Math::PlanePath::PentSpiralSkewed3pm

Usage

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

Overview

This path makes a pentagonal (five-sided) spiral with points skewed so as to fit a square grid and fully cover the plane.

          10 ...             2
         /  \  \
       11  3  9 20           1
      /  /  \  \  \
    12  4  1--2  8 19    <- Y=0
      \  \       |  |
       13  5--6--7 18       -1
         \          |
          14-15-16-17       -2
     ^  ^  ^  ^  ^  ^
    -2 -1 X=0 1  2  3 ...

The pattern is similar to the "SquareSpiral" but cuts three corners which makes each cycle is faster. Each cycle is just 5 steps longer than the previous (where it's 8 for a "SquareSpiral").

                38             n_start => 0
             39 21 37  ...
          40 22  9 20 36 57
       41 23 10  2  8 19 35 56
    42 24 11  3  0  1  7 18 34 55
       43 25 12  4  5  6 17 33 54
          44 26 13 14 15 16 32 53
             45 27 28 29 30 31 52
                46 47 48 49 50 51

Functions

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

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

"$path = Math::PlanePath::PentSpiral->new (n_start => $n)"

Create and return a new path object.

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

Return the point number for coordinates "$x,$y". $x and $y are each rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1.

Oeis

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

    n_start=1 (the default)
      A192136    N on X axis, (5*n^2 - 3*n + 2)/2
      A140066    N on Y axis
      A116668    N on X negative axis, (5n^2 + n + 2)/2
      A134238    N on Y negative axis
      A158187    N on North-West diagonal, 10*n^2 + 1
      A005891    N on South-East diagonal, centred pentagonals
    n_start=0
      A000566    N on X axis, heptagonal numbers
      A005476    N on Y axis
      A005475    N on X negative axis
      A147875    N on Y negative axis, second heptagonals
      A033583    N on North-West diagonal, 10*n^2
      A028895    N on South-East diagonal, 5*triangular

Home page

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