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 into coordinates $x,$y in the plane. The current classes include

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

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

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

PeanoCurve             self-similar base-3 quadrant traversal
HilbertCurve           self-similar base-2 quadrant traversal
ZOrderCurve            replicating Z shapes
ImaginaryBase          replicating in four directions

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

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

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

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

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

CoprimeColumns         coprime X,Y
File                   points from a disk file

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

Number Types

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

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

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

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

FUNCTIONS

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The return may be an over-estimate of the range, and many of the points between $n_lo and $n_hi may go outside the rectangle, but the range at least bounds N.

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

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

$bool = $path->x_negative()

$bool = $path->y_negative()

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

$n = $path->n_start()

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

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

$arms = $path->arms_count()

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

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

$str = $path->figure()

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

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

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

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

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

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

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

type is a string, one of

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

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.

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

Scaling and Orientation

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

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

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

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

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

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

Loop Step

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

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

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

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

There are various interesting properties of primes in quadratic progressions. Some quadratics seem to have more primes than others, eg. "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 by the roots making a no-primes gap.

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

Self-Similar Powers

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

Base          Path
----          ----
  2         HilbertCurve, ZOrderCurve (default),
              ImaginaryBase (default),
              DragonCurve, DragonRounded, DragonMidpoint,
  3         PeanoCurve (default), GosperIslands, GosperSide
              SierpinskiTriangle,
              SierpinskiArrowhead, SierpinskiArrowheadCentres,
  4         KochCurve, KochPeaks, KochSnowflakes, KochSquareflakes
  8         QuadricCurve, QuadricIslands
parameter   PeanoCurve, ZOrderCurve, ImaginaryBase

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

Triangular Lattice

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

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

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

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

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

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

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

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

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

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

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

The sqrt(3) factor can be worked into a hypotenuse radial distance calculation as follows if comparing distances from the origin of points at different angles. See for instance TriangularHypot taking triangular points by radial distance.

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