# Math-PlanePath Gallery

See also `examples/numbers.pl` in the Math-PlanePath sources for a sample program printing text numbers. Module links are to the online man pages, and see the full list of online man pages.

### TriangleSpiralSkewed

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

### KnightSpiral

See `Math::PlanePath::KnightSpiral`. See also the Knight's Tour Art page at borderchess.org.

### GreekKeySpiral

turns=2 (the default)

turns=5

turns=1

See `Math::PlanePath::GreekKeySpiral`. Have a look also at Jo Edkins Greek Key pages.

### Corner

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

### CornerAlternating

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

### StaircaseAlternating

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`.

### PyramidRows

(See PyramidRows.)

### CellularRule

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

### CellularRule57

See `Math::PlanePath::CellularRule57`. The second image is rule 57 mirrored, which is rule 99.

### CellularRule190

See `Math::PlanePath::CellularRule190`. The second image is rule 190 mirrored, which is rule 246.

### VogelFloret

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

### PeanoCurve

In the usual base 3 ternary, and higher radix=7.
See `Math::PlanePath::PeanoCurve`, and Peano's 1890 paper.

### PeanoDiagonals

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).

### HilbertCurve and HilbertSpiral

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

### ZOrderCurve

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,

### GrayCode

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

### WunderlichSerpentine

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.

### WunderlichMeander

See `Math::PlanePath::WunderlichMeander`, and Wunderlich's 1972 paper from this page in German scanned pdf.

### BetaOmega

See `Math::PlanePath::BetaOmega`, and papers by Jens-Michael Wierum: definition cached at citeseer, and CCCG paper.

### AR2W2Curve

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.

### KochelCurve

See `Math::PlanePath::KochelCurve`, and the paper by Herman Haverkort.

### DekkingCurve and DekkingCentres

See `Math::PlanePath::DekkingCurve` and `Math::PlanePath::DekkingCentres`.

### ImaginaryHalf

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

### DigitGroups

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`.)

### GosperReplicate

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`.

### QuintetReplicate

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

### KochSquareflakes

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

### SierpinskiTriangle

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`.

### SierpinskiCurve

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

### SierpinskiCurveStair

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

### DragonRounded

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

### DragonMidpoint

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

### AlternatePaper

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

### AlternatePaperMidpoint

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

### TerdragonRounded

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

### TerdragonMidpoint

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

### TerdragonRounded

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

### R5DragonCurve

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

### R5DragonMidpoint

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

### ComplexPlus

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

### ComplexMinus

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`.)

### ComplexRevolving

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

### TriangularHypot

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

### PythagoreanTree

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.

### CfracDigits

radix=2 first few points showing growth pattern

### GcdRationals

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.

### RationalsTree

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`.

### ChanTree

k=3 as points and tree lines
k=4 and k=5 as points
See `Math::PlanePath::ChanTree` and paper by "Analogs of the Stern Sequence", Integers 2011, .

### DivisibleColumns

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

### PowerArray

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

### UlamWarburton

First few points to show the shape.
Tree structure.
parts=2 line segments.
parts=1 tree structure.
See `Math::PlanePath::UlamWarburton`.

### UlamWarburtonQuarter

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`.

### OneOfEight

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`.

### ToothpickTree and ToothpickReplicate

parts=3 as tree.
parts=octant as tree.
parts=wedge as toothpicks.
See `Math::PlanePath::ToothpickTree` and `Math::PlanePath::ToothpickReplicate`.