similar half-plane
    AR2W2Curve             2x2 self-similar of four parts
    KochelCurve            3x3 self-similar of two parts
    DekkingCurve           5x5 self-similar, edges
    DekkingCentres         5x5 self-similar, centres
    CincoCurve             5x5 self-similar

    ImaginaryBase          replicate in four directions
    ImaginaryHalf          half-plane replicate three directions
    CubicBase              replicate in three directions
    SquareReplicate        3x3 replicating squares
    CornerReplicate        2x2 replicating "U"
    LTiling                self-simlar L shapes
    DigitGroups            digits grouped by zeros
    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 "+" traversal
    QuintetCentres           likewise but centres of squares
    QuintetReplicate       self-similar "+" tiling

    DragonCurve            paper folding
    DragonRounded          paper folding rounded corners
    DragonMidpoint         paper folding segment midpoints
    AlternatePaper         alternating direction folding
    AlternatePaperMidpoint alternating direction folding, midpoints
    TerdragonCurve         ternary dragon
    TerdragonRounded       ternary dragon rounded corners
    TerdragonMidpoint      ternary dragon segment midpoints
    R5DragonCurve          radix-5 dragon curve
    R5DragonMidpoint       radix-5 dragon curve midpoints
    CCurve                 "C" curve
    ComplexPlus            base i+realpart
    ComplexMinus           base i-realpart, including twindragon
    ComplexRevolving       revolving base i+1

    SierpinskiCurve        self-similar right-triangles
    SierpinskiCurveStair   self-similar right-triangles, stair-step
    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 between X and Y axes
    DiagonalsAlternating   diagonals Y to X and back again
    DiagonalsOctant        diagonals between Y axis and X=Y centre
    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
    CellularRule57         cellular automaton (rule 99 mirror too)
    CellularRule190        cellular automaton (rule 246 mirror too)
    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
    ChanTree               rationals X/Y multi-child tree
    CfracDigits            continued fraction 0<X/Y<1 by digits
    CoprimeColumns         coprime X,Y
    DivisibleColumns       X divisible by Y
    WythoffArray           Fibonacci recurrences
    PowerArray             powers in rows
    File                   points from a disk file

And in the separate Math-PlanePath-Toothpick distribution

    ToothpickTree          pattern of toothpicks
    ToothpickReplicate     same by replication rather than tree
    ToothpickUpist         toothpicks only growing upwards
    ToothpickSpiral        toothpicks around the origin

    LCornerTree            L-shape corner growth
    LCornerReplicate       same by replication rather than tree
    HTree                  H shapes replicated

The paths are object oriented to allow parameters, though many have none. See examples/ in the Math-PlanePath sources for a sample printout of numbers from 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 fractions but expect rounding-off for big floating point exponents.

Floating point infinities (when available) give NaN or infinite returns of some kind (some unspecified kind as yet). n_to_xy() on negative infinity is an empty return, the same as other negative $n.

Floating point NaNs (when available) give NaN, infinite, or empty/undef returns, but again of some unspecified kind as yet.

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
    Number::Fraction    1.14 or higher for abs()

A few classes might truncate a bignum or a fraction to a float as yet. In general the intention is to make the calculations generic enough to act on any sensible number type. Recent enough versions of the bignum modules might be required, perhaps BigInt of Perl 5.8 or higher for ** exponentiation operator.

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


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.

Coordinate Methods


Return X,Y coordinates of point $n on the path. If there's no point $n then the return is an empty list. For example

    my ($x,$y) = $path->n_to_xy (-123)
      or next;   # 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.

($dx,$dy) = $path->n_to_dxdy ($n)

Return the change in X and Y going from point $n to point $n+1, or for paths with multiple arms from $n to $n+$arms_count (thus advancing one point along the arm of $n).

