`examples/numbers.pl`

in the Math-PlanePath sources for a
sample program printing text numbers.

(See
`Math::PlanePath::SquareSpiral`

.)

(See
`Math::PlanePath::HexSpiral`

and
`Math::PlanePath::HexSpiralSkewed`

.)

(See
`Math::PlanePath::PentSpiral`

,
and
`Math::PlanePath::HeptSpiral`

.)

(See
`Math::PlanePath::DiamondSpiral`

and
`Math::PlanePath::AztecDiamondRings`

.)

(See
`Math::PlanePath::TriangleSpiral`

.)

In the default skew="left", and also "right", "up" and "down. See
`Math::PlanePath::TriangleSpiralSkewed`

.

(See
`Math::PlanePath::AnvilSpiral`

.)

(See
`Math::PlanePath::OctagramSpiral`

.)

See
`Math::PlanePath::KnightSpiral`

. See also the
Knight's Tour Art page at
borderchess.org.

(See
`Math::PlanePath::CretanLabyrinth`

.)

(See
`Math::PlanePath::SquareArms`

,
`Math::PlanePath::DiamondArms`

and
`Math::PlanePath::HexArms`

.)

turns=2 (the default)

turns=5

turns=1

See
`Math::PlanePath::GreekKeySpiral`

. Have a look also at
Jo Edkins Greek Key
pages.

(See
`Math::PlanePath::Diagonals`

and
`Math::PlanePath::DiagonalsAlternating`

.)

(See
`Math::PlanePath::DiagonalsOctant`

.)

The second image is with wider=4. See
`Math::PlanePath::Corner`

.

See
`Math::PlanePath::Staircase`

.

The first image lines don't show direction, so it looks the same as the plain
Staircase but is going alternately up and back. The second image is with the
end_type="square" option. See
`Math::PlanePath::StaircaseAlternating`

.

(See
`Math::PlanePath::MPeaks`

.)

(See
`Math::PlanePath::PyramidSides`

.)

(See
PyramidRows.)

Samples of rule=30 and rule=73. See
`Math::PlanePath::CellularRule`

.

See
`Math::PlanePath::CellularRule54`

.

See
`Math::PlanePath::CellularRule57`

. The second image is rule 57 mirrored, which is rule 99.

See
`Math::PlanePath::CellularRule190`

. The second image is rule 190 mirrored, which is rule
246.

(See
`Math::PlanePath::SacksSpiral`

.)

Default phi, and other rotations sqrt 2 and sqrt 5. See
`Math::PlanePath::VogelFloret`

.

(See
`Math::PlanePath::TheodorusSpiral`

and
`Math::PlanePath::ArchimedeanChords`

.)

(See
`Math::PlanePath::PixelRings`

.)

(See
`Math::PlanePath::FilledRings`

.)

(See
`Math::PlanePath::MultipleRings`

.)

In the usual base 3 ternary, and higher radix=7.

See
`Math::PlanePath::PeanoCurve`

,
and
Peano's 1890
paper.

See
`Math::PlanePath::HilbertCurve`

and
`Math::PlanePath::HilbertSpiral`

, and
Hilbert's 1891
paper.

If you ever wanted a jumper which is everywhere continuous but nowhere differentiable, try woolly thoughts.

In the default radix=2, and higher radix=5. See
`Math::PlanePath::ZOrderCurve`

.

Here's a cute image of the
fibbinary
numbers plotted on ZOrderCurve radix 2,

In the default binary, and higher radix=4.
See
`Math::PlanePath::GrayCode`

.

The default "alternating", and also "coil" order and radix=7 "alternating".
See
`Math::PlanePath::WunderlichSerpentine`

, and Wunderlich's 1972
paper
from
this page in German,
scanned
pdf.

See
`Math::PlanePath::WunderlichMeander`

, and Wunderlich's 1972
paper
from
this page in German
scanned
pdf.

See
`Math::PlanePath::BetaOmega`

, and papers by
Jens-Michael Wierum:
definition
cached at citeseer, and
CCCG paper.

start_shape=A1 (the default)

start_shape=D2

start_shape=B2

start_shape=B1rev

start_shape=D1rev

start_shape=A2rev

See
`Math::PlanePath::AR2W2Curve`

, and
the
paper by Asano, Ranjan, Roos, Welzl and Widmayer.

