++ed by:
EGOR

1 PAUSE user

Kevin Ryde

NAME

Math::PlanePath::MathImagePyramidReplicate -- replicating squares

SYNOPSIS

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

DESCRIPTION

This is a self-similar replicating pyramid shape made from 4 points each,

                                               4

                                               3

                                               2

                                               1

                                           <- Y=0

                                              -1

                                              -2

                                              -3

                                              -4

                     ^
    -4  -3  -2  -1  X=0  1   2   3   4

The base shape is the initial N=0 to N=8 section,

      +---+
      | 2 |
  +---+---+---+
  | 3 | 0 | 1 |
  +---+---+---+

It then repeats inverted to make a similar shape but upside-down,

        +---+---+---+---+---+---+---+
        | 5   4   7 | 2 |13  12  15 |
        +---+   +---+   +---+   +---+
            | 6 | 3   0   1 |14 |
            +---+---+---+---+---+
                | 9   8  11 |
                +---+   +---+
                    |10 |
                    +---+

Level Ranges

A given replication extends to ...

    Nlevel = 4^level - 1
    - ... <= X <= ...
    - ... <= Y <= ...

Complex Base

This pattern corresponds to expressing a complex integer X+i*Y in base b=...

    X+Yi = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]

using complex digits a[i] encoded in N in integer base 4 ...

    a[i] digit     N digit
    ----------     -------
         0            0
         1            1
         i            2
        -1            3

FUNCTIONS

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

$path = Math::PlanePath::MathImagePyramidReplicate->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 0 and if $n < 0 then the return is an empty list.

SEE ALSO

Math::PlanePath, Math::PlanePath::CornerReplicate, Math::PlanePath::SquareReplicate, Math::PlanePath::LTiling, Math::PlanePath::GosperReplicate, Math::PlanePath::QuintetReplicate