    +  $n+1 == $next_x,$next_y
    |                    $dx = $next_x - $x
    +  $n == $x,$y       $dy = $next_y - $y

$n can be fractional and in that case the dX,dY is from that fractional $n position to $n+1 (or $n+$arms).

           frac $n+1 == $next_x,$next_y
    integer *---+----
            |  /
            | /
            |/                 $dx = $next_x - $x
    integer *

In both cases n_to_dxdy() is the difference $dx=$next_x-$x, $dy=$next_y-$y. Currently for most paths it's merely two n_to_xy() calls to calculate the two points, but some paths can calculate a dX,dY with a little less work.

$rsquared = $path->n_to_radius ($n)
$rsquared = $path->n_to_rsquared ($n)

Return the radial distance R=sqrt(X^2+Y^2) of point $n, or the radius squared R^2=X^2+Y^2. If there's no point $n then the return is undef.

For a few paths these might be calculated with less work than n_to_xy(). For example the SacksSpiral is simply R^2=N, or for example the MultipleRings path with its default step=6 has an integer radius for integer $n whereas $x,$y are fractional (and inexact).


Return the N point number at 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 fill 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) and give undef.

@n_list = $path->xy_to_n_list ($x,$y)

Return a list of N point numbers at coordinates $x,$y. If there's nothing at $x,$y then return an empty list.

    my @n_list = $path->xy_to_n(20,20);

Most paths have just a single N for a given X,Y but some such as DragonCurve and TerdragonCurve have multiple N's at a given X,Y and this method returns all of them.

$bool = $path->xy_is_visited ($x,$y)

Return true if $x,$y is visited. This is equivalent to


Some paths cover the plane and for them xy_is_visited() is always true. For others it might be less work to just test a point than to calculate its $n.


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 might be an over-estimate of the N range required to cover the rectangle. Even if the range is exact the nature of the path may mean many points between $n_lo and $n_hi are outside the rectangle. But the range is at least a lower and upper bound on the N values which occur in the rectangle. Classes which can 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 might be merely the starting $path->n_start(), which is fine if the origin is in the desired rectangle but away from the origin might actually start higher.

$x1,$y1 and $x2,$y2 can be fractional. 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 (which makes a foreach do no loops). But rect_to_n_range() may not always notice there's no points in the rectangle and might instead return some over-estimate.

Descriptive Methods

$n = $path->n_start()

Return the first N in the path. The start is usually either 0 or 1 according to what is most natural for the path. Some paths have an n_start parameter to control the numbering.

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

$f = $path->n_frac_discontinuity()

Return the fraction of N at which there may be 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 then the return is undef. That means for example fractions between N=7 to N=8 give smooth continuous X,Y values (of some kind).

This is mainly of interest for drawing line segments between 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.

$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 starting from $path->n_start() and incrementing by 4 each time.

$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 extend into negative X coordinates and/or negative Y coordinates, respectively.

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

$x = $path->x_minimum()
$y = $path->y_minimum()
$x = $path->x_maximum()
$y = $path->y_maximum()

Return the minimum or maximum of the X or Y coordinate reached by integer N values in the path. If there's no minimum or maximum then return undef.

$dx = $path->dx_minimum()
$dx = $path->dx_maximum()
$dy = $path->dy_minimum()
$dy = $path->dy_maximum()

Return the minimum or maximum change dX, dY occurring in the path for integer N to N+1. For a multi-arm path the change is N to N+arms so it's the change along the same arm.

Various paths which go by rows have non-decreasing Y. For them dy_minimum() is 0.

$adx = $path->absdx_minimum()
$adx = $path->absdx_maximum()
$ady = $path->absdy_minimum()
$ady = $path->absdy_maximum()

Return the minimum or maximum change abs(dX) or abs(dY) occurring in the path for integer N to N+1. For a multi-arm path the change is N to N+arms so it's the change along the same arm.

absdx_maximum() is simply max(dXmax,-dXmin), the biggest change either positive or negative. absdy_maximum() similarly.

absdx_minimum() is 0 if dX=0 occurs anywhere in the path, which means any vertical step. If X always changes then absdx_minimum() will be something bigger than 0. absdy_minimum() likewise 0 if any horizontal dY=0, or bigger if Y always changes.

