Math-PlanePath Gallery

(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.

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

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.

Second image has corners rounded to show the pattern.
See Math::PlanePath::PeanoDiagonals, and E.H. Moore's 1900 paper figure 3 for this sort of drawing (here is transpose).

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.

See Math::PlanePath::HilbertSides.

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::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.

See Math::PlanePath::AlternateTerdragon.

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

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.

LCornerTree

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, 2018, 2020, 2021 Kevin Ryde.

(Back to the Math-PlanePath home page.)