1Math::PlanePath::GosperUIsselranCdosn(t3r)ibuted Perl DoMcautmhe:n:tPaltainoenPath::GosperIslands(3)
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NAME

6       Math::PlanePath::GosperIslands -- concentric Gosper islands
7

SYNOPSIS

9        use Math::PlanePath::GosperIslands;
10        my $path = Math::PlanePath::GosperIslands->new;
11        my ($x, $y) = $path->n_to_xy (123);
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DESCRIPTION

14       This path is integer versions of the Gosper island at successive
15       levels, arranged as concentric rings on a triangular lattice (see
16       "Triangular Lattice" in Math::PlanePath).  Each island is the outline
17       of a self-similar tiling of the plane by hexagons, and the sides are
18       self-similar expansions of line segments
19
20                 35----34                                           8
21                /        \
22           ..-36          33----32          29----28                7
23                                  \        /        \
24                                   31----30          27----26       6
25                                                             \
26                                                              25    5
27
28                                                           78       4
29                                                             \
30                          11-----10                           77    3
31                         /         \                         /
32                 13----12           9---- 8                76       2
33                /                          \                 \
34              14           3---- 2           7                ...   1
35                \        /        \
36                 15     4           1    24                     <- Y=0
37                /        \                 \
38              16           5----- 6         23                     -1
39                \                          /
40                 17----18          21----22                        -2
41                         \        /
42                          19----20                                 -3
43
44
45
46
47              -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9 10 11
48
49   Hexagon Replications
50       The islands can be thought of as a self-similar tiling of the plane by
51       hexagonal shapes.  The innermost hexagon N=1 to N=6 is replicated six
52       times to make the next outline N=7 to N=24.
53
54                       *---*
55                      /     \
56                 *---*       *---*
57                /     \     /     \
58               *       *---*       *
59                \     /     \     /
60                 *---*       *---*
61                /     \     /     \
62               *       *---*       *
63                \     /     \     /
64                 *---*       *---*
65                      \     /
66                       *---*
67
68       Then that shape is in turn replicated six times around itself to make
69       the next level.
70
71                               *---E
72                              /     \
73                         *---*       *---*       *---*
74                        /                 \     /     \
75                       *                   *---*       *---*
76                        \                 /                 \
77                         *               *                   D
78                        /                 \                 /
79                   *---*                   *               *
80                  /     \                 /                 \
81             *---*       *---*       *---*                   *
82            /                 \     /     \                 /
83           *                   *---*       *---*       *---*
84            \                 /                 \     /     \
85             *               *                   C---*       *---*
86            /                 \                 /                 \
87           *                   *       O       *                   *
88            \                 /                 \                 /
89             *---*       *---*                   *               *
90                  \     /     \                 /                 \
91                   *---*       *---*       *---*                   *
92                  /                 \     /     \                 /
93                 *                   *---*       *---*       *---*
94                  \                 /                 \     /
95                   *               *                   *---*
96                  /                 \                 /
97                 *                   *               *
98                  \                 /                 \
99                   *---*       *---*                   *
100                        \     /     \                 /
101                         *---*       *---*       *---*
102                                          \     /
103                                           *---*
104
105       The shapes here and at higher level are like hexagons with wiggly
106       sides.  The sides are symmetric, so they mate on opposite sides but the
107       join "corner" is not on the X axis (after the first level).  For
108       example the point marked "C" (N=7) is above the axis at X=5,Y=1.  The
109       next replication joins at point "D" (N=25) at X=11,Y=5.
110
111   Side and Radial Lines
112       The sides at each level is a self-similar line segment expansion,
113
114                                        *---*
115             *---*     becomes         /
116                                  *---*
117
118       This expanding side shape is also the radial line or spoke from the
119       origin out to "corner" join points.  At level 0 they're straight lines,
120
121              ...----*
122                    / \
123                   /   \ side
124                  /     \
125                 /       \
126                O---------*
127                         /
128                        /
129
130       Then at higher levels they're wiggly, such as level 1,
131
132           ...--*
133               / \
134              *   *---*
135               \       \