$sum = $path->sumxy_minimum()
$sum = $path->sumxy_maximum()

Return the minimum or maximum values taken by coordinate sum X+Y reached by integer N values in the path. If there's no minimum or maximum then return undef.

S=X+Y is an anti-diagonal. A path which is always right and above some anti-diagonal has a minimum. Some paths might be entirely left and below and so have a maximum, though that's unusual.

                          \        Path always above
                           \ |     has minimum S=X+Y
      Path always below      |\
      has maximum S=X+Y      | \
                                \  S=X+Y
$sum = $path->sumabsxy_minimum()
$sum = $path->sumabsxy_maximum()

Return the minimum or maximum values taken by coordinate sum abs(X)+abs(Y) reached by integer N values in the path. A minimum always exists but if there's no maximum then return undef.

SumAbs=abs(X)+abs(Y) is sometimes called the "taxi-cab" or "Manhatten" distance, being how far to travel through a square-grid city to get to X,Y. sumabsxy_minimum() is then how close to the origin the path extends.

SumAbs can also be interpreted geometrically as numbering the anti-diagonals of the quadrant containing X,Y, which is equivalent to asking which diamond shape X,Y falls on. sumabsxy_minimum() is then the smallest such diamond reached by the path.

        /|\       SumAbs = which diamond X,Y falls on
       / | \
      /  |  \
      \  |  /
       \ | /
$diffxy = $path->diffxy_minimum()
$diffxy = $path->diffxy_maximum()

Return the minimum or maximum values taken by coordinate difference X-Y reached by integer N values in the path. If there's no minimum or maximum then return undef.

D=X-Y is a leading diagonal. A path which is always right and below such a diagonal has a minimum, for example HypotOctant. A path which is always left and above some diagonal has a maximum D=X-Y. For example various wedge-like paths such as PyramidRows in its default step=2, and "upper octant" paths have a maximum.

                                 /   D=X-Y
        Path always below     | /
        has maximum D=X-Y     |/
                            / |      Path always above
                           /         has minimum D=X-Y
$absdiffxy = $path->absdiffxy_minimum()
$absdiffxy = $path->absdiffxy_maximum()

Return the minimum or maximum values taken by abs(X-Y) for integer N in the path. The minimum is 0 or more. If there's maximum then return undef.

abs(X-Y) can be interpreted geometrically as the distance away from the X=Y diagonal and measured at right-angles to that line.

     d=abs(X-Y)  X=Y line
           ^    /
            \  /
            /  \
           /    \
          o      v
         /         d=abs(X-Y)

Paths which visit the X=Y line (or approach it as an infimum) have absdiffxy_minimum() = 0. Otherwise absdiffxy_minimum() is how close they come to the line.

If the path is entirely below the X=Y line so X>=Y then X-Y>=0 and absdiffxy_minimum() is the same as diffxy_minimum(). If the path is entirely below the X=Y line then absdiffxy_minimum() is - diffxy_maximum().

$dsumxy = $path->dsumxy_minimum()
$dsumxy = $path->dsumxy_maximum()
$ddiffxy = $path->ddiffxy_minimum()
$ddiffxy = $path->ddiffxy_maximum()

Return the minimum or maximum change dSum or dDiffXY occurring in the path for integer N to N+1. For a multi-arm path the change is N to N+arms so it's the change along the same arm.

$rsquared = $path->rsquared_minimum()
$rsquared = $path->rsquared_maximum()

Return the minimum or maximum Rsquared = X^2+Y^2 reached by integer N values in the path. If there's no minimum or maximum then return undef.

Rsquared is always >= 0 so it always has a minimum. The minimum will be more than 0 for paths which don't include the origin X=0,Y=0.