See
`Math::PlanePath::KochelCurve`

, and
the paper by
Herman Haverkort.

See

`Math::PlanePath::DekkingCurve`

and
`Math::PlanePath::DekkingCentres`

.

See
`Math::PlanePath::CincoCurve`

. And see
Fortran
90 code by John Dennis.

In the default radix=2, and higher radix=5.

(See
`Math::PlanePath::ImaginaryBase`

.)

In the default radix=2, and higher radix=5. Second row is the digit order
variations XXY, YXX, XnYX, XnXY, YXnX.

See
`Math::PlanePath::ImaginaryHalf`

.

In the default radix=2, and higher radix=5.

(See
`Math::PlanePath::CubicBase`

.)

(See
`Math::PlanePath::CornerReplicate`

.)

(See
`Math::PlanePath::SquareReplicate`

.)

(See
`Math::PlanePath::LTiling`

.)

In the default radix=2, and higher radix=5. These samples are drawn to just 0
to 2047 and 0 to 15624 respectively to show some of the shape, since
continuing on they fill the entire plane. (See
`Math::PlanePath::DigitGroups`

.)

(See
`Math::PlanePath::FibonacciWordFractal`

.)

(See
`Math::PlanePath::Flowsnake`

and
`Math::PlanePath::FlowsnakeCentres`

.)

See Ed Schouten's hexagon centres code too.

Drawn just N=0 to N=2400 (=7^4-1) to show the shape, since continuing on fills
the entire plane.
See
`Math::PlanePath::GosperReplicate`

.

(See
`Math::PlanePath::GosperIslands`

and
`Math::PlanePath::GosperSide`

.)

(See
`Math::PlanePath::QuintetCurve`

and
`Math::PlanePath::QuintetCentres`

.)

Drawn just 0 to 3124 (5^5-1) to show the shape, since continuing on fills the
entire plane. See
`Math::PlanePath::QuintetReplicate`

.

See
`Math::PlanePath::KochCurve`

,
`Math::PlanePath::KochPeaks`

and
`Math::PlanePath::KochSnowflakes`

, and Koch's 1904 paper
at archive.org
(pages 145-174)

Second image is with the "inward" option. See
`Math::PlanePath::KochSquareflakes`

.

(See
`Math::PlanePath::QuadricCurve`

and
`Math::PlanePath::QuadricIslands`

.)

align=triangular (the default)

align=right

align=left, showing X<0

align=diagonal

See
`Math::PlanePath::SierpinskiTriangle`

.

align=triangular (the default)

align=right

align=left, showing X<0

align=diagonal

See
`Math::PlanePath::SierpinskiArrowhead`

.

Alignments "triangular", "right" "left", "diagonal". See
`Math::PlanePath::SierpinskiArrowheadCentres`

.

Default arms=1 and a full arms=8. See
`Math::PlanePath::SierpinskiCurve`

.

Default arms=1 and a full arms=8. See
`Math::PlanePath::SierpinskiCurveStair`

.

(See
`Math::PlanePath::HIndexing`

.)

(See
`Math::PlanePath::DragonCurve`

.)

1 arm and 3 arms. See
`Math::PlanePath::DragonRounded`

.

1 arm and 4 arms. See
`Math::PlanePath::DragonMidpoint`

.

The second image has the vertices rounded off to show the pattern. See
`Math::PlanePath::AlternatePaper`

.

1 arm and 8 arms. See
`Math::PlanePath::AlternatePaperMidpoint`

.

See
`Math::PlanePath::TerdragonCurve`

.

1 arm and 6 arms. See
`Math::PlanePath::TerdragonRounded`

.

1 arm and 6 arms. See
`Math::PlanePath::TerdragonMidpoint`

.

1 arm and 4 arms. The second and third images have the vertices rounded off
to show the pattern. See
`Math::PlanePath::R5DragonCurve`

.

1 arm and 4 arms. See
`Math::PlanePath::R5DragonMidpoint`

.

(See
`Math::PlanePath::CCurve`

.)

Default i+1, and with realpart=2 for i+2 . See
`Math::PlanePath::ComplexPlus`

.

Default i-1, and with realpart=2 for i-2. These samples are points 0 to 1023
and 0 to 3124 respectively to show the shape, since continuing on they fill
the entire plane.

