1Math::PlanePath::SierpiUnssekriCCuornvterSitbauitre(d3M)Paetrhl::DPolcaunmeePnattaht:i:oSnierpinskiCurveStair(3)
2
3
4

NAME

6       Math::PlanePath::SierpinskiCurveStair -- Sierpinski curve with
7       stair-step diagonals
8

SYNOPSIS

10        use Math::PlanePath::SierpinskiCurveStair;
11        my $path = Math::PlanePath::SierpinskiCurveStair->new (arms => 2);
12        my ($x, $y) = $path->n_to_xy (123);
13

DESCRIPTION

15       This is a variation on the "SierpinskiCurve" with stair-step diagonal
16       parts.
17
18           10  |                                  52-53
19               |                                   |  |
20            9  |                               50-51 54-55
21               |                                |        |
22            8  |                               49-48 57-56
23               |                                   |  |
24            7  |                         42-43 46-47 58-59 62-63
25               |                          |  |  |        |  |  |
26            6  |                      40-41 44-45       60-61 64-65
27               |                       |                          |
28            5  |                      39-38 35-34       71-70 67-66
29               |                          |  |  |        |  |  |
30            4  |                12-13    37-36 33-32 73-72 69-68    92-93
31               |                 |  |              |  |              |  |
32            3  |             10-11 14-15       30-31 74-75       90-91 94-95
33               |              |        |        |        |        |        |
34            2  |              9--8 17-16       29-28 77-76       89-88 97-96
35               |                 |  |              |  |              |  |
36            1  |        2--3  6--7 18-19 22-23 26-27 78-79 82-83 86-87 98-99
37               |        |  |  |        |  |  |  |        |  |  |  |        |
38           Y=0 |     0--1  4--5       20-21 24-25       80-81 84-85       ...
39               |
40               +-------------------------------------------------------------
41                  ^
42                 X=0 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19
43
44       The tiling is the same as the "SierpinskiCurve", but each diagonal is a
45       stair step horizontal and vertical.  The correspondence is
46
47           SierpinskiCurve        SierpinskiCurveStair
48
49                       7--                   12--
50                     /                        |
51                    6                     10-11
52                    |                      |
53                    5                      9--8
54                     \                        |
55              1--2     4             2--3  6--7
56            /     \  /               |  |  |
57           0        3             0--1  4--5
58
59       The "SierpinskiCurve" N=0 to N=3 corresponds to N=0 to N=5 here.  N=7
60       to N=12 which is a copy of the N=0 to N=5 base.  Point N=6 is an extra
61       in between the parts.  The next such extra is N=19.
62
63   Diagonal Length
64       The "diagonal_length" option can make longer diagonals, still in stair-
65       step style.  For example
66
67                    diagonal_length => 4
68           10  |                                 36-37
69               |                                  |  |
70            9  |                              34-35 38-39
71               |                               |        |
72            8  |                           32-33       40-41
73               |                            |              |
74            7  |                        30-31             42-43
75               |                         |                    |
76            6  |                     28-29                   44-45
77               |                      |                          |
78            5  |                     27-26                   47-46
79               |                         |                    |
80            4  |                8--9    25-24             49-48    ...
81               |                |  |        |              |        |
82            3  |             6--7 10-11    23-22       51-50    62-63
83               |             |        |        |        |        |
84            2  |          4--5       12-13    21-20 53-52    60-61
85               |          |              |        |  |        |
86            1  |       2--3             14-15 18-19 54-55 58-59
87               |       |                    |  |        |  |
88           Y=0 |    0--1                   16-17       56-57
89               |
90               +------------------------------------------------------
91                 ^
92                X=0 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
93
94       The length is reckoned from N=0 to the end of the first side N=8, which
95       is X=1 to X=5 for length 4 units.
96
97   Arms
98       The optional "arms" parameter can give up to eight copies of the curve,
99       each advancing successively.  For example
100
101           arms => 8
102
103              98-90 66-58       57-65 89-97            5
104                  |  |  |        |  |  |
105           99    82-74 50-42 41-49 73-81    96         4
106            |              |  |              |
107           91-83       26-34 33-25       80-88         3
108               |        |        |        |
109           67-75       18-10  9-17       72-64         2
110            |              |  |              |
111           59-51 27-19     2  1    16-24 48-56         1
112               |  |  |              |  |  |
113              43-35 11--3     .  0--8 32-40       <- Y=0
114
115              44-36 12--4        7-15 39-47           -1
116               |  |  |              |  |  |
117           60-52 28-20     5  6    23-31 55-63        -2
118            |              |  |              |
119           68-76       21-13 14-22       79-71        -3
120               |        |        |        |
121           92-84       29-37 38-30       87-95        -4
122                           |  |
123                 85-77 53-45 46-54 78-86              -5
124                  |  |  |        |  |  |
125                 93 69-61       62-70 94              -6
126
127                              ^
128           -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6
129
130       The multiples of 8 (or however many arms) N=0,8,16,etc is the original
131       curve, and the further mod 8 parts are the copies.
132
133       The middle "." shown is the origin X=0,Y=0.  It would be more
134       symmetrical to have the origin the middle of the eight arms, which
135       would be X=-0.5,Y=-0.5 in the above, but that would give fractional X,Y
136       values.  Apply an offset X+0.5,Y+0.5 to centre if desired.
137
138   Level Ranges
139       The N=0 point is reckoned as level=0, then N=0 to N=5 inclusive is
140       level=1, etc.  Each level is 4 copies of the previous and an extra 2
141       points between.
142
143           LevelPoints[k] = 4*LevelPoints[k-1] + 2   starting LevelPoints[0]=1
144                          = 2 + 2*4 + 2*4^2 + ... + 2*4^(k-1) + 1*4^k
145                          = (5*4^k - 2)/3
146
147           Nlevel[k] = LevelPoints[k] - 1         since starting at N=0
148                     = 5*(4^k - 1)/3
149                     = 0, 5, 25, 105, 425, 1705, 6825, 27305, ...    (A146882)
150
151       The width along the X axis of a level doubles each time, plus an extra
152       distance 3 between.
153
154           LevelWidth[k] = 2*LevelWidth[k-1] + 3     starting LevelWidth[0]=0
155                         = 3 + 3*2 + 3*2^2 + ... + 3*2^(k-1) + 0*2^k
156                         = 3*(2^k - 1)
157
158           Xlevel[k] = 1 + LevelWidth[k]
159                     = 3*2^k - 2
160                     = 1, 4, 10, 22, 46, 94, 190, 382, ...           (A033484)
161
162   Level Ranges with Diagonal Length
163       With "diagonal_length" = L, level=0 is reckoned as having L many points
164       instead of just 1.
165
166           LevelPoints[k] = 2 + 2*4 + 2*4^2 + ... + 2*4^(k-1) + L*4^k
167                          = ( (3L+2)*4^k - 2 )/3
168
169           Nlevel[k] = LevelPoints[k] - 1
170                     = ( (3L+2)*4^k - 5 )/3
171
172       The width of level=0 becomes L-1 instead of 0.
173
174           LevelWidth[k] = 2*LevelWidth[k-1] + 3     starting LevelWidth[0]=L-1
175                         = 3 + 3*2 + 3*2^2 + ... + 3*2^(k-1) + (L-1)*2^k
176                         = (L+2)*2^k - 3
177
178           Xlevel[k] = 1 + LevelWidth[k]
179                     = (L+2)*2^k - 2
180
181       Level=0 as L many points can be thought of as a little block which is
182       replicated in mirror image to make level=1.  For example the diagonal 4
183       example above becomes
184
185                       8  9            diagonal_length => 4
186                       |  |
187                    6--7 10-11
188                    |        |
189                 .  5       12  .
190
191              2--3             14-15
192              |                    |
193           0--1                   16-17
194
195       The spacing between the parts is had in the tiling by taking a margin
196       of 1/2 at the base and 1 horizontally left and right.
197
198   Level Fill
199       The curve doesn't visit all the points in the eighth of the plane below
200       the X=Y diagonal.  In general Nlevel+1 many points of the triangular
201       area Xlevel^2/4 are visited, for a filled fraction which approaches a
202       constant
203
204                         Nlevel          4*(3L+2)
205           FillFrac = ------------   ->  ---------
206                      Xlevel^2 / 4       3*(L+2)^2
207
208       For example the default L=1 has FillFrac=20/27=0.74.  Or L=2
209       FillFrac=2/3=0.66.  As the diagonal length increases the fraction
210       decreases due to the growing holes in the pattern.
211