RSquared generally has no maximum since the paths usually extend infinitely in some direction. rsquared_maximum() returns undef in that case.

($dx,$dy) = $path->dir_minimum_dxdy()
($dx,$dy) = $path->dir_maximum_dxdy()

Return a vector which is the minimum or maximum angle taken by a step integer N to N+1, or for a multi-arm path N to N+arms so it's the change along the same arm. Directions are reckoned anti-clockwise around from the X axis.

                  |  *  dX=2,dY=2
    dX=-1,dY=1  * | /
            ------+----*  dX=1,dY=0
                  * dX=0,dY=-1

A path which is always goes N,S,E,W such as the SquareSpiral has minimum East dX=1,dY=0 and maximum South dX=0,dY=-1.

Paths which go diagonally may have different limits. For example the KnightSpiral goes in 2x1 steps and so has minimum East-North-East dX=2,dY=1 and maximum East-South-East dX=2,dY=-1.

If the path has directions approaching 360 degrees then dir_maximum_dxdy() is 0,0 to mean a full circle as a supremum. For example MultipleRings.

If the path only ever goes East then the maximum is East dX=1,dY=0, and the minimum the same. This isn't particularly interesting, but arises for example in the Columns path height=0.

$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.

Tree Methods

Some paths are structured like a tree where each N has a parent and possibly some children.

                / | \
             456 999 458
            /        / \
          1000    1001 1005

The N numbering and any relation to X,Y positions varies among the paths. Some are numbered by rows in breadth-first style and some have children with X,Y positions adjacent to their parent, but that shouldn't be assumed, only that there's a parent-child relation down from some set of top nodes.

@n_children = $path->tree_n_children($n)

Return a list of N values which are the child nodes of $n, or return an empty list if $n has no children.

There could be no children either because $path is not a tree or because there's no children at a particular $n.

$num = $path->tree_n_num_children($n)

Return the number of children of $n, or 0 if $n has no children, or undef if $n < n_start() (ie. before the start of the path).

If the tree is considered as a directed graph then this is the "out-degree" of $n.

$n_parent = $path->tree_n_parent($n)

Return the parent node of $n, or undef if it has no parent.

There is no parent at the top of the tree, or one of multiple tops, or if $path is not a tree.

$n_root = $path->tree_n_root ($n)

Return the N which is root node of $n. This is the top of the tree as by following tree_n_parent() repeatedly until no more parent.

The return is undef if there's no such $n or $path is not a tree.

$depth = $path->tree_n_to_depth($n)

Return the depth of node $n, or undef if there's no point $n. The top of the tree is depth=0, then its children are depth=1, etc.

The depth is a count of how many parent, grandparent, etc, levels are above $n, ie. until reaching tree_n_to_parent() returning undef. For non-tree paths tree_n_to_parent() is always undef and tree_n_to_depth() is always 0.

$n_lo = $path->tree_depth_to_n($depth)
$n_hi = $path->tree_depth_to_n_end($depth)
($n_lo, $n_hi) = $path->tree_depth_to_n_range ($depth)

Return the first or last N, or both those N, for tree level $depth in the path. If there's no such $depth or $path is not a tree then return undef, or for tree_depth_to_n_range() return an empty list.

The points $n_lo through $n_hi might not necessarily all be at $depth. It's possible for depths to be interleaved or intermixed in the point numbering. But many paths are breadth-wise successive rows and for them $n_lo to $n_hi inclusive is all $depth.

$n_hi can only exist if the row has a finite number of points. That's true of all current paths, but perhaps allowance should be made for $n_hi as undef or some such if there is no maximum N for some row.

$num = $path->tree_depth_to_width ($depth)

Return the number of points at $depth in the tree. If there's no such $depth or $path is not a tree then return undef.

