Math::PlanePath::WythoffArray(3pm)

NAME

Math::PlanePath::WythoffArray -- table of Fibonacci recurrences

SYNOPSIS

 use Math::PlanePath::WythoffArray;
 my $path = Math::PlanePath::WythoffArray->new;
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path is the Wythoff array by David R. Morrison

"A Stolarsky Array of Wythoff Pairs", in Collection of Manuscripts Related to the Fibonacci Sequence, pages 134 to 136, The Fibonacci Association, 1980. <http://www.math.ucsb.edu/~drm/papers/stolarsky.pdf>

It's an array of Fibonacci recurrences which positions each N according to Zeckendorf base trailing zeros.

     15  |  40   65  105  170  275  445  720 1165 1885 3050 4935
     14  |  38   62  100  162  262  424  686 1110 1796 2906 4702
     13  |  35   57   92  149  241  390  631 1021 1652 2673 4325
     12  |  33   54   87  141  228  369  597  966 1563 2529 4092
     11  |  30   49   79  128  207  335  542  877 1419 2296 3715
     10  |  27   44   71  115  186  301  487  788 1275 2063 3338
      9  |  25   41   66  107  173  280  453  733 1186 1919 3105
      8  |  22   36   58   94  152  246  398  644 1042 1686 2728
      7  |  19   31   50   81  131  212  343  555  898 1453 2351
      6  |  17   28   45   73  118  191  309  500  809 1309 2118
      5  |  14   23   37   60   97  157  254  411  665 1076 1741
      4  |  12   20   32   52   84  136  220  356  576  932 1508
      3  |   9   15   24   39   63  102  165  267  432  699 1131
      2  |   6   10   16   26   42   68  110  178  288  466  754
      1  |   4    7   11   18   29   47   76  123  199  322  521
    Y=0  |   1    2    3    5    8   13   21   34   55   89  144
         +-------------------------------------------------------
           X=0    1    2    3    4    5    6    7    8    9   10

All rows have the Fibonacci style recurrence

    W(X+1) = W(X) + W(X-1)
    eg. X=4,Y=2 is N=42=16+26, sum of the two values to its left

X axis N=1,2,3,5,8,etc is the Fibonacci numbers. The row Y=1 above them N=4,7,11,18,etc is the Lucas numbers.

Y axis N=1,4,6,9,12,etc is the "spectrum" of the golden ratio, meaning its multiples rounded down to an integer.

    phi = (sqrt(5)+1)/2
    spectrum(k) = floor(phi*k)
    N on Y axis = Y + spectrum(Y+1)
    Eg. Y=5  N=5+floor((5+1)*phi)=14

The recurrence in each row starts as if the row was preceded by two values Y,spectrum(Y+1) which can be thought of adding to be Y+spectrum(Y+1) on the Y axis, then Y+2*spectrum(Y+1) in the X=1 column, etc.

If the first two values in a row have a common factor then that factor remains in all subsequent sums. For example the Y=2 row starts with two even numbers N=6,N=10 so all N values in the row are even.

Every N from 1 upwards occurs precisely once in the table. The recurrence means that in each row N grows roughly as a power phi^X, the same as the Fibonacci numbers. This means they become large quite quickly.

Zeckendorf Base

The N values are arranged according to trailing zero bits when N is represented in the Zeckendorf base. The Zeckendorf base expresses N as a sum of Fibonacci numbers, choosing at each stage the largest possible Fibonacci. For example

    Fibonacci numbers F[0]=1, F[1]=2, F[2]=3, F[3]=5, etc
    45 = 34 + 8 + 3
       = F[7] + F[4] + F[2]
       = 10010100        1-bits at 7,4,2

Here F[] is indexed by bit positions starting 0 for the least signficiant (which would be Fibonacci(2) in the usual Fibonacci indexing).

The Wythoff array written in Zeckendorf base bits is

      8 | 1000001 10000010 100000100 1000001000 10000010000
      7 |  101001  1010010  10100100  101001000  1010010000
      6 |  100101  1001010  10010100  100101000  1001010000
      5 |  100001  1000010  10000100  100001000  1000010000
      4 |   10101   101010   1010100   10101000   101010000
      3 |   10001   100010   1000100   10001000   100010000
      2 |    1001    10010    100100    1001000    10010000
      1 |     101     1010     10100     101000     1010000
    Y=0 |       1       10       100       1000       10000
        +---------------------------------------------------
              X=0        1         2          3           4

The X coordinate is the number of trailing zeros, or equivalently the index of the lowest Fibonacci used in the sum. For example in the X=3 column all the N's there have F[3]=5 as their lowest term.

The Y coordinate is formed by removing the trailing "0100..00", ie. all trailing zeros plus the "01" above them. For example,

    N = 45 = Zeck 10010100
                      ^^^^ strip low zeros and "01" above them
    Y = Zeck(1001) = F[3]+F[0] = 5+1 = 6

The Zeckendorf form never has consecutive "11" bits, because after subtracting an F[k] the remainder is smaller than the next lower F[k-1]. Numbers with no concecutive "11" bits are sometimes called the fibbinary numbers (see Math::NumSeq::Fibbinary).

