Reference

Math::PlanePath::PowerArray(3pm)

NAME

Math::PlanePath::PowerArray -- array by powers

SYNOPSIS

 use Math::PlanePath::PowerArray;
 my $path = Math::PlanePath::PowerArray->new (radix => 2);
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This is a split of N into an odd part and power of 2,

     12  |   25    50   100   200   400   800  1600  3200  6400
     11  |   23    46    92   184   368   736  1472  2944  5888
     10  |   21    42    84   168   336   672  1344  2688  5376
      9  |   19    38    76   152   304   608  1216  2432  4864
      8  |   17    34    68   136   272   544  1088  2176  4352
      7  |   15    30    60   120   240   480   960  1920  3840
      6  |   13    26    52   104   208   416   832  1664  3328
      5  |   11    22    44    88   176   352   704  1408  2816
      4  |    9    18    36    72   144   288   576  1152  2304
      3  |    7    14    28    56   112   224   448   896  1792
      2  |    5    10    20    40    80   160   320   640  1280
      1  |    3     6    12    24    48    96   192   384   768
    Y=0  |    1     2     4     8    16    32    64   128   256
         +------------------------------------------------------
            X=0     1     2     3     4     5     6     7     8

For N=odd*2^k, the coordinates are X=k, Y=(odd-1)/2. The X coordinate is how many factors of 2 can be divided out. The Y coordinate counts odd integers 1,3,5,7,etc as 0,1,2,3,etc. This is clearer by writing N values in binary,

    N values in binary
      6  |  1101     11010    110100   1101000  11010000 110100000
      5  |  1011     10110    101100   1011000  10110000 101100000
      4  |  1001     10010    100100   1001000  10010000 100100000
      3  |   111      1110     11100    111000   1110000  11100000
      2  |   101      1010     10100    101000   1010000  10100000
      1  |    11       110      1100     11000    110000   1100000
    Y=0  |     1        10       100      1000     10000    100000
         +---------------------------------------------------------
             X=0         1         2         3         4         5

Column X=0 is all the odd numbers, column X=1 is exactly one low 0-bit, and so on.

Radix

The "radix" parameter can do the same dividing out in a higher base. For example radix 3 divides out factors of 3,

     radix => 3
      8  |   13    39   117   351  1053  3159  9477 28431
      7  |   11    33    99   297   891  2673  8019 24057
      6  |   10    30    90   270   810  2430  7290 21870
      5  |    8    24    72   216   648  1944  5832 17496
      4  |    7    21    63   189   567  1701  5103 15309
      3  |    5    15    45   135   405  1215  3645 10935
      2  |    4    12    36   108   324   972  2916  8748
      1  |    2     6    18    54   162   486  1458  4374
    Y=0  |    1     3     9    27    81   243   729  2187
         +------------------------------------------------
            X=0     1     2     3     4     5     6     7

N=1,3,9,27,etc on the X axis is the powers of 3.

N=1,2,4,5,7,etc on the Y axis is the integers N = 1or2 mod 3, ie. those not a multiple of 3. Notice if Y = 1or2 mod 4 then the N values in that row are all even, or if Y = 0or3 mod 4 then the N values are all odd.

    radix => 3,  N values in ternary
      6  |   101     1010    10100   101000  1010000 10100000
      5  |    22      220     2200    22000   220000  2200000
      4  |    21      210     2100    21000   210000  2100000
      3  |    12      120     1200    12000   120000  1200000
      2  |    11      110     1100    11000   110000  1100000
      1  |     2       20      200     2000    20000   200000
    Y=0  |     1       10      100     1000    10000   100000
         +----------------------------------------------------
             X=0        1        2        3        4        5

Boundary Length

The points N=1 to N=2^k-1 inclusive have a boundary length

    boundary = 2^k + 2k   = 4,8,14,24,42,76,...   (OEIS A100314)

For example N=1 to N=7 is

    +---+
    | 7 |
    +   +
    | 5 |
    +   +---+
    | 3   6 |
    +       +---+
    | 1   2   4 |
    +---+---+---+

The height is the odd numbers, so 2^(k-1). The width is the power k. So total boundary 2*height+2*width = 2^k + 2k.

If N=2^k is included then it's on the X axis and so add 2 for boundary = 2^k + 2k + 2 (OEIS 2*A052968).

For another radix the calculation is similar

    boundary = 2 * (radix-1) * radix^(k-1) + 2*k

For example radix=3, N=1 to N=8 is

    8
    7
    5
    4
    2  6
    1  3

The height is the non-multiples of the radix, so (radix-1)/radix * radix^k. The width is the power k. Total boundary = 2*height+2*width.