$height = $path->tree_n_to_subheight($n)

Return the height of the sub-tree starting at $n, or undef if infinite. The height of a tree is the longest distance down to a leaf node. For example,

    ...                      N     subheight
       6    7   8            0       undef
        \    \ /             1       undef
         3    4   5          2         2
          \    \ /           3       undef
           1    2            4         1
            \  /             5         0
              0             ...

At N=0 and all the left side the tree continues infinitely so the sub-height is infinite (so undef). For N=2 the sub-height is 2 because the longest path down is 2 levels (to N=4 then N=7 or N=8). For a leaf node such as N=5 the sub-height is 0.

Tree Descriptive Methods

$num = $path->tree_num_roots()

Return the number of root nodes in $path. If $path is not a tree then return 0. Many tree paths have a single root and for them the return is 1.

@n_list = $path->tree_root_n_list()

Return a list of the N values which are the root nodes in $path. If $path is not a tree then this is an empty list. There are tree_num_roots() many return values.

$num = $path->tree_num_children_minimum()
$num = $path->tree_num_children_maximum()
@nums = $path->tree_num_children_list()

Return the possible number of children of the nodes of $path, either the minimum, maximum, or a list of all possible number of children.

For tree_num_children_list() the list of values is in increasing order, so the first value is tree_num_children_minimum() and the last is tree_num_children_maximum().

$bool = $path->tree_any_leaf()

Return true if there are any leaf nodes in the tree, meaning any N for which tree_n_num_children() is 0.

This is the same as tree_num_children_minimum()==0 since if NumChildren=0 occurs then there are leaf nodes.

Some trees may have no leaf nodes, for example in the complete binary tree of RationalsTree every node always has 2 children.

Parameter Methods

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


      name        =>    parameter key arg for new()
      share_key   =>    string, or undef
      description =>    human readable string
      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 reach 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",

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 PlanePath 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.

    { wider => { name => "wider",
                 type => "integer",


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) {
      print "$n  $x,$y\n";

The separate n_to_xy() calls were motivated by plotting just some N points of a path, such as just the primes or the perfect squares. Successive positions in paths could perhaps be done more efficiently in an iterator style. 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 could increment instead of recalculate.

If interested only in a particular rectangle or similar region then iterating has the disadvantage that it may stray outside the target region for a long time, making an iterator much less useful than it seems. For wild paths it can be better to apply xy_to_n() by rows or similar across the desired region.

Math::NumSeq::PlanePathCoord etc offer the PlanePath coordinates, directions, turns, etc as sequences. The iterator forms there simply make repeated calls to n_to_xy() etc.

Scaling and Orientation

The paths generally make a first move to the right and go anti-clockwise around from the X axis, unless there's some more natural orientation. Anti-clockwise is the usual direction for mathematical spirals.

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. 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 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 too.

The Rows and Columns paths are 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 for example Transform::Canvas, and 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
       2/2      DiagonalsOctant (2 rows for +2)
        2       SacksSpiral, PyramidSides, Corner, PyramidRows (default)
        4       DiamondSpiral, AztecDiamondRings, Staircase
       4/2      CellularRule54, CellularRule57,
                  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),
                  FilledRings (average 2*pi)
        7       HeptSpiralSkewed
        8       SquareSpiral, PyramidSpiral
      16/2      StaircaseAlternating (up and back for +16)
        9       TriangleSpiral, TriangleSpiralSkewed
       12       AnvilSpiral
       16       OctagramSpiral, ToothpickSpiral
      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)
     128/4      CretanLabyrinth (4 loops for +128)
      216       HexArms (each arm)

    totient     CoprimeColumns, DiagonalRationals
    numdivisors DivisibleColumns
    various     CellularRule

    parameter   MultipleRings, PyramidRows

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 the 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 squares N=(base*base)^level. Or some multiple or relationship to such a power for things like KochPeaks and GosperIslands.

