Math::PlanePath::VogelFloret -- circular pattern like a sunflower
use Math::PlanePath::VogelFloret; my $path = Math::PlanePath::VogelFloret->new; my ($x, $y) = $path->n_to_xy (123); # other rotations $path = Math::PlanePath::VogelFloret->new (rotation_type => 'sqrt2');
The is an implementation of Helmut Vogel's model for the arrangement of seeds in the head of a sunflower. Integer points are on a spiral at multiples of the golden ratio phi = (1+sqrt(5))/2,
27 19 24 14 11 22 16 6 29 30 9 3 8 1 21 17 . 4 13 25 2 5 12 7 26 10 18 20 15 23 31 28
The polar coordinates for a point N are
R = sqrt(N) * radius_factor angle = N / (phi**2) in revolutions, 1==full circle = N * -phi modulo 1, with since 1/phi^2 = 2-phi theta = 2*pi * angle in radians
Going from point N to N+1 adds an angle 0.382 revolutions around (anti-clockwise, the usual spiralling direction), which means just over 1/3 of a circle. Or equivalently it's -0.618 back (clockwise) which is phi=1.618 ignoring the integer part since that's a full circle -- only the fractional part determines the position.
radius_factor is a scaling 0.6242 designed to put the closest points 1 apart. The closest are N=1 and N=4. See "Packing" below.
rotation_type parameter selects other possible floret forms.
$path = Math::PlanePath::VogelFloret->new (rotation_type => 'sqrt2');
The current types are as follows. The
radius_factor for each keeps points at least 1 apart so unit circles don't overlap.
rotation_type rotation_factor radius_factor "phi" 2-phi = 0.3820 0.624 "sqrt2" sqrt(2) = 0.4142 0.680 "sqrt3" sqrt(3) = 0.7321 0.756 "sqrt5" sqrt(5) = 0.2361 0.853
The "sqrt2" floret is quite similar to phi, but doesn't pack as tightly. Custom rotations can be made with
# R = sqrt(N) * radius_factor # angle = N * rotation_factor in revolutions # theta = 2*pi * angle in radians # $path = Math::PlanePath::VogelFloret->new (rotation_factor => sqrt(37), radius_factor => 2.0);
rotation_factor should be an irrational number. A rational like P/Q merely results in Q many straight lines and doesn't spread the points enough to suit R=sqrt(N). Irrationals which are very close to simple rationals behave that way too. (Of course all floating point values are implicitly rationals, but are fine within the limits of floating point accuracy.)
The "noble numbers" (A+B*phi)/(C+D*phi) with A*D-B*C=1, A<B, C<D behave similar to the basic phi. Their continued fraction expansion begins with some arbitrary values and then becomes a repeating "1" the same as phi. The effect is some spiral arms near the origin then the phi-ness dominating for large N.
Each point is at an increasing distance sqrt(N) from the origin. This sqrt based on how many unit figures will fit within that distance. The area within radius R is
T = pi * R^2 area of circle R
so if N figures each of area A are packed into that space then the radius R is proportional to sqrt(N),
N*A = T = pi * R^2 R = sqrt(N) * sqrt(A/pi)
The tightest possible packing for unit circles is a hexagonal honeycomb grid, each of area A = sqrt(3)/2 = 0.866. That would be factor sqrt(A/pi) = 0.525. The phi floret packing is not as tight as that, needing radius factor 0.624 as described above.
Generally the tightness of the packing depends on the fractions which closely approximate the rotation factor. If the terms of the continued fraction expansion are large then there's large regions of spiral arcs with gaps between. The density in such regions is low and a big radius factor is needed to keep the points apart. If the continued fraction terms are ever increasing then there may be no radius factor big enough to always keep the points a minimum distance apart ... or something like that.
The terms of the continued fraction for phi are all 1 and is therefore, in that sense, among all irrationals, the value least well approximated by rationals.
1 phi = 1 + ------ 1 + 1 ------ ^ 1 + 1 | --- | ^ 1 + 1 | | ---- | | ^ ... terms -+---+---+
sqrt(3) is 1,2 repeating. sqrt(13) is 3s repeating.
