1Math::PlanePath::QuinteUtsCeurrvCeo(n3t)ributed Perl DocMuamtehn:t:aPtliaonnePath::QuintetCurve(3)
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6 Math::PlanePath::QuintetCurve -- self-similar "plus" shaped curve
7
9 use Math::PlanePath::QuintetCurve;
10 my $path = Math::PlanePath::QuintetCurve->new;
11 my ($x, $y) = $path->n_to_xy (123);
12
14 This path is Mandelbrot's "quartet" trace of spiralling self-similar
15 "+" shape,
16 125--... 93--92 11
18 | | |
19 123-124 94 91--90--89--88 10
20 | | |
21 122-121-120 103-102 95 82--83 86--87 9
22 | | | | | | |
23 115-116 119 104 101-100--99 96 81 84--85 8
24 | | | | | | |
25 113-114 117-118 105 32--33 98--97 80--79--78 7
26 | | | | |
27 112-111-110-109 106 31 34--35--36--37 76--77 6
28 | | | | |
29 108-107 30 43--42 39--38 75 5
30 | | | | |
31 25--26 29 44 41--40 73--74 4
32 | | | | |
33 23--24 27--28 45--46--47 72--71--70--69--68 3
34 | | |
35 22--21--20--19--18 49--48 55--56--57 66--67 2
36 | | | | |
37 5---6---7 16--17 50--51 54 59--58 65 1
38 | | | | | | |
39 0---1 4 9---8 15 52--53 60--61 64 <- Y=0
40 | | | | | |
41 2---3 10--11 14 62--63 -1
42 | |
43 12--13 -2
44 ^
46 X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
47 As per
49 Benoit B. Mandelbrot, "The Fractal Geometry of Nature", W. H. Freeman
51 and Co., 1983, ISBN 0-7167-1186-9, section 7, "Harnessing the Peano
52 Monster Curves", pages 72-73.
53 Mandelbrot calls this a "quartet", taken as 4 parts around a further
55 middle part (like 4 players around a table). The module name "quintet"
56 here is a mistake, though it does suggest the base-5 nature of the
57 curve.
58 The base figure is the initial N=0 to N=5.
60 5
62 |
63 |
64 0---1 4 base figure
65 | |
66 | |
67 2---3
68 It corresponds to a traversal of the following "+" shape,
70 .... 5
72 . |
73 . <|
74 |
75 0----1 .. 4 ....
76 v | | .
77 . |> |> .
78 . | | .
79 .... 2----3 ....
80 . v .
81 . .
82 . .
83 . .. .
84 The "v" and ">" notches are the side the figure is directed at the
86 higher replications. The 0, 2 and 3 sub-curves are the right hand side
87 of the line and are a plain repetition of the base figure. The 1 and 4
88 parts are to the left and are a reversal. The first such reversal is
89 seen in the sample above as N=5 to N=10.
90 .....
91 . .
92 5---6---7 ...
94 . . | .
95 . | . reversed figure
96 ... 9---8 ...
97 | .
98 | .
99 10 ...
100 Mandelbrot gives the expansion without designating start and end. The
102 start is chosen here so the expansion has sub-curve 0 forward (not
103 reverse). This ensures the next expansion has the curve the same up to
104 the preceding level, and extending from there.
105 In the base figure it can be seen the N=5 endpoint is rotated up around
107 from the N=0 to N=1 direction. This makes successive higher levels
108 slowly spiral around.
109 base b = 2 + i
111 N = 5^level
112 angle = level * arg(b) = level*atan(1/2)
113 = level * 26.56 degrees
114 In the sample shown above N=125 is level=3 and has spiralled around to
116 angle 3*26.56=79.7 degrees. The next level goes to X negative in the
117 second quadrant. A full circle around the plane is approximately level
118 14.
119 Arms
121 The optional "arms => $a" parameter can give 1 to 4 copies of the
122 curve, each advancing successively. For example "arms=>4" is as
123 follows. N=4*k points are the plain curve, and N=4*k+1, N=4*k+2 and
124 N=4*k+3 are rotated copies of it.
125 69--65 ...