    Base          Path
    ----          ----
      2         HilbertCurve, HilbertSpiral, ZOrderCurve (default),
                  GrayCode (default), BetaOmega, AR2W2Curve,
                  SierpinskiCurve, HIndexing, SierpinskiCurveStair,
                  ImaginaryBase (default), ImaginaryHalf (default),
                  CubicBase (default) CornerReplicate,
                  ComplexMinus (default), ComplexPlus (default),
                  ComplexRevolving, DragonCurve, DragonRounded,
                  DragonMidpoint, AlternatePaper, AlternatePaperMidpoint,
                  CCurve, DigitGroups (default), PowerArray (default)
      3         PeanoCurve (default), WunderlichSerpentine (default),
                  WunderlichMeander, KochelCurve,
                  GosperIslands, GosperSide
                  SierpinskiTriangle, SierpinskiArrowhead,
                  TerdragonCurve, TerdragonRounded, TerdragonMidpoint,
                  UlamWarburton, UlamWarburtonQuarter (each level)
      4         KochCurve, KochPeaks, KochSnowflakes, KochSquareflakes,
      5         QuintetCurve, QuintetCentres, QuintetReplicate,
                  DekkingCurve, DekkingCentres, CincoCurve,
                  R5DragonCurve, R5DragonMidpoint
      7         Flowsnake, FlowsnakeCentres, GosperReplicate
      8         QuadricCurve, QuadricIslands
      9         SquareReplicate
    Fibonacci   FibonacciWordFractal, WythoffArray
    parameter   PeanoCurve, WunderlichSerpentine, ZOrderCurve, GrayCode,
                  ImaginaryBase, ImaginaryHalf, CubicBase, ComplexPlus,
                  ComplexMinus, DigitGroups, PowerArray

Many number sequences plotted on these self-similar 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

     / \ / \ / \ / \ / \ /
     \ / \ / \ / \ / \ / \
     / \ / \ / \ / \ / \ /
     \ / \ / \ / \ / \ / \
     / \ / \ / \ / \ / \ /

This is done in integer X,Y on a square grid by using every second square and offsetting alternate rows. This means sum X+Y even, ie. X,Y either both even or both odd, not of opposite parity.

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

The X axis the and diagonals X=Y and X=-Y divide the plane into six equal parts in this grid.

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

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

The resulting triangles are flatter than they should be. The triangle base is width=2 and top is height=1, whereas it would be height=sqrt(3) for an equilateral triangle. 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 before rotating or the result will be skewed. 60 degree rotations can be made within the integer X,Y coordinates directly as follows, all giving integer X,Y 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.


The formulas section in the POD of each class describes some of the calculations. This might be of interest even if the code is not.

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. Another way is to number vertically on a 60 degree angle with coordinates i,j,

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

These coordinates are sometimes used for hexagonal grids in board games etc. 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 in that it doesn't number anything i,j alone can't already, but it has 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 like the i,j above but with k worked in too.

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

N to dX,dY -- Fractional

n_to_dxdy() is the change from N to N+1, and is designed both for integer N and fractional N. For fractional N it can be convenient to calculate a dX,dY at floor(N) and at floor(N)+1 and then combine the two in proportion to frac(N).

                      N+1    \
                     /|       |
                    / |       |
                   /  |       | frac
                  /   |       |
                 /    |       |
                /     |      /
    this_dX  dX,dY     next_dX
    this_dY            next_dY

         frac   1-frac

    int = int(N)
    frac = N - int    0 <= frac < 1

    this_dX,this_dY  at int
    next_dX,next_dY  at int+1

    at fractional N
      dX = this_dX * (1-frac) + next_dX * frac
      dY = this_dY * (1-frac) + next_dY * frac

This is combination of this_dX,this_dY and next_dX,next_dY in proportion to the distances from positions N to int+1 and from int+1 to N+1.

The formulas can be rearranged to

    dX = this_dX + frac*(next_dX - this_dX)
    dY = this_dY + frac*(next_dY - this_dY)

which is like dX,dY at the integer position plus fractional part of a turn or change to the next dX,dY.

