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.
$n
$x,$y
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.
Math::PlanePath::Something
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 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 CoprimeColumns coprime X,Y DivisibleColumns X divisible by Y WythoffArray Fibonacci recurrences PowerArray powers in rows 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 sample printout of numbers from selected paths or all paths.
examples/numbers.pl
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. Calculations which split an input into digits of some base don't loop infinitely on infinities.
n_to_xy()
Floating point nans (when available) 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()
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 to act on any sensible number type. Recent enough versions of the bignum modules might be required, perhaps 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.
undef
$x
$y
In the following Foo is one of the various subclasses, see the list above and under "SEE ALSO".
Foo
$path = Math::PlanePath::Foo->new (key=>value, ...)
Create and return a new path object. Optional key/value parameters may control aspects of the object.
($x,$y) = $path->n_to_xy ($n)
Return X,Y coordinates of point $n on the path. If there's no point $n then the return is an empty list. For example
my ($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.
$path->n_start()
($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 by one along the arm of $n).
$n+1
$n+$arms_count
+ $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).
$n+$arms
frac $n+1 == $next_x,$next_y v integer *---+---- | / | / |/ $dx = $next_x - $x frac + $n == $x,$y $dy = $next_y - $y | 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.
n_to_dxdy()
$dx=$next_x-$x, $dy=$next_y-$y
$rsquared = $path->n_to_rsquared ($n)
Return the radial distance X^2+Y^2 of point $n, this being the radial distance R=hypot(X,Y). If there's no point $n then the return is undef.
For a few paths this can be calculated with less work than n_to_xy(). For example the SacksSpiral is simply R^2==N.
$n = $path->xy_to_n ($x,$y)
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 = $path->xy_to_n(20,20);
Most paths have just a single N for a given X,Y but some such as DragonCurve and TerdragonCurve have multiple N's at a given X,Y and method returns all of them.
($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,
$x1
$y1
$x2
$y2
my ($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, and many of the points between $n_lo and $n_hi might be outside the rectangle even when the range is exact. But the range is at least an 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_lo
$n_hi
$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() might not always notice there's no points in the rectangle and instead return some over-estimate.
$n_lo=1
$n_hi=0
foreach
rect_to_n_range()
$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.
n_start
Some classes have secret dubious undocumented support for N values below this (zero or negative), but n_start() is the intended starting point.
n_start()
$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.
$f
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.
$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.
$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.
$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.
@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. The could be no children either because $path is not a tree or because there's no children at a particular $n.
$path
$num = $path->tree_n_num_children($n)
Return the number of children of $n, or 0 if $n has no children.
$n_parent = $path->tree_n_parent($n)
Return the parent node of $n, or undef if it has no parent. There could be no parent either because $path is not a tree or because $n is the top of the tree (or one of the tops).
$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
type
"integer" "enum" "boolean" "string" "filename"
"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
choices
{ name => "flavour", type => "enum", choices => ["strawberry","chocolate"], }
minimum and/or maximum are omitted if there's no hard limit on the parameter.
minimum
maximum
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.
share_key
name
$hashref = Math::PlanePath::Foo->parameter_info_hash()
Return a hashref mapping parameter names $info->{'name'} to their $info records.
$info->{'name'}
$info
{ 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) { 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 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.
sqrt()
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.
xy_to_n()
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.
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.
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 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.
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, SierpinskiArrowheadCentres, TerdragonCurve, TerdragonRounded, TerdragonMidpoint, UlamWarburton, UlamWarburtonQuarter (each level) 4 KochCurve, KochPeaks, KochSnowflakes, KochSquareflakes, LTiling, 5 QuintetCurve, QuintetCentres, QuintetReplicate, 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.
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 and 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 \ / \ / \ / ----------------- X=0 / \ / \ / \
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 it 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.
For a triangular lattice the rotation formulas above allow calculations to be done in the rectangular X,Y coordinates, those being 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
Such 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
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::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::CoprimeColumns, Math::PlanePath::DivisibleColumns, Math::PlanePath::WythoffArray, Math::PlanePath::PowerArray, Math::PlanePath::File
Math::NumSeq::PlanePathCoord, Math::NumSeq::PlanePathDelta, Math::NumSeq::PlanePathTurn, Math::NumSeq::PlanePathN
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 (module Image::Magick) demo scripts lsys.pl and tree.pl
http://user42.tuxfamily.org/math-planepath/index.html
http://user42.tuxfamily.org/math-planepath/gallery.html
Copyright 2010, 2011, 2012 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/>.
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.