1Math::PlanePath::VogelFUlsoerretC(o3n)tributed Perl DocuMmaetnht:a:tPiloannePath::VogelFloret(3)
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6 Math::PlanePath::VogelFloret -- circular pattern like a sunflower
7
9 use Math::PlanePath::VogelFloret;
10 my $path = Math::PlanePath::VogelFloret->new;
11 my ($x, $y) = $path->n_to_xy (123);
12 # other rotations
14 $path = Math::PlanePath::VogelFloret->new
15 (rotation_type => 'sqrt2');
16
18 The is an implementation of Helmut Vogel's model for the arrangement of
19 seeds in the head of a sunflower. Integer points are on a spiral at
20 multiples of the golden ratio phi = (1+sqrt(5))/2,
21 27 19
23 24
24 14 11
26 22 16
27 6 29
28 30 9 3
30 8
31 1 21
32 17 .
33 4
34 13
35 25 2 5
36 12
37 7 26
38 10
39 18
40 20 15
41 23 31
43 28
44 The polar coordinates for a point N are
46 R = sqrt(N) * radius_factor
48 angle = N / (phi**2) in revolutions, 1==full circle
49 = N * -phi modulo 1, with since 1/phi^2 = 2-phi
50 theta = 2*pi * angle in radians
51 Going from point N to N+1 adds an angle 0.382 revolutions around (anti-
53 clockwise, the usual spiralling direction), which means just over 1/3
54 of a circle. Or equivalently it's -0.618 back (clockwise) which is
55 phi=1.618 ignoring the integer part since that's a full circle -- only
56 the fractional part determines the position.
57 "radius_factor" is a scaling 0.6242 designed to put the closest points
59 1 apart. The closest are N=1 and N=4. See "Packing" below.
60 Other Rotation Types
62 An optional "rotation_type" parameter selects other possible floret
63 forms.
64 $path = Math::PlanePath::VogelFloret->new
66 (rotation_type => 'sqrt2');
67 The current types are as follows. The "radius_factor" for each keeps
69 points at least 1 apart so unit circles don't overlap.
70 rotation_type rotation_factor radius_factor
72 "phi" 2-phi = 0.3820 0.624
73 "sqrt2" sqrt(2) = 0.4142 0.680
74 "sqrt3" sqrt(3) = 0.7321 0.756
75 "sqrt5" sqrt(5) = 0.2361 0.853
76 The "sqrt2" floret is quite similar to phi, but doesn't pack as
78 tightly. Custom rotations can be made with "rotation_factor" and
79 "rotation_factor" parameters,
80 # R = sqrt(N) * radius_factor
82 # angle = N * rotation_factor in revolutions
83 # theta = 2*pi * angle in radians
84 #
85 $path = Math::PlanePath::VogelFloret->new
86 (rotation_factor => sqrt(37),
87 radius_factor => 2.0);
88 Usually "rotation_factor" should be an irrational number. A rational
90 like P/Q merely results in Q many straight lines and doesn't spread the
91 points enough to suit R=sqrt(N). Irrationals which are very close to
92 simple rationals behave that way too. (Of course all floating point
93 values are implicitly rationals, but are fine within the limits of
94 floating point accuracy.)
95 The "noble numbers" (A+B*phi)/(C+D*phi) with A*D-B*C=1, A<B, C<D behave
97 similar to the basic phi. Their continued fraction expansion begins
98 with some arbitrary values and then becomes a repeating "1" the same as
99 phi. The effect is some spiral arms near the origin then the phi-ness
100 dominating for large N.
101 Packing
103 Each point is at an increasing distance sqrt(N) from the origin. This
104 sqrt based on how many unit figures will fit within that distance. The
105 area within radius R is
106 T = pi * R^2 area of circle R
108 so if N figures each of area A are packed into that space then the
110 radius R is proportional to sqrt(N),
111 N*A = T = pi * R^2
113 R = sqrt(N) * sqrt(A/pi)
114 The tightest possible packing for unit circles is a hexagonal honeycomb
116 grid, each of area A = sqrt(3)/2 = 0.866. That would be factor
117 sqrt(A/pi) = 0.525. The phi floret packing is not as tight as that,
118 needing radius factor 0.624 as described above.
