1Math::PlanePath::KochSnUoswefrlaCkoenst(r3i)buted Perl DMoactuhm:e:nPtlaatnieoPnath::KochSnowflakes(3)
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NAME

6       Math::PlanePath::KochSnowflakes -- Koch snowflakes as concentric rings
7

SYNOPSIS

9        use Math::PlanePath::KochSnowflakes;
10        my $path = Math::PlanePath::KochSnowflakes->new;
11        my ($x, $y) = $path->n_to_xy (123);
12

DESCRIPTION

14       This path traces out concentric integer versions of the Koch snowflake
15       at successively greater iteration levels.
16
17                                      48                                6
18                                     /  \
19                             50----49    47----46                       5
20                               \              /
21                    54          51          45          42              4
22                   /  \        /              \        /  \
23           56----55    53----52                44----43    41----40     3
24             \                                                  /
25              57                      12                      39        2
26             /                       /  \                       \
27           58----59          14----13    11----10          37----38     1
28                   \           \       3      /           /
29                    60          15  1----2   9          36         <- Y=0
30                   /                          \           \
31           62----61           4---- 5    7---- 8           35----34    -1
32             \                       \  /                       /
33              63                       6                      33       -2
34                                                                \
35           16----17    19----20                28----29    31----32    -3
36                   \  /        \              /        \  /
37                    18          21          27          30             -4
38                               /              \
39                             22----23    25----26                      -5
40                                     \  /
41                                      24                               -6
42
43                                       ^
44           -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9
45
46       The initial figure is the triangle N=1,2,3 then for the next level each
47       straight side expands to 3x longer and a notch like N=4 through N=8,
48
49             *---*     becomes     *---*   *---*
50                                        \ /
51                                         *
52
53       The angle is maintained in each replacement, for example the segment
54       N=5 to N=6 becomes N=20 to N=24 at the next level.
55
56   Triangular Coordinates
57       The X,Y coordinates are arranged as integers on a square grid per
58       "Triangular Lattice" in Math::PlanePath, except the Y coordinates of
59       the innermost triangle which is
60
61                         N=3     X=0, Y=+2/3
62                          *
63                         / \
64                        /   \
65                       /     \
66                      /   o   \
67                     /         \
68                N=1 *-----------* N=2
69           X=-1, Y=-1/3      X=1, Y=-1/3
70
71       These values are not integers, but they're consistent with the centring
72       and scaling of the higher levels.  If all-integer is desired then
73       rounding gives Y=0 or Y=1 and doesn't overlap the subsequent points.
74
75   Level Ranges
76       Counting the innermost triangle as level 0, each ring is
77
78           Nstart = 4^level
79           length = 3*4^level    many points
80
81       For example the outer ring shown above is level 2 starting N=4^2=16 and
82       having length=3*4^2=48 points (through to N=63 inclusive).
83
84       The X range at a given level is the initial triangle baseline iterated
85       out.  Each level expands the sides by a factor of 3 so
86
87            Xlo = -(3^level)
88            Xhi = +(3^level)
89
90       For example level 2 above runs from X=-9 to X=+9.  The Y range is the
91       points N=6 and N=12 iterated out.  Ylo in level 0 since there's no
92       downward notch on that innermost triangle.
93
94           Ylo = / -(2/3)*3^level if level >= 1
95                 \ -1/3           if level == 0
96           Yhi = +(2/3)*3^level
97
98       Notice that for each level the extents grow by a factor of 3 but the
99       notch introduced in each segment is not big enough to go past the
100       corner positions.  They can equal the extents horizontally, for example
101       in level 1 N=14 is at X=-3 the same as the corner N=4, and on the right
102       N=10 at X=+3 the same as N=8, but they don't go past.
103
104       The snowflake is an example of a fractal curve with ever finer
105       structure.  The code here can be used for that by going from N=Nstart
106       to N=Nstart+length-1 and scaling X/3^level Y/3^level to give a 2-wide
107       1-high figure of desired fineness.  See examples/koch-svg.pl for a
108       complete program doing that as an SVG image file.
109
110   Area
111       The area of the snowflake at a given level can be calculated from the
112       area under the Koch curve per "Area" in Math::PlanePath::KochCurve
113       which is the 3 sides, and the central triangle
114
115                        *          ^ Yhi
116                       / \         |          height = 3^level
117                      /   \        |
118                     /     \       |
119                    *-------*      v
120
121                    <------->      width = 3^level - (- 3^level) = 2*3^level
122                   Xlo      Xhi
123
124           triangle_area = width*height/2 = 9^level
125
126           snowflake_area[level] = triangle_area[level] + 3*curve_area[level]
127                                 = 9^level + 3*(9^level - 4^level)/5
128                                 = (8*9^level - 3*4^level) / 5
129
130       If the snowflake is conceived as a fractal of fixed initial triangle
131       size and ever-smaller notches then the area is divided by that central
132       triangle area 9^level,
133
134           unit_snowflake[level] = snowflake_area[level] / 9^level
135                                 = (8 - 3*(4/9)^level) / 5
136                                 -> 8/5      as level -> infinity
137
138       Which is the well-known 8/5 * initial triangle area for the fractal
139       snowflake.
140

FUNCTIONS

142       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
143       classes.
144
145       "$path = Math::PlanePath::KochSnowflakes->new ()"
146           Create and return a new path object.
147
148   Level Methods
149       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
150           Return per "Level Ranges" above,
151
152               (4**$level,
153                4**($level+1) - 1)
154

FORMULAS

156   Rectangle to N Range
157       As noted in "Level Ranges" above, for a given level
158
159                 -(3^level) <= X <= 3^level
160           -(2/3)*(3^level) <= Y <= (2/3)*(3^level)
161
162       So the maximum X,Y in a rectangle gives
163
164           level = ceil(log3(max(abs(x1), abs(x2), abs(y1)*3/2, abs(y2)*3/2)))
165
166       and the last point in that level is
167
168           Nlevel = 4^(level+1) - 1
169
170       Using this as an N range is an over-estimate, but an easy calculation.
171       It's not too difficult to trace down for an exact range
172

OEIS

174       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
175       the Koch snowflake include the following.  See "OEIS" in
176       Math::PlanePath::KochCurve for entries related to a single Koch side.
177
178           <http://oeis.org/A164346> (etc)
179
180           A164346   number of points in ring n, being 3*4^n
181           A178789   number of acute angles in ring n, 4^n + 2
182           A002446   number of obtuse angles in ring n, 2*4^n - 2
183
184       The acute angles are those of +/-120 degrees and the obtuse ones +/-240
185       degrees.  Eg. in the outer ring=2 shown above the acute angles are at
186       N=18, 22, 24, 26, etc.  The angles are all either acute or obtuse, so
187       A178789 + A002446 = A164346.
188

SEE ALSO

190       Math::PlanePath, Math::PlanePath::KochCurve, Math::PlanePath::KochPeaks
191
192       Math::PlanePath::QuadricIslands
193

HOME PAGE

195       <http://user42.tuxfamily.org/math-planepath/index.html>
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LICENSE

198       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 Kevin Ryde
199
200       Math-PlanePath is free software; you can redistribute it and/or modify
201       it under the terms of the GNU General Public License as published by
202       the Free Software Foundation; either version 3, or (at your option) any
203       later version.
204
205       Math-PlanePath is distributed in the hope that it will be useful, but
206       WITHOUT ANY WARRANTY; without even the implied warranty of
207       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
208       General Public License for more details.
209
210       You should have received a copy of the GNU General Public License along
211       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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215perl v5.32.0                      2020-07-28Math::PlanePath::KochSnowflakes(3)
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