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. 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.)

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.

(See Math::PlanePath::SierpinskiCurve.)

(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 and FB tree lines.
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, and H. Lee Price's paper at arxiv.org.

(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 bigger 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 or 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.

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.

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 Kevin Ryde.

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