119 Generally the tightness of the packing depends on the fractions which
121 closely approximate the rotation factor. If the terms of the continued
122 fraction expansion are large then there's large regions of spiral arcs
123 with gaps between. The density in such regions is low and a big radius
124 factor is needed to keep the points apart. If the continued fraction
125 terms are ever increasing then there may be no radius factor big enough
126 to always keep the points a minimum distance apart ... or something
127 like that.
128 The terms of the continued fraction for phi are all 1 and is therefore,
130 in that sense, among all irrationals, the value least well approximated
131 by rationals.
132 1
134 phi = 1 + ------
135 1 + 1
136 ------
137 ^ 1 + 1
138 | ---
139 | ^ 1 + 1
140 | | ----
141 | | ^ ...
142 terms -+---+---+
143 sqrt(3) is 1,2 repeating. sqrt(13) is 3s repeating.
145 Fibonacci and Lucas Numbers
147 The Fibonacci numbers F(k) = 1,1,2,3,5,8,13,21, etc and Lucas number
148 L(k) = 2,1,3,4,7,11,18, etc form almost straight lines on the X axis of
149 the phi floret. This occurs because N*-phi is close to an integer for
150 those N. For example N=13 has angle 13*-phi = -21.0344, the fractional
151 part -0.0344 puts it just below the X axis.
152 Both F(k) and L(k) grow exponentially (as phi^k) which soon outstrips
154 the sqrt in the R radial distance so they become widely spaced apart
155 along the X axis.
156 For interest, or for reference, the angle F(k)*phi is in fact roughly
158 the next Fibonacci number F(k+1), per the well-known limit F(k+1)/F(k)
159 -> phi as k->infinity,
160 angle = F(k)*-phi
162 = -F(k+1) + epsilon
163 The Lucas numbers similarly with L(k)*phi close to L(k+1). The
165 "epsilon" approaches zero quickly enough in both cases that the
166 resulting Y coordinate approaches zero. This can be calculated as
167 follows, writing
168 beta = -1/phi =-0.618
170 Since abs(beta)<1 the powers beta^k go to zero.
172 F(k) = (phi^k - beta^k) / (phi - beta) # an integer
174 angle = F(k) * -phi
176 = - (phi*phi^k - phi*beta^k) / (phi - beta)
177 = - (phi^(k+1) - beta^(k+1)
178 + beta^(k+1) - phi*beta^k) / (phi - beta)
179 = - F(k+1) - (phi-beta)*beta^k / (phi - beta)
180 = - F(k+1) - beta^k
181 frac(angle) = - beta^k = 1/(-phi)^k
183 The arc distance away from the X axis at radius R=sqrt(F(k)) is then as
185 follows, simplifying using phi*(-beta)=1 and phi - beta = sqrt(5). The
186 Y coordinate vertical distance is a little less than the arc distance.
187 arcdist = 2*pi * R * frac(angle)
189 = 2*pi * sqrt((phi^k - beta^k)/sqrt(5)) * 1/(-phi)^k
190 = - (-1)^k * 2*pi * sqrt((1/phi^2k*phi^k - beta^3k)/sqrt(5))
191 = - (-1)^k * 2*pi * sqrt((1/phi^k - 1/(-phi)^3k)/sqrt(5))
192 -> 0 as k -> infinity
193 Essentially the radius increases as phi^(k/2) but the angle frac
195 decreases as (1/phi)^k so their product goes to zero. The (-1)^k in
196 the formula puts the points alternately just above and just below the X
197 axis.
198 The calculation for the Lucas numbers is very similar, with term
200 +(beta^k) instead of -(beta^k) and an extra factor sqrt(5).