FUNCTIONS

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

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

Create and return a new path object.

"($x,$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 < 0" 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" then there's no N and the return is "undef".

N values grow rapidly with $x. Pass in a number type such as "Math::BigInt" to preserve 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. Within each column, increasing Y is increasing N. So in a rectangle the lower left corner is the minimum N and the upper right is the maximum N.

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

N to Turn Left or Right

The turn left or right is given by

    radix = 2     left at N==0 mod radix and N==1mod4, right otherwise
    radix >= 3    left at N==0 mod radix
                  right at N=1 or radix-1 mod radix
                  straight otherwise

The points N!=0 mod radix are on the Y axis and those N==0 mod radix are off the axis. For that reason the turn at N==0 mod radix is to the left,

    |
    C--
       ---
    A--__ --        point B is N=0 mod radix,
    |    --- B      turn left A-B-C is left

For radix>=3 the turns at A and C are to the right, since the point before A and after C is also on the Y axis. For radix>=4 there's of run of points on the Y axis which are straight.

For radix=2 the "B" case N=0 mod 2 applies, but for the A,C points in between the turn alternates left or right.

    1--     N=1 mod 4             3--      N=3 mod 4
     \ --   turn left              \ --    turn right
      \  --                         \  --
       2   --                        2   --
             --                            --
               --                            --
                 0                             4

Points N=2 mod 4 are X=1 and Y=N/2 whereas N=0 mod 4 has 2 or more trailing 0 bits so X>1 and Y<N/2.

    N mod 4      turn
    -------     ------
       0        left         for radix=2
       1         left
       2        left
       3         right

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A007814> (etc)

    radix=2
      A007814    X coordinate, count low 0-bits of N
      A006519    2^X
      A025480    Y coordinate of N-1, ie. seq starts from N=0
      A003602    Y+1, being k for which N=(2k-1)*2^m
      A153733    2*Y of N-1, strip low 1 bits
      A000265    2*Y+1, strip low 0 bits
      A094267    dX, change count low 0-bits
      A050603    abs(dX)
      A108715    dY, change in Y coordinate
      A000079    N on X axis, powers 2^X
      A057716    N not on X axis, the non-powers-of-2
      A005408    N on Y axis (X=0), the odd numbers
      A003159    N in X=even columns, even trailing 0 bits
      A036554    N in X=odd columns
      A014480    N on X=Y diagonal, (2n+1)*2^n
      A118417    N on X=Y+1 diagonal, (2n-1)*2^n
                   (just below X=Y diagonal)
      A054582    permutation N by diagonals, upwards
      A209268      inverse
      A135764    permutation N by diagonals, downwards
      A249725      inverse
      A075300    permutation N-1 by diagonals, upwards
      A117303    permutation N at transpose X,Y
      A100314    boundary length for N=1 to N=2^k-1 inclusive
                   being  2^k+2k
      A131831      same, after initial 1
      A052968    half boundary length N=1 to N=2^k inclusive
                   being  2^(k-1)+k+1
    radix=3
      A007949    X coordinate, power-of-3 dividing N
      A000244    N on X axis, powers 3^X
      A001651    N on Y axis (X=0), not divisible by 3
      A016051    N on Y column X=1
      A051063    N on Y column X=1
      A007417    N in X=even columns, even trailing 0 digits
      A145204    N in X=odd columns (extra initial 0)
      A141396    permutation, N by diagonals down from Y axis
      A191449    permutation, N by diagonals up from X axis
      A135765    odd N by diagonals, deletes the Y=1,2mod4 rows
      A000975    Y at N=2^k, being binary "10101..101"
    radix=4
      A000302    N on X axis, powers 4^X
    radix=5
      A112765    X coordinate, power-of-5 dividing N
      A000351    N on X axis, powers 5^X
    radix=6
      A122841    X coordinate, power-of-6 dividing N
    radix=10
      A011557    N on X axis, powers 10^X
      A067251    N on Y axis, not a multiple of 10
      A151754    Y coordinate of N=2^k, being floor(2^k*9/10)

SEE ALSO

Math::PlanePath, Math::PlanePath::WythoffArray, Math::PlanePath::ZOrderCurve

David M. Bradley "Counting Ordered Pairs", Mathematics Magazine, volume 83, number 4, October 2010, page 302, DOI 10.4169/002557010X528032. <http://www.math.umaine.edu/~bradley/papers/JStor002557010X528032.pdf>

HOME PAGE

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LICENSE

Copyright 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 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.

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

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