FUNCTIONS

213       See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path
214       classes.
215
216       "$path = Math::PlanePath::SierpinskiCurveStair->new ()"
217       "$path = Math::PlanePath::SierpinskiCurveStair->new (diagonal_length =>
218       $L, arms => $A)"
219           Create and return a new path object.
220
221       "($x,$y) = $path->n_to_xy ($n)"
222           Return the X,Y coordinates of point number $n on the path.  Points
223           begin at 0 and if "$n < 0" then the return is an empty list.
224
225           Fractional positions give an X,Y position along a straight line
226           between the integer positions.
227
228       "$n = $path->n_start()"
229           Return 0, the first N in the path.
230
231   Level Methods
232       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
233           Return "(0, ((3*$diagonal_length +2) * 4**$level - 5)/3" as per
234           "Level Ranges with Diagonal Length" above.
235

OEIS

237       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
238       this path include
239
240           <http://oeis.org/A146882> (etc)
241
242           A146882   Nlevel, for level=1 up
243           A033484   Xmax and Ymax in level, being 3*2^n - 2
244

SEE ALSO

246       Math::PlanePath, Math::PlanePath::SierpinskiCurve
247

HOME PAGE

249       <http://user42.tuxfamily.org/math-planepath/index.html>
250

LICENSE

252       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 Kevin Ryde
253
254       Math-PlanePath is free software; you can redistribute it and/or modify
255       it under the terms of the GNU General Public License as published by
256       the Free Software Foundation; either version 3, or (at your option) any
257       later version.
258
259       Math-PlanePath is distributed in the hope that it will be useful, but
260       WITHOUT ANY WARRANTY; without even the implied warranty of
261       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
262       General Public License for more details.
263
264       You should have received a copy of the GNU General Public License along
265       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
266
267
268
269perl v5.32.0                      2020-M0a7t-h2:8:PlanePath::SierpinskiCurveStair(3)
Impressum