201 L(k) = phi^k + beta^k
203 angle = L(k) * -phi
205 = -phi*phi^k - phi*beta^k
206 = -phi^(k+1) - beta^(k+1) + beta^(k+1) - phi*beta^k
207 = -L(k) + beta^k * (beta - phi)
208 = -L(k) - sqrt(5) * beta^k
209 frac(angle) = -sqrt(5) * beta^k = -sqrt(5) / (-phi)^k
211 arcdist = 2*pi * R * frac(angle)
213 = 2*pi * sqrt(L(k)) * sqrt(5)*beta^k
214 = 2*pi * sqrt(phi^k + 1/(-phi)^k) * sqrt(5)*beta^k
215 = (-1)*k * 2*pi * sqrt(5) * sqrt((-beta)^2k * phi^k + beta^3k)
216 = (-1)*k * 2*pi * sqrt(5) * sqrt((-beta)^k + beta^3k)
217 Spectrum
219 The spectrum of a real number is its multiples, each rounded down to an
220 integer. For example the spectrum of phi is
221 floor(phi), floor(2*phi), floor(3*phi), floor(4*phi), ...
223 1, 3, 4, 6, ...
224 When plotted on the Vogel floret these integers are all in the first
226 1/phi = 0.618 of the circle.
227 61 53
229 69 40 45 58
230 48 32 71
231 56 27 37
232 35 19 24 50
233 43 14 11 63
234 64 22 6 16 29
235 30 3 42
236 51 9 1 8 21
237 72 17 4 . 55
238 38
239 59 25 12
240 46 33
242 67
244 This occurs because
246 angle = N * 1/phi^2
248 = N * (1-1/phi)
249 = N * -1/phi # modulo 1
250 = floor(int*phi) * -1/phi # N=spectrum
251 = (int*phi - frac) * -1/phi # 0<frac<1
252 = int + frac*1/phi
253 = frac * 1/phi # modulo 1
254 So the angle is a fraction from 0 to 1/phi=0.618 of a revolution. In
256 general for a "rotation_factor=t" with 0<t<1 the spectrum of 1/t falls
257 within the first 0 to t angle.
258 Fibonacci Word
260 The Fibonacci word 0,1,0,0,1,0,1,0,0,1,etc is the least significant bit
261 of the Zeckendorf base representation of i, starting from i=0. Plotted
262 at N=i on the "VogelFloret" gives
263 1 0
265 1 1 1 0 0 Fibonacci word
266 1 1 1 0 0
267 1 1 0 0 0
268 1 1 1 1 1 0 0 0 0
269 1 1 1 1 0 0 0
270 1 1 1 1 . 0 0 0
271 1 1 1 0 0 0 0
272 1 0 0 0 0 0
273 1 1 0 0 0 0
274 0 0 0 0 0 0 0 0
275 0 0 0 0
276 0 0
277 This pattern occurs because the Fibonacci word, among its various
279 possible definitions, is 0 or 1 according to whether i+1 occurs in the
280 spectrum of phi (1,3,4,6,8,etc) or not. So for example at i=5 the
281 value is 0 because i+1=6 is in the spectrum of phi, then at i=6 the
282 value is 1 because i+1=7 is not.
283 The "+1" for i to spectrum has the effect of rotating the spectrum
285 pattern described above by -0.381 (one rotation factor back). So the
286 Fibonacci word "0"s are from angle -0.381 to -0.381+0.618=0.236 and the
287 rest "1"s. 0.236 is close to 1/4, hence the "0"s to "1"s line just
288 before the Y axis.
289 Repdigits in Decimal
291 Some of the decimal repdigits 11, 22, ..., 99, 111, ..., 999, etc make
292 nearly straight radial lines on the phi floret. For example 11, 66,
293 333, 888 make a line upwards to the right.
294 11 and 66 are at the same polar angle because the difference is 55 and
296 55*phi = 88.9919 is nearly an integer meaning the angle is nearly
297 unchanged when added. Similarly 66 to 333 difference 267 has 267*phi =
298 432.015, or 333 to 888 difference 555 has 555*phi = 898.009. The 55 is
299 a Fibonacci number, the 123 between 99 and 222 is a Lucas number, and
300 267 = 144+123 = F(12)+L(10).
301 The differences 55 and 555 apply to pairs 22 and 77, 33 and 88, 666 and
303 1111, etc, making four straightish arms. 55 and 555 themselves are
304 near the X axis.
