1Math::PlanePath::GosperUSsiedre(C3o)ntributed Perl DocumMeanttha:t:iPolnanePath::GosperSide(3)
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6 Math::PlanePath::GosperSide -- one side of the Gosper island
7
9 use Math::PlanePath::GosperSide;
10 my $path = Math::PlanePath::GosperSide->new;
11 my ($x, $y) = $path->n_to_xy (123);
12
14 This path is a single side of the Gosper island, in integers
15 ("Triangular Lattice" in Math::PlanePath).
16 20-... 14
18 /
19 18----19 13
20 /
21 17 12
22 \
23 16 11
24 /
25 15 10
26 \
27 14----13 9
28 \
29 12 8
30 /
31 11 7
32 \
33 10 6
34 /
35 8---- 9 5
36 /
37 6---- 7 4
38 /
39 5 3
40 \
41 4 2
42 /
43 2---- 3 1
44 /
45 0---- 1 <- Y=0
46 ^
48 X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 ...
49 The path slowly spirals around counter clockwise, with a lot of
51 wiggling in between. The N=3^level point is at
52 N = 3^level
54 angle = level * atan(sqrt(3)/5)
55 = level * 19.106 degrees
56 radius = sqrt(7) ^ level
57 A full revolution for example takes roughly level=19 which is about
59 N=1,162,000,000.
60 Both ends of such levels are in fact sub-spirals, like an "S" shape.
62 The path is both the sides and the radial spokes of the "GosperIslands"
64 path, as described in "Side and Radial Lines" in
65 Math::PlanePath::GosperIslands. Each N=3^level point is the start of a
66 "GosperIslands" ring.
67 The path is the same as the "TerdragonCurve" except the turns here are
69 by 60 degrees each, whereas "TerdragonCurve" is by 120 degrees. See
70 Math::PlanePath::TerdragonCurve for the turn sequence and total
71 direction formulas etc.
72
74 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
75 classes.
76 "$path = Math::PlanePath::GosperSide->new ()"
78 Create and return a new path object.
79 "($x,$y) = $path->n_to_xy ($n)"
81 Return the X,Y coordinates of point number $n on the path. Points
82 begin at 0 and if "$n < 0" then the return is an empty list.
83 Fractional $n gives a point on the straight line between integer N.
85 Level Methods
87 "($n_lo, $n_hi) = $path->level_to_n_range($level)"
88 Return "(0, 3**$level)".
89
91 Level Endpoint
92 The endpoint of each level N=3^k is at
93 X + Y*i*sqrt(3) = b^k
95 where b = 2 + w = 5/2 + sqrt(3)/2*i
96 where w=1/2 + sqrt(3)/2*i sixth root of unity
97 X(k) = ( 5*X(k-1) - 3*Y(k-1) )/2 for k>=1
99 Y(k) = ( X(k-1) + 5*Y(k-1) )/2
100 starting X(0)=2 Y(0)=0
101 X(k) = 5*X(k-1) - 7*X(k-2) for k>=2
103 starting X(0)=2 X(1)=5
104 = 2, 5, 11, 20, 23, -25, -286, -1255, -4273, -12580, -32989,..
105 Y(k) = 5*Y(k-1) - Y*X(k-2) for k>=2
107 starting Y(0)=0 Y(1)=1
108 = 0, 1, 5, 18, 55, 149, 360, 757, 1265, 1026, -3725, ...
109 (A099450)
110 The curve base figure is XY(k)=XY(k-1)+rot60(XY(k-1))+XY(k-1) giving
112 XY(k) = (2+w)^k = b^k where w is the sixth root of unity giving the
113 rotation by +60 degrees.
114 The mutual recurrences are similar with the rotation done by (X-3Y)/2,
116 (Y+X)/2 per "Triangular Lattice" in Math::PlanePath. The separate
117 recurrences are found by using the first to get Y(k-1) = -2/3*X(k) +
118 5/3*X(k-1) and substitute into the other to get X(k+1). Similar the
119 other way around for Y(k+1).
120
122 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
123 this path include
124 <http://oeis.org/A099450> (etc)
126 A229215 direction 1,2,3,-1,-2,-3 (clockwise)
128 A099450 Y at N=3^k (for k>=1)
129 Also the turn sequence is the same as the terdragon curve, see "OEIS"
131 in Math::PlanePath::TerdragonCurve for the several turn forms, N
132 positions of turns, etc.
133
135 Math::PlanePath, Math::PlanePath::GosperIslands,
136 Math::PlanePath::TerdragonCurve, Math::PlanePath::KochCurve
137 Math::Fractal::Curve
139
141 <http://user42.tuxfamily.org/math-planepath/index.html>
142
144 Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
145 Kevin Ryde
146 Math-PlanePath is free software; you can redistribute it and/or modify
148 it under the terms of the GNU General Public License as published by
149 the Free Software Foundation; either version 3, or (at your option) any
150 later version.
151 Math-PlanePath is distributed in the hope that it will be useful, but
153 WITHOUT ANY WARRANTY; without even the implied warranty of
154 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
155 General Public License for more details.
156 You should have received a copy of the GNU General Public License along
158 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
159perl v5.34.0 2021-07-22 Math::PlanePath::GosperSide(3)