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Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::SierpinskiCurveStair \-\- Sierpinski curve with stair\-step diagonals .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::SierpinskiCurveStair; \& my $path = Math::PlanePath::SierpinskiCurveStair\->new (arms => 2); \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This is a variation on the \f(CW\*(C`SierpinskiCurve\*(C'\fR with stair-step diagonal parts. .PP .Vb 10 \& 10 | 52\-53 \& | | | \& 9 | 50\-51 54\-55 \& | | | \& 8 | 49\-48 57\-56 \& | | | \& 7 | 42\-43 46\-47 58\-59 62\-63 \& | | | | | | | \& 6 | 40\-41 44\-45 60\-61 64\-65 \& | | | \& 5 | 39\-38 35\-34 71\-70 67\-66 \& | | | | | | | \& 4 | 12\-13 37\-36 33\-32 73\-72 69\-68 92\-93 \& | | | | | | | \& 3 | 10\-11 14\-15 30\-31 74\-75 90\-91 94\-95 \& | | | | | | | \& 2 | 9\-\-8 17\-16 29\-28 77\-76 89\-88 97\-96 \& | | | | | | | \& 1 | 2\-\-3 6\-\-7 18\-19 22\-23 26\-27 78\-79 82\-83 86\-87 98\-99 \& | | | | | | | | | | | | | \& Y=0 | 0\-\-1 4\-\-5 20\-21 24\-25 80\-81 84\-85 ... \& | \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& ^ \& X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 .Ve .PP The tiling is the same as the \f(CW\*(C`SierpinskiCurve\*(C'\fR, but each diagonal is a stair step horizontal and vertical. The correspondence is .PP .Vb 1 \& SierpinskiCurve SierpinskiCurveStair \& \& 7\-\- 12\-\- \& / | \& 6 10\-11 \& | | \& 5 9\-\-8 \& \e | \& 1\-\-2 4 2\-\-3 6\-\-7 \& / \e / | | | \& 0 3 0\-\-1 4\-\-5 .Ve .PP The \f(CW\*(C`SierpinskiCurve\*(C'\fR N=0 to N=3 corresponds to N=0 to N=5 here. N=7 to N=12 which is a copy of the N=0 to N=5 base. Point N=6 is an extra in between the parts. The next such extra is N=19. .SS "Diagonal Length" .IX Subsection "Diagonal Length" The \f(CW\*(C`diagonal_length\*(C'\fR option can make longer diagonals, still in stair-step style. For example .PP .Vb 10 \& diagonal_length => 4 \& 10 | 36\-37 \& | | | \& 9 | 34\-35 38\-39 \& | | | \& 8 | 32\-33 40\-41 \& | | | \& 7 | 30\-31 42\-43 \& | | | \& 6 | 28\-29 44\-45 \& | | | \& 5 | 27\-26 47\-46 \& | | | \& 4 | 8\-\-9 25\-24 49\-48 ... \& | | | | | | \& 3 | 6\-\-7 10\-11 23\-22 51\-50 62\-63 \& | | | | | | \& 2 | 4\-\-5 12\-13 21\-20 53\-52 60\-61 \& | | | | | | \& 1 | 2\-\-3 14\-15 18\-19 54\-55 58\-59 \& | | | | | | \& Y=0 | 0\-\-1 16\-17 56\-57 \& | \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& ^ \& X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 .Ve .PP The length is reckoned from N=0 to the end of the first side N=8, which is X=1 to X=5 for length 4 units. .SS "Arms" .IX Subsection "Arms" The optional \f(CW\*(C`arms\*(C'\fR parameter can give up to eight copies of the curve, each advancing successively. For example .PP .Vb 1 \& arms => 8 \& \& 98\-90 66\-58 57\-65 89\-97 5 \& | | | | | | \& 99 82\-74 50\-42 41\-49 73\-81 96 4 \& | | | | \& 91\-83 26\-34 33\-25 80\-88 3 \& | | | | \& 67\-75 18\-10 9\-17 72\-64 2 \& | | | | \& 59\-51 27\-19 2 1 16\-24 48\-56 1 \& | | | | | | \& 43\-35 11\-\-3 . 0\-\-8 32\-40 <\- Y=0 \& \& 44\-36 12\-\-4 7\-15 39\-47 \-1 \& | | | | | | \& 60\-52 28\-20 5 6 23\-31 55\-63 \-2 \& | | | | \& 68\-76 21\-13 14\-22 79\-71 \-3 \& | | | | \& 92\-84 29\-37 38\-30 87\-95 \-4 \& | | \& 85\-77 53\-45 46\-54 78\-86 \-5 \& | | | | | | \& 93 69\-61 62\-70 94 \-6 \& \& ^ \& \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 .Ve .PP The multiples of 8 (or however many arms) N=0,8,16,etc is the original curve, and the further mod 8 parts are the copies. .PP The middle \*(L".