127 | | |
128 ..-117-113-109 73 61--57--53--49 120
129 | | | |
130 101-105 77 25--29 41--45 100-104 116
131 | | | | | | | |
132 97--93 81 21 33--37 92--96 108-112
133 | | | |
134 50--46 89--85 17--13-- 9 88--84--80--76--72
135 | | | |
136 54 42--38 10-- 6 1-- 5 20--24--28 64--68
137 | | | | | | |
138 58 30--34 14 2 0-- 4 16 36--32 60
139 | | | | | | |
140 66--62 26--22--18 7-- 3 8--12 40--44 56
141 | | | |
142 70--74--78--82--86 11--15--19 87--91 48--52
143 | | | |
144 110-106 94--90 39--35 23 83 95--99
145 | | | | | | | |
146 114 102--98 47--43 31--27 79 107-103
147 | | | |
148 118 51--55--59--63 75 111-115-119-..
149 | | |
150 ... 67--71
151 The curve is essentially an ever expanding "+" shape with one corner at
153 the origin. Four such shapes pack as follows,
154 +---+
156 | |
157 +---@--- +---+
158 | | B |
159 +---+ +---+ +---@
160 | C | | |
161 +---+ +---O---+ +---+
162 | | | A |
163 @---+ +---+ +---+
164 | D | |
165 +---+ +---@---+
166 | |
167 +---+
168 At higher replication levels the sides are wiggly and spiralling and
170 the centres of each rotate around, but their sides are symmetric and
171 mesh together perfectly to fill the plane.
172
174 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
175 classes.
176 "$path = Math::PlanePath::QuintetCurve->new ()"
178 "$path = Math::PlanePath::QuintetCurve->new (arms => $a)"
179 Create and return a new path object.
180 "($x,$y) = $path->n_to_xy ($n)"
182 Return the X,Y coordinates of point number $n on the path. Points
183 begin at 0 and if "$n < 0" then the return is an empty list.
184 Fractional positions give an X,Y position along a straight line
186 between the integer positions.
187 "$n = $path->n_start()"
189 Return 0, the first N in the path.
190 "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
192 In the current code the returned range is exact, meaning $n_lo and
193 $n_hi are the smallest and biggest in the rectangle, but don't rely
194 on that yet since finding the exact range is a touch on the slow
195 side. (The advantage of which though is that it helps avoid very
196 big ranges from a simple over-estimate.)
197 Level Methods
199 "($n_lo, $n_hi) = $path->level_to_n_range($level)"
200 Return "(0, 5**$level)", or for multiple arms return "(0, $arms *
201 5**$level)".
202 There are 5^level + 1 points in a level, numbered starting from 0.
204 On the second and subsequent arms the origin is omitted (so as not
205 to repeat that point) and so just 5^level for them, giving
206 5^level+1 + (arms-1)*5^level = arms*5^level + 1 many points
207 starting from 0.
208
210 X,Y to N
211 The current approach uses the "QuintetCentres" "xy_to_n()". Because
212 the tiling in "QuintetCurve" and "QuintetCentres" is the same, the X,Y
213 coordinates for a given N are no more than 1 away in the grid.
214 The way the two lowest shapes are arranged in fact means that for a
216 "QuintetCurve" N at X,Y then the same N on the "QuintetCentres" is at
217 one of three locations
218 X, Y same
220 X, Y+1 up
221 X-1, Y+1 up and left
222 X-1, Y left
223 This is so even when the "arms" multiple paths are in use (the same
225 arms in both coordinates).
226 Is there an easy way to know which of the four offsets is right? The
228 current approach is to give each to "QuintetCentres" to make an N, put
229 that N back through "n_to_xy()" to see if it's the target $n.
230
232 Math::PlanePath, Math::PlanePath::QuintetCentres,
233 Math::PlanePath::QuintetReplicate, Math::PlanePath::Flowsnake
234
236 <http://user42.tuxfamily.org/math-planepath/index.html>
237
239 Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 Kevin Ryde
240 This file is part of Math-PlanePath.
242 Math-PlanePath is free software; you can redistribute it and/or modify
244 it under the terms of the GNU General Public License as published by
245 the Free Software Foundation; either version 3, or (at your option) any
246 later version.
247 Math-PlanePath is distributed in the hope that it will be useful, but
249 WITHOUT ANY WARRANTY; without even the implied warranty of
250 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
251 General Public License for more details.
252 You should have received a copy of the GNU General Public License along
254 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
255perl v5.32.0 2020-07-28 Math::PlanePath::QuintetCurve(3)