The Fibonacci numbers F(k) = 1,1,2,3,5,8,13,21, etc and Lucas number L(k) = 2,1,3,4,7,11,18, etc form almost straight lines on the X axis of the phi floret. This occurs because N*-phi is close to an integer for those N. For example N=13 has angle 13*-phi = -21.0344, the fractional part -0.0344 puts it just below the X axis.
Both F(k) and L(k) grow exponentially (as phi^k) which soon outstrips the sqrt in the R radial distance so they become widely spaced apart along the X axis.
For interest, or for reference, the angle F(k)*phi is in fact roughly the next Fibonacci number F(k+1), per the well-known limit F(k+1)/F(k) -> phi as k->infinity,
angle = F(k)*-phi = -F(k+1) + epsilon
The Lucas numbers similarly with L(k)*phi close to L(k+1). The "epsilon" approaches zero quickly enough in both cases that the resulting Y coordinate approaches zero. This can be calculated as follows, writing
beta = -1/phi =-0.618
Since abs(beta)<1 the powers beta^k go to zero.
F(k) = (phi^k - beta^k) / (phi - beta) # an integer angle = F(k) * -phi = - (phi*phi^k - phi*beta^k) / (phi - beta) = - (phi^(k+1) - beta^(k+1) + beta^(k+1) - phi*beta^k) / (phi - beta) = - F(k+1) - (phi-beta)*beta^k / (phi - beta) = - F(k+1) - beta^k frac(angle) = - beta^k = 1/(-phi)^k
The arc distance away from the X axis at radius R=sqrt(F(k)) is then as follows, simplifying using phi*(-beta)=1 and phi - beta = sqrt(5). The Y coordinate vertical distance is a little less than the arc distance.
arcdist = 2*pi * R * frac(angle) = 2*pi * sqrt((phi^k - beta^k)/sqrt(5)) * 1/(-phi)^k = - (-1)^k * 2*pi * sqrt((1/phi^2k*phi^k - beta^3k)/sqrt(5)) = - (-1)^k * 2*pi * sqrt((1/phi^k - 1/(-phi)^3k)/sqrt(5)) -> 0 as k -> infinity
Essentially the radius increases as phi^(k/2) but the angle frac decreases as (1/phi)^k so their product goes to zero. The (-1)^k in the formula puts the points alternately just above and just below the X axis.
The calculation for the Lucas numbers is very similar, with term +(beta^k) instead of -(beta^k) and an extra factor sqrt(5).
L(k) = phi^k + beta^k angle = L(k) * -phi = -phi*phi^k - phi*beta^k = -phi^(k+1) - beta^(k+1) + beta^(k+1) - phi*beta^k = -L(k) + beta^k * (beta - phi) = -L(k) - sqrt(5) * beta^k frac(angle) = -sqrt(5) * beta^k = -sqrt(5) / (-phi)^k arcdist = 2*pi * R * frac(angle) = 2*pi * sqrt(L(k)) * sqrt(5)*beta^k = 2*pi * sqrt(phi^k + 1/(-phi)^k) * sqrt(5)*beta^k = (-1)*k * 2*pi * sqrt(5) * sqrt((-beta)^2k * phi^k + beta^3k) = (-1)*k * 2*pi * sqrt(5) * sqrt((-beta)^k + beta^3k)
The spectrum of a real number is its multiples, each rounded down to an integer. For example the spectrum of phi is
floor(phi), floor(2*phi), floor(3*phi), floor(4*phi), ... 1, 3, 4, 6, ...
When plotted on the Vogel floret these integers are all in the first 1/phi = 0.618 of the circle.
61 53 69 40 45 58 48 32 71 56 27 37 35 19 24 50 43 14 11 63 64 22 6 16 29 30 3 42 51 9 1 8 21 72 17 4 . 55 38 59 25 12 46 33 67
This occurs because
angle = N * 1/phi^2 = N * (1-1/phi) = N * -1/phi # modulo 1 = floor(int*phi) * -1/phi # N=spectrum = (int*phi - frac) * -1/phi # 0<frac<1 = int + frac*1/phi = frac * 1/phi # modulo 1
So the angle is a fraction from 0 to 1/phi=0.618 of a revolution. In general for a
rotation_factor=t with 0<t<1 the spectrum of 1/t falls within the first 0 to t angle.