\*(R" shown is the origin X=0,Y=0. It would be more symmetrical to have the origin the middle of the eight arms, which would be X=\-0.5,Y=\-0.5 in the above, but that would give fractional X,Y values. Apply an offset X+0.5,Y+0.5 to centre if desired. .SS "Level Ranges" .IX Subsection "Level Ranges" The N=0 point is reckoned as level=0, then N=0 to N=5 inclusive is level=1, etc. Each level is 4 copies of the previous and an extra 2 points between. .PP .Vb 3 \& LevelPoints[k] = 4*LevelPoints[k\-1] + 2 starting LevelPoints[0]=1 \& = 2 + 2*4 + 2*4^2 + ... + 2*4^(k\-1) + 1*4^k \& = (5*4^k \- 2)/3 \& \& Nlevel[k] = LevelPoints[k] \- 1 since starting at N=0 \& = 5*(4^k \- 1)/3 \& = 0, 5, 25, 105, 425, 1705, 6825, 27305, ... (A146882) .Ve .PP The width along the X axis of a level doubles each time, plus an extra distance 3 between. .PP .Vb 3 \& LevelWidth[k] = 2*LevelWidth[k\-1] + 3 starting LevelWidth[0]=0 \& = 3 + 3*2 + 3*2^2 + ... + 3*2^(k\-1) + 0*2^k \& = 3*(2^k \- 1) \& \& Xlevel[k] = 1 + LevelWidth[k] \& = 3*2^k \- 2 \& = 1, 4, 10, 22, 46, 94, 190, 382, ... (A033484) .Ve .SS "Level Ranges with Diagonal Length" .IX Subsection "Level Ranges with Diagonal Length" With \f(CW\*(C`diagonal_length\*(C'\fR = L, level=0 is reckoned as having L many points instead of just 1. .PP .Vb 2 \& LevelPoints[k] = 2 + 2*4 + 2*4^2 + ... + 2*4^(k\-1) + L*4^k \& = ( (3L+2)*4^k \- 2 )/3 \& \& Nlevel[k] = LevelPoints[k] \- 1 \& = ( (3L+2)*4^k \- 5 )/3 .Ve .PP The width of level=0 becomes L\-1 instead of 0. .PP .Vb 3 \& LevelWidth[k] = 2*LevelWidth[k\-1] + 3 starting LevelWidth[0]=L\-1 \& = 3 + 3*2 + 3*2^2 + ... + 3*2^(k\-1) + (L\-1)*2^k \& = (L+2)*2^k \- 3 \& \& Xlevel[k] = 1 + LevelWidth[k] \& = (L+2)*2^k \- 2 .Ve .PP Level=0 as L many points can be thought of as a little block which is replicated in mirror image to make level=1. For example the diagonal 4 example above becomes .PP .Vb 5 \& 8 9 diagonal_length => 4 \& | | \& 6\-\-7 10\-11 \& | | \& . 5 12 . \& \& 2\-\-3 14\-15 \& | | \& 0\-\-1 16\-17 .Ve .PP The spacing between the parts is had in the tiling by taking a margin of 1/2 at the base and 1 horizontally left and right. .SS "Level Fill" .IX Subsection "Level Fill" The curve doesn't visit all the points in the eighth of the plane below the X=Y diagonal. In general Nlevel+1 many points of the triangular area Xlevel^2/4 are visited, for a filled fraction which approaches a constant .PP .Vb 3 \& Nlevel 4*(3L+2) \& FillFrac = \-\-\-\-\-\-\-\-\-\-\-\- \-> \-\-\-\-\-\-\-\-\- \& Xlevel^2 / 4 3*(L+2)^2 .Ve .PP For example the default L=1 has FillFrac=20/27=0.74. Or L=2 FillFrac=2/3=0.66. As the diagonal length increases the fraction decreases due to the growing holes in the pattern. .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for the behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::SierpinskiCurveStair\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::SierpinskiCurveStair\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::SierpinskiCurveStair->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::SierpinskiCurveStair\->new (diagonal_length => $L, arms => $A)""" 4 .el .IP "\f(CW$path = Math::PlanePath::SierpinskiCurveStair\->new (diagonal_length => $L, arms => $A)\fR" 4 .IX Item "$path = Math::PlanePath::SierpinskiCurveStair->new (diagonal_length => $L, arms => $A)" .PD Create and return a new path object. .ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4 .el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4 .IX Item "($x,$y) = $path->n_to_xy ($n)" Return the X,Y coordinates of point number \f(CW$n\fR on the path. Points begin at 0 and if \f(CW\*(C`$n < 0\*(C'\fR then the return is an empty list. .Sp Fractional positions give an X,Y position along a straight line between the integer positions. .ie n .IP """$n = $path\->n_start()""" 4 .el .IP "\f(CW$n = $path\->n_start()\fR" 4 .IX Item "$n = $path->n_start()" Return 0, the first N in the path. .SS "Level Methods" .IX Subsection "Level Methods" .ie n .IP """($n_lo, $n_hi) = $path\->level_to_n_range($level)""" 4 .el .IP "\f(CW($n_lo, $n_hi) = $path\->level_to_n_range($level)\fR" 4 .IX Item "($n_lo, $n_hi) = $path->level_to_n_range($level)" Return \f(CW\*(C`(0, ((3*$diagonal_length +2) * 4**$level \- 5)/3\*(C'\fR as per \*(L"Level Ranges with Diagonal Length\*(R" above. .SH "OEIS" .IX Header "OEIS" Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include .Sp .RS 4 (etc) .RE .PP .Vb 2 \& A146882 Nlevel, for level=1 up \& A033484 Xmax and Ymax in level, being 3*2^n \- 2 .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::SierpinskiCurve .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2011, 2012, 2013, 2014 Kevin Ryde .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .