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Kevin Ryde
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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 Vogel floret arranges integer points in a spiral with points based on the golden ratio phi = (1+sqrt(5))/2 and resembling the pattern of seeds found in the head of a sunflower,

                27       19

                14          11
          22                         16
                       6                   29

    30           9           3
                       1                   21
          17              .
       25                 2     5
                    7                      26
             20       15

                            23       31

Most of the PlanePaths are implicitly quadratic, but the VogelFloret is instead based on integer multiples of phi (or other selected rotation factor).

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 and since 1/phi^2 = 2-phi
    theta = 2*pi * angle        in radians

Each point N+1 is at an angle 0.382 counter-clockwise around from the preceding point N, which is 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.

Other Rotation Types

An optional 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 rotation_factor and rotation_factor parameters,

    # 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);

Usually 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 is 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

    N*A = T = pi * R^2
    R = sqrt(N) * sqrt(A/pi)

The tightest possible packing for unit circle figures is a hexagonal honeycomb grid each of area A = sqrt(3)/2 = 0.866, for a factor sqrt(A/pi) = 0.525. The phi floret packing is not as tight as that, needing radius factor 0.624 per above.

Generally the tightness of the packing for a given rotation factor depends on what fractions closely approximate that 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 denominators are ever increasing then there may be no 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 in that sense is, among all irrationals, the value least well approximated by rationals.

    phi = 1 + ------
              1 +  1
              ^   1 +  1
              |       ---
              |   ^   1 +  1
              |   |       ----
              |   |   ^   ...
       terms -+---+---+

sqrt(3) is 1,2 repeating. sqrt(13) is 3s repeating.

Fibonacci and Lucas Numbers

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 calculation F(k)*phi is in fact 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). 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 and since abs(beta)<1 the powers beta^k go to zero.

    F(k) = (phi^k - beta^k) / (phi - beta)

    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))
              approaches 0 as k -> infinity

Basically 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)

Repdigits in Decimal

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 55 and 555 differences apply to between pairs 22 to 77, 33 to 88, 666 to 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 about a quarter turn (9*0.024 = 0.219) upwards.

Repdigits in Other Bases

By choosing a radix so that "11" (or similar repunit) is close to the X axis, spirals like the decimal 11111 above can be created. This includes when "11" is a Fibonacci number or Lucas number, such as base 12 making "11" equal to 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 "11"=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 the 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

The available rotation_type values are listed above (see "Other Rotation Types"). radius_factor can be given on top of a type to scale it differently.

If a rotation_factor is given then the default radius_factor is not yet quite settled. 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.

For $n < 0 the return is an empty list, it being considered there are no negative points in the spiral.

$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.

The builtin 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 undef.

With rotation_factor and 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.


Math::PlanePath, Math::PlanePath::SacksSpiral, Math::PlanePath::TheodorusSpiral




Copyright 2010, 2011 Kevin Ryde

This file is part of Math-PlanePath.

Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

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