Stripping low zeros is similar to what the "PowerArray" does with low zero digits in an ordinary base such as binary (see Math::PlanePath::PowerArray). Doing it in the Zeckendorf base is like taking out powers of the golden ratio phi=1.618.

Turn Sequence

The path turns

    straight     at N=2 and N=10
    right        N="...101" in Zeckendorf base
    left         otherwise

For example at N=12 the path turns to the right, since N=13 is on the right hand side of the vector from N=11 to N=12. It's almost 180-degrees around and back, but on the right hand side.

      4  | 12
      3  | 
      2  | 
      1  |       11   
    Y=0  |                13
         +--------------------
          X=0  1  2  3  4  5

This happens because N=12 is Zeckendorf "10101" which ends "..101". For such an ending N-1 is "..100" and N+1 is "..1000". So N+1 has more trailing zeros and hence bigger X smaller Y than N-1 has. The way the curve grows in a "concave" fashion means that therefore N+1 is on the right-hand side.

    | N                        N ending "..101"
    |  
    |                          N+1 bigger X smaller Y
    |      N-1                     than N-1
    |               N+1   
    +--------------------

Cases for N ending "..000", "..010" and "..100" can be worked through to see that everything else turns left (or the initial N=2 and N=10 go straight ahead).

On the Y axis all N values end "..01", with no trailing 0s. As noted above stripping that "01" from N gives the Y coordinate. Those N ending "..101" are therefore at Y coordinates which end "..1", meaning "odd" Y in Zeckendorf base.

X,Y Start

Options "x_start => $x" and "y_start => $y" give a starting position for the array. For example to start at X=1,Y=1

      4  |    9  15  24  39  63         x_start => 1
      3  |    6  10  16  26  42         y_start => 1
      2  |    4   7  11  18  29 
      1  |    1   2   3   5   8 
    Y=0  | 
         +----------------------
         X=0  1   2   3   4   5

This can be helpful to work in rows and columns numbered from 1 instead of from 0. Numbering from X=1,Y=1 corresponds to the array in Morrison's paper above.

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.

"$path = Math::PlanePath::WythoffArray->new ()"

"$path = Math::PlanePath::WythoffArray->new (x_start => $x, y_start => $y)": Create and return a new path object. The default "x_start" and "y_start" are 0.

"($x,$y) = $path->n_to_xy ($n)": Return the X,Y coordinates of point number $n on the path. Points begin at 1 and if "$n < 1" then the return is an empty list.

"$n = $path->xy_to_n ($x,$y)"

Return the N point number at coordinates "$x,$y". If "$x<0" or "$y<0" (or the "x_start" or "y_start" options) then there's no N and the return is "undef".

N values grow rapidly with $x. Pass in a bignum type such as "Math::BigInt" for full precision.

"($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)": The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in the rectangle.

FORMULAS

Rectangle to N Range

Within each row increasing X is increasing N, and in each column increasing Y is increasing N. So in any rectangle the minimum N is in the lower left corner and the maximum N is in the upper right corner.

    |               N max
    |     ----------+
    |    |  ^       |
    |    |  |       |
    |    |   ---->  |
    |    +----------
    |   N min
    +-------------------

OEIS

The Wythoff array is in Sloane's Online Encyclopedia of Integer Sequences in various forms,

<http://oeis.org/A035614> (etc)

    x_start=0,y_start=0 (the defaults)
      A035614     X, column numbered from 0
      A191360     X-Y, the diagonal containing N
      A019586     Y, the row containing N
      A083398     max diagonal X+Y+1 for points 1 to N
    x_start=1,y_start=1
      A035612     X, column numbered from 1
      A003603     Y, vertical para-budding sequence
      A143299     Zeckendorf bit count in row Y
      A185735     left-justified row addition
      A186007     row subtraction
      A173028     row multiples
      A173027     row of n * Fibonacci numbers
      A220249     row of n * Lucas numbers
    A003622     N on Y axis, odd Zeckendorfs "..1"
    A020941     N on X=Y diagonal
    A139764     N dropped down to X axis, ie. N value on the X axis,
                  being lowest Fibonacci used in the Zeckendorf form
    A000045     N on X axis, Fibonacci numbers skipping initial 0,1
    A000204     N on Y=1 row, Lucas numbers skipping initial 1,3
    A001950     N+1 of N on Y axis, anti-spectrum of phi
    A022342     N not on Y axis, even Zeckendorfs "..0"
    A000201     N+1 of N not on Y axis, spectrum of phi
    A003849     bool 1,0 if N on Y axis or not, being the Fibonacci word
    A035336     N in second column
    A160997     total N along anti-diagonals X+Y=k
    A188436     turn 1=right,0=left or straight, skip initial five 0s
    A134860     N positions of right turns, Zeckendorf "..101"
    A003622     Y coordinate of right turns, Zeckendorf "..1"
    A114579     permutation N at transpose Y,X
    A083412     permutation N by Diagonals from Y axis downwards
    A035513     permutation N by Diagonals from X axis upwards
    A064274       inverse permutation

HOME PAGE

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

LICENSE

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.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.