N to dX,dY -- Self-Similar

For most of the self-similar paths such as HilbertCurve the change dX,dY is determined by following the state table transitions down through either all digits of N, or to the last non-9 digit, ie. drop any low digits equal to radix-1.

Generally paths which are the edges of some tiling use all digits, and those which are the centres of a tiling stop at the lowest non-9. This can be seen for example in the DekkingCurve using all digits, whereas its DekkingCentres variant stops at the lowest non-24.

Perhaps this all-digits vs low-non-9 even characterizes path style as edges or centres of a tiling, when a path is specified in some way that a tiling is not quite obvious.


The mandatory methods for a PlanePath subclass are

    xy_to_n_list()     if multiple N's map to an X,Y

It sometimes happens that one of n_to_xy() or xy_to_n() is easier than the other but both should be implemented.

n_to_xy() should do something sensible on fractional N. The suggestion is to make it an X,Y proportionally between integer N positions. It can be along a straight line or an arc as best suits the path. A straight line can be done simply by two calculations of the surrounding integer points, until it's clear how to work the fraction into the code directly.

xy_to_n_list() has a base implementation calling plain xy_to_n() to give a single N at X,Y. If a path has multiple Ns at an X,Y (eg. DragonCurve) then it should implement xy_to_n_list() to return all those Ns and also implement a plain xy_to_n() returning the first of them.

rect_to_n_range() can initially be any convenient over-estimate. It should give N big enough that from there onwards all points are sure to be beyond the given X,Y rectangle.

The following descriptive methods have base implementations

    n_start()           1
    class_x_negative()  \ 1, so whole plane
    class_y_negative()  /
    x_negative()        calls class_x_negative()
    y_negative()        calls class_x_negative()

The base n_start() starts at N=1. Paths which treat N as digits of some radix or where there's self-similar replication are often best started from N=0 instead since doing so puts nice powers-of-2 etc on the axes or diagonals.

    use constant n_start => 0;    # digit or replication style

Paths which use only parts of the plane should define class_x_negative() and/or class_y_negative() to false. For example if only the first quadrant X>=0,Y>=0 then

    use constant class_x_negative => 0;
    use constant class_y_negative => 0;

If negativeness varies with path parameters then x_negative() and/or y_negative() follow those parameters and the class_() forms are whether any set of parameters ever gives negative.

The following methods have base implementations calling n_to_xy(). A subclass can implement them directly if they can be done more efficiently.

    n_to_dxdy()           calls n_to_xy() twice
    n_to_rsquared()       calls n_to_xy()
    n_to_radius()         sqrt of n_to_rsquared()

SacksSpiral is an example of an easy n_to_rsquared(). Or TheodorusSpiral is only slightly trickier. Unless a path has some sort of easy X^2+Y^2 then it might as well let the base implementation call n_to_xy().

The way n_to_dxdy() supports fractional N can be a little tricky. One way is to calculate on the integers below and above and combine as described in "N to dX,dY -- Fractional". For some paths the calculation of turn or direction at ceil(N) can be worked into a calculation of the direction at floor(N) so taking not much more work.

The following method has a base implementation calling xy_to_n(). A subclass can implement is directly if it can be done more efficiently.

    xy_is_visited()     defined(xy_to_n($x,$y))

Paths such as SquareSpiral which fill the plane have xy_is_visited() always true, so for them

    use constant xy_is_visited => 1;

For a tree path the following methods are mandatory


The other tree methods have base implementations,


Calls tree_n_children() and counts the number of return values. Many trees can count the children with less work than calculating outright, for example RationalsTree is simply always 2 for N>=Nstart.


Calls tree_depth_to_n($depth+1)-1. This assumes that the depth level ends where the next begins. This is true for the various breadth-wise tree traversals, but anything interleaved etc will need its own implementation.


Calls tree_depth_to_n() and tree_depth_to_n_end(). For some paths the row start and end, or start and width, might be calculated together more efficiently.