(See
`Math::PlanePath::ComplexMinus`

.)

This sample is points 0 to 1023 to show the shape, since continuing on it
fills the entire plane. (See
`Math::PlanePath::ComplexRevolving`

.)

(See
Hypot
and
`Math::PlanePath::HypotOctant`

.)

In the default "even" points, and "odd", "all", "hex", "hex_rotated" and
"hex_centred". See
`Math::PlanePath::TriangularHypot`

.)

UAD tree lines, high to low and low to high

FB and UMT tree lines.

UArD tree rows in AB and PQ

AB points.

AC points.

BC points (see the POD on why it's straight lines).

SM points, short and medium legs.

SC points, short leg and hypotenuse, 0 < X < sqrt(1/2)*Y.

MC points, medium leg and hypotenuse, wedge sqrt(1/2)*Y < X < Y.

See
`Math::PlanePath::PythagoreanTree`

. Also see H. Lee Price's
paper at arxiv.org on FB, and
my mathematical write-up on UMT.

(See
`Math::PlanePath::DiagonalRationals`

.)

(See
`Math::PlanePath::FactorRationals`

.)

radix=2 (the default)

radix=2 first few points showing growth pattern

radix=3 lines

radix=4 lines

See
`Math::PlanePath::CfracDigits`

and the
paper by
Jeffrey Shallit.

pairs_order="rows" (the default)

pairs_order="rows_reverse"

pairs_order="rows" to N=68*67/2 showing growth pattern.

pairs_order=diagonals_down to N=47^2 showing growth pattern.

Notice in the last two images how growth rows and diagonals are sheared down to wedges of successive integer part int(X/Y). The wedges are slope X=2*Y, X=3*Y, etc. The diagonals case nicely covers the quadrilateral X≤d, X+Y≤2*d.

See
`Math::PlanePath::GcdRationals`

and Lance Fortnow's
blog
entry.

Points visited

SB as tree and by rows

CW as tree.

HCS, AYT as trees.

Bird, Drib as trees.

L as tree.

See
`Math::PlanePath::RationalsTree`

.

(See
`Math::PlanePath::FractionsTree`

.)

k=3 as points and tree lines

k=4 and k=5 as points

See
`Math::PlanePath::ChanTree`

and
paper by
Song Heng Chan
"Analogs of the Stern Sequence",
Integers 2011,
online at ejcnt.

(See
`Math::PlanePath::CoprimeColumns`

.)

With divisor_type="all" and "proper". See
`Math::PlanePath::DivisibleColumns`

.

(See
`Math::PlanePath::WythoffArray`

.)

(See
`Math::PlanePath::WythoffPreliminaryTriangle`

.)

In the default radix=2, and higher radix=5. See
`Math::PlanePath::PowerArray`

.

First few points to show the shape.

Tree structure.

parts=2 line segments.

parts=1 tree structure.

See
`Math::PlanePath::UlamWarburton`

.

parts=1 points.

parts=octant points.

parts=octant_up lines.

This is the first few points to show the shape. Continuing on fills 6/16 of
the plane. See
`Math::PlanePath::UlamWarburtonQuarter`

.

The following in the separate Math-PlanePath-Toothpick distribution.

parts=4 and parts=1

parts=octant and parts=octant+1

parts=octant_up and parts=octant_up+1

parts=wedge and parts=wedge+1

parts=diagonal

parts=diagonal-1

See
`Math::PlanePath::LCornerTree`

.

(See
`Math::PlanePath::LCornerReplicate`

.)

parts=4

parts=1

parts=octant

parts=wedge

parts=3mid

parts=3side

parts=1 non-leaf nodes.

The non-leaf image highlights the branching from the diagonals. Notice that
on each branch the sub-branches on the "near" side of the branch are 1
position earlier than on the "far" side. See
`Math::PlanePath::OneOfEight`

.

(See
`Math::PlanePath::HTree`

.)

parts=3 as tree.

parts=octant as tree.

parts=wedge as toothpicks.

See
`Math::PlanePath::ToothpickTree`

and
`Math::PlanePath::ToothpickReplicate`

.

(See
`Math::PlanePath::ToothpickUpist`

.)

(See
`Math::PlanePath::ToothpickSpiral`

.)

This page Copyright 2011, 2012, 2013, 2014 Kevin Ryde.