`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`

. Have a look also at 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 lines don't show the 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, or other rotations such as sqrt 2 or 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, or higher base such as 7.

See
`Math::PlanePath::PeanoCurve`

, and
Peano's 1890
paper.

See
`Math::PlanePath::HilbertCurve`

and
`Math::PlanePath::HilbertSpiral`

. And see
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, or higher such as 5. See
`Math::PlanePath::ZOrderCurve`

.

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

In the default binary, or higher such as 4.
See
`Math::PlanePath::GrayCode`

.

The default "alternating", or "coil" order, or alternating with radix 7. See
`Math::PlanePath::WunderlichSerpentine`

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

See
`Math::PlanePath::WunderlichMeander`

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

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, or higher such as 5.

(See
`Math::PlanePath::ImaginaryBase`

.)

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

See
`Math::PlanePath::ImaginaryHalf`

.

In the default radix=2, or higher such as 5.

(See
`Math::PlanePath::CubicBase`

.)

(See
`Math::PlanePath::CornerReplicate`

.)

(See
`Math::PlanePath::SquareReplicate`

.)

(See
`Math::PlanePath::LTiling`

.)

In the default radix=2, or higher such as 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`

.)

Have a look at 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`

.)

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`

.)

One arm and three arms. See
`Math::PlanePath::DragonRounded`

.

(See
`Math::PlanePath::DragonMidpoint`

.)

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

.

One arm or eight arms. See
`Math::PlanePath::AlternatePaperMidpoint`

.

See
`Math::PlanePath::TerdragonCurve`

.

One arm or six arms. See
`Math::PlanePath::TerdragonRounded`

.

One arm or six arms. See
`Math::PlanePath::TerdragonMidpoint`

.

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

.

(See
`Math::PlanePath::R5DragonMidpoint`

.)

(See
`Math::PlanePath::CCurve`

.)

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

.

Default i-1, or with i-2 realpart=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, or "odd", "all", "hex", "hex_rotated" or
"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`

. See H. Lee Price's
paper at arxiv.org, and my
mathematical writeup of the UMT tree.

(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 or 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 or 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" or "proper". See
`Math::PlanePath::DivisibleColumns`

.

(See
`Math::PlanePath::WythoffArray`

.)

In the default radix=2, or higher such as 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`

.

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.