Returns tree_depth_to_n_end() - tree_depth_to_n() + 1. This suits breadth-wise style paths where all points at $depth are in a contiguous block. Any path not like that will need its own tree_depth_to_width().

tree_num_children_minimum(), tree_num_children_maximum()

Return the first and last values of tree_num_children_list() as the minimum and maximum.


Calls tree_num_children_minimum(). If the minimum num_children is 0 then there's leaf nodes.


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::AnvilSpiral, Math::PlanePath::OctagramSpiral, Math::PlanePath::KnightSpiral, Math::PlanePath::CretanLabyrinth

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

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

Math::PlanePath::PeanoCurve, Math::PlanePath::WunderlichSerpentine, Math::PlanePath::WunderlichMeander, Math::PlanePath::HilbertCurve, Math::PlanePath::HilbertSpiral, Math::PlanePath::ZOrderCurve, Math::PlanePath::GrayCode, Math::PlanePath::AR2W2Curve, Math::PlanePath::BetaOmega, Math::PlanePath::KochelCurve, Math::PlanePath::DekkingCurve, Math::PlanePath::DekkingCentres, Math::PlanePath::CincoCurve

Math::PlanePath::ImaginaryBase, Math::PlanePath::ImaginaryHalf, Math::PlanePath::CubicBase, Math::PlanePath::SquareReplicate, Math::PlanePath::CornerReplicate, Math::PlanePath::LTiling, 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::SierpinskiCurveStair, Math::PlanePath::HIndexing

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

Math::PlanePath::DragonCurve, Math::PlanePath::DragonRounded, Math::PlanePath::DragonMidpoint, Math::PlanePath::AlternatePaper, Math::PlanePath::AlternatePaperMidpoint, Math::PlanePath::TerdragonCurve, Math::PlanePath::TerdragonRounded, Math::PlanePath::TerdragonMidpoint, Math::PlanePath::R5DragonCurve, Math::PlanePath::R5DragonMidpoint, Math::PlanePath::CCurve

Math::PlanePath::ComplexPlus, Math::PlanePath::ComplexMinus, Math::PlanePath::ComplexRevolving

Math::PlanePath::Rows, Math::PlanePath::Columns, Math::PlanePath::Diagonals, Math::PlanePath::DiagonalsAlternating, Math::PlanePath::DiagonalsOctant, Math::PlanePath::Staircase, Math::PlanePath::StaircaseAlternating, Math::PlanePath::Corner

Math::PlanePath::PyramidRows, Math::PlanePath::PyramidSides, Math::PlanePath::CellularRule, Math::PlanePath::CellularRule54, Math::PlanePath::CellularRule57, Math::PlanePath::CellularRule190, Math::PlanePath::UlamWarburton, Math::PlanePath::UlamWarburtonQuarter

Math::PlanePath::DiagonalRationals, Math::PlanePath::FactorRationals, Math::PlanePath::GcdRationals, Math::PlanePath::RationalsTree, Math::PlanePath::FractionsTree, Math::PlanePath::ChanTree, Math::PlanePath::CfracDigits, Math::PlanePath::CoprimeColumns, Math::PlanePath::DivisibleColumns, Math::PlanePath::WythoffArray, Math::PlanePath::PowerArray, Math::PlanePath::File

Math::PlanePath::LCornerTree, Math::PlanePath::LCornerReplicate, Math::PlanePath::ToothpickTree, Math::PlanePath::ToothpickReplicate, Math::PlanePath::ToothpickUpist, Math::PlanePath::ToothpickSpiral, Math::PlanePath::OneOfEight, Math::PlanePath::HTree


math-image, displaying various sequences on these paths.

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

Other Ways To Do It

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

PerlMagick (module Image::Magick) demo scripts and



Copyright 2010, 2011, 2012, 2013 Kevin Ryde

This file is part of Math-PlanePath.




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