136                *   *---C
137               /   /   /
138              O---*   ...
139
140       Or level 2,
141
142                *---E
143             ...   / \
144                  *   *---*       *---*
145                   \       \     /     \
146                    *       *---*       *---*
147                   /                         \
148                  *                       *---D
149                   \                     /   /
150                    *---*           *---*   ...
151                         \         /
152                          *       *
153                         /         \
154                        *           *
155                         \         /
156                          *   *---C
157                         /   /
158                        O---*
159
160       The lines become ever wigglier at higher levels, and in fact become "S"
161       shapes with the ends spiralling around and in (and in fact middle
162       sections likewise S and spiralling, to a lesser extent).
163
164       The spiralling means that the X axis is crossed multiple times at
165       higher levels.  For example in level 11 X>0,Y=0 occurs 22 times between
166       N=965221 and N=982146.  Likewise on diagonal lines X=Y and X=-Y which
167       are "sixths" of the initial hexagon.
168
169       The self-similar spiralling means the Y axis too is crossed multiple
170       times at higher levels.  In level 11 again X=0,Y>0 is crossed 7 times
171       between N=1233212 and N=1236579.  (Those N's are bigger than the X axis
172       crossing, because the Nstart position at level 11 has rotated around
173       some 210 degrees to just under the negative X axis.)
174
175       In general any radial straight line is crossed many times way
176       eventually.
177
178   Level Ranges
179       Counting the inner hexagon as level=0, the side length and whole ring
180       is
181
182           sidelen = 3^level
183           ringlen = 6 * 3^level
184
185       The starting N for each level is the total points preceding it, so
186
187           Nstart = 1 + ringlen(0) + ... + ringlen(level-1)
188                  = 3^(level+1) - 2
189
190       For example level=2 starts at Nstart=3^3-2=25.
191
192       The way the side/radial lines expand as described above makes the
193       Nstart position rotate around at each level.  N=7 is at X=5,Y=1 which
194       is angle
195
196           angle = arctan(1*sqrt(3) / 5)
197                 = 19.106.. degrees
198
199       The sqrt(3) factor as usual turns the flattened integer X,Y coordinates
200       into proper equilateral triangles.  The further levels are then
201       multiples of that angle.  For example N=25 at X=11,Y=5 is 2*angle, or
202       N=79 at X=20,Y=18 at 3*angle.
203
204       The N=7 which is the first radial replication at X=5,Y=1, scaled to
205       unit sided equilateral triangles, has distance from the origin
206
207           d1 = hypot(5, 1*sqrt(3)) / 2
208              = sqrt(7)
209
210       The subsequent levels are powers of that sqrt(7),
211
212           Xstart,Ystart of the Nstart(level) position
213
214           d = hypot(Xstart,Ystart*sqrt(3))/2
215             = sqrt(7) ^ level
216
217       This multiple of the angle and powering of the radius means the Nstart
218       points are on a logarithmic spiral.
219
220   Fractal Island
221       The Gosper island is usually conceived as a fractal, with the initial
222       hexagon in a fixed position and the sides having the line segment
223       substitution described above, for ever finer levels of "S" spiralling
224       wiggliness.  The code here can be used for that by rotating the Nstart
225       position back to the X axis and scaling down to a desired unit radius.
226
227           Xstart,Ystart of the Nstart(level) position
228
229           scale factor = 0.5 / hypot(Ystart*sqrt(3), Xstart)
230           rotate angle = - atan2 (Ystart*sqrt(3), Xstart)
231
232       This scale and rotate puts the Nstart point at X=1,Y=0 and further
233       points of the ring around from that.  Remember the sqrt(3) factor on Y
234       for all points to turn the integer coordinates into proper equilateral
235       triangles.
236
237       Notice the line segment substitution doesn't change the area of the
238       initial hexagon.  Effectively (and not to scale),
239
240                                        *
241                                       / \
242           *-------*    becomes   *   /   *
243                                   \ /
244                                    *
245
246       So the area lost below is gained above (or vice versa).  The result is
247       a line of ever greater length enclosing an unchanging area.
248

FUNCTIONS

250       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
251       classes.
252
253       "$path = Math::PlanePath::GosperIslands->new ()"
254           Create and return a new path object.
255
256       "($x,$y) = $path->n_to_xy ($n)"
257           Return the X,Y coordinates of point number $n on the path.  Points
258           begin at 1 and if "$n < 0" then the return is an empty list.
259

SEE ALSO

261       Math::PlanePath, Math::PlanePath::KochSnowflakes,
262       Math::PlanePath::GosperSide
263

HOME PAGE

265       <http://user42.tuxfamily.org/math-planepath/index.html>
266

LICENSE

268       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
269       Kevin Ryde
270
271       Math-PlanePath is free software; you can redistribute it and/or modify
272       it under the terms of the GNU General Public License as published by
273       the Free Software Foundation; either version 3, or (at your option) any
274       later version.
275
276       Math-PlanePath is distributed in the hope that it will be useful, but
277       WITHOUT ANY WARRANTY; without even the implied warranty of
278       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
279       General Public License for more details.
280
281       You should have received a copy of the GNU General Public License along
282       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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286perl v5.34.0                      2022-01-21 Math::PlanePath::GosperIslands(3)
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