305 A separate spiral arm arises from 11111 falling moderately close to the
307 X axis since 11111*-phi = -17977.9756, or about 0.024 of a circle
308 upwards. The subsequent 22222, 33333, 44444, etc make a little arc of
309 nine values going upwards that much each time for a total about a
310 quarter turn 9*0.024 = 0.219.
311 Repdigits in Other Bases
313 By choosing a radix so that "11" (or similar repunit) in that radix is
314 close to the X axis, spirals like the decimal 11111 above can be
315 created. This includes when "11" in the base is a Fibonacci number or
316 Lucas number, such as base 12 so "11" base 12 is 13. If "11" is near
317 the negative X axis then there's two spiral arms, one going out on the
318 X negative side and one X positive, eg. base 16 has 0x11=17 which is
319 near the negative X axis. A four-arm shape can be formed similarly if
320 "11" is near the Y axis, eg. base 107.
321
323 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
324 classes.
325 "$path = Math::PlanePath::VogelFloret->new ()"
327 "$path = Math::PlanePath::VogelFloret->new (key => value, ...)"
328 Create and return a new path object.
329 The default is Vogel's phi floret. Optional parameters can vary
331 the pattern,
332 rotation_type => string, choices above
334 rotation_factor => number
335 radius_factor => number
336 The available "rotation_type" values are listed above (see "Other
338 Rotation Types"). "radius_factor" can be given together with
339 "rotation_type" to have its rotation, but scale the radius
340 differently.
341 If a "rotation_factor" is given then the default "radius_factor" is
343 not specified yet. Currently it's 1.0, but perhaps something
344 suiting at least the first few N positions would be better.
345 "($x,$y) = $path->n_to_xy ($n)"
347 Return the X,Y coordinates of point number $n on the path.
348 $n can be any value "$n >= 0" and fractions give positions on the
350 spiral in between the integer points, though the principle interest
351 for the floret is where the integers fall.
352 For "$n < 0" the return is an empty list, it being considered there
354 are no negative points in the spiral.
355 "$rsquared = $path->n_to_rsquared ($n)"
357 Return the radial distance R^2 of point $n, or "undef" if there's
358 no point $n. As per the formulas above this is simply
359 $n * $radius_factor**2
361 "$n = $path->xy_to_n ($x,$y)"
363 Return an integer point number for coordinates "$x,$y". Each
364 integer N is considered the centre of a circle of diameter 1 and an
365 "$x,$y" within that circle returns N.
366 The builtin "rotation_type" choices are scaled so no two points are
368 closer than 1 apart so the circles don't overlap, but they also
369 don't cover the plane and if "$x,$y" is not within one of those
370 circles then the return is "undef".
371 With "rotation_factor" and "radius_factor" parameters it's possible
373 for unit circles to overlap. In the current code the return is the
374 largest N covering "$x,$y", but perhaps that will change.
375 "$str = $path->figure ()"
377 Return "circle".
378
380 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
381 this path include
382 <http://oeis.org/A000201> (etc)
384 A000201 spectrum of phi, N in first 0.618 of circle
386 A003849 Fibonacci word, values 0,1
387
389 Math::PlanePath, Math::PlanePath::SacksSpiral,
390 Math::PlanePath::TheodorusSpiral
391 Math::NumSeq::FibonacciWord, Math::NumSeq::Fibbinary
393
395 <http://user42.tuxfamily.org/math-planepath/index.html>
396
398 Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
399 2020 Kevin Ryde
400 This file is part of Math-PlanePath.
402 Math-PlanePath is free software; you can redistribute it and/or modify
404 it under the terms of the GNU General Public License as published by
405 the Free Software Foundation; either version 3, or (at your option) any
406 later version.
407 Math-PlanePath is distributed in the hope that it will be useful, but
409 WITHOUT ANY WARRANTY; without even the implied warranty of
410 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
411 General Public License for more details.
412 You should have received a copy of the GNU General Public License along
414 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
415perl v5.32.1 2021-01-27 Math::PlanePath::VogelFloret(3)