The Fibonacci word 0,1,0,0,1,0,1,0,0,1,etc is the least significant bit of the Zeckendorf base representation of i, starting from i=0. Plotted at N=i on the
1 0 1 1 1 0 0 Fibonacci word 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 . 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
This pattern occurs because the Fibonacci word, among its various possible definitions, is 0 or 1 according to whether i+1 occurs in the spectrum of phi (1,3,4,6,8,etc) or not. So for example at i=5 the value is 0 because i+1=6 is in the spectrum of phi, then at i=6 the value is 1 because i+1=7 is not.
The "+1" for i to spectrum has the effect of rotating the spectrum pattern described above by -0.381 (one rotation factor back). So the Fibonacci word "0"s are from angle -0.381 to -0.381+0.618=0.236 and the rest "1"s. 0.236 is close to 1/4, hence the "0"s to "1"s line just before the Y axis.
Some of the decimal repdigits 11, 22, ..., 99, 111, ..., 999, etc make nearly straight radial lines on the phi floret. For example 11, 66, 333, 888 make a line upwards to the right.
11 and 66 are at the same polar angle because the difference is 55 and 55*phi = 88.9919 is nearly an integer meaning the angle is nearly unchanged when added. Similarly 66 to 333 difference 267 has 267*phi = 432.015, or 333 to 888 difference 555 has 555*phi = 898.009. The 55 is a Fibonacci number, the 123 between 99 and 222 is a Lucas number, and 267 = 144+123 = F(12)+L(10).
The differences 55 and 555 apply to pairs 22 and 77, 33 and 88, 666 and 1111, etc, making four straightish arms. 55 and 555 themselves are near the X axis.
A separate spiral arm arises from 11111 falling moderately close to the X axis since 11111*-phi = -17977.9756, or about 0.024 of a circle upwards. The subsequent 22222, 33333, 44444, etc make a little arc of nine values going upwards that much each time for a total about a quarter turn 9*0.024 = 0.219.
By choosing a radix so that "11" (or similar repunit) in that radix is close to the X axis, spirals like the decimal 11111 above can be created. This includes when "11" in the base is a Fibonacci number or Lucas number, such as base 12 so "11" base 12 is 13. If "11" is near the negative X axis then there's two spiral arms, one going out on the X negative side and one X positive, eg. base 16 has 0x11=17 which is near the negative X axis. A four-arm shape can be formed similarly if "11" is near the Y axis, eg. base 107.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::VogelFloret->new ()
$path = Math::PlanePath::VogelFloret->new (key => value, ...)
Create and return a new path object.
The default is Vogel's phi floret. Optional parameters can vary the pattern,
rotation_type => string, choices above rotation_factor => number radius_factor => number
rotation_type values are listed above (see "Other Rotation Types").
radius_factor can be given together with
rotation_type to have its rotation, but scale the radius differently.
rotation_factor is given then the default
radius_factor is not specified yet. Currently it's 1.0, but perhaps something suiting at least the first few N positions would be better.
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number
$n on the path.
$n can be any value
$n >= 0 and fractions give positions on the spiral in between the integer points, though the principle interest for the floret is where the integers fall.
$n < 0 the return is an empty list, it being considered there are no negative points in the spiral.
$rsquared = $path->n_to_rsquared ($n)
Return the radial distance R^2 of point
undef if there's no point
$n. As per the formulas above this is simply
$n * $radius_factor**2
$n = $path->xy_to_n ($x,$y)
Return an integer point number for coordinates
$x,$y. Each integer N is considered the centre of a circle of diameter 1 and an
$x,$y within that circle returns N.
rotation_type choices are scaled so no two points are closer than 1 apart so the circles don't overlap, but they also don't cover the plane and if
$x,$y is not within one of those circles then the return is
radius_factor parameters it's possible for unit circles to overlap. In the current code the return is the largest N covering
$x,$y, but perhaps that will change.
$str = $path->figure ()
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
A000201 spectrum of phi, N in first 0.618 of circle A003849 Fibonacci word, values 0,1
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde
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