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The curve is named for the two base shapes

         Beta                     Omega

           *---*                  *---*
           |   |                  |   |
         --*   *                --*   *--
               |

The beta is made from three betas and an omega sub-parts.  The omega is made
from four betas.  In each case the sub-parts are suitably rotated,
transposed or reversed, so expanding to

    Beta = 3*Beta+Omega      Omega = 4*Beta

      *---*---*---*            *---*---*---*
      |           |            |           |
      *---*   *---*            *---*   *---*
          |   |                    |   |
    --*   *   *---*          --*   *   *   *--
      |   |       |            |   |   |   |
      *---*   *---*            *---*   *---*
              |

The sub-parts represent successive ever-smaller substitutions.  They have
the effect of making the start a beta going alternately up or down.  For
this integer version the start direction is kept fixed as a beta going
upwards and the higher levels then alternate up and down from there.

=head2 Level Ranges

Reckoning the initial N=0 to N=3 as level 1, a replication level extends to

    Nlevel = 4^level - 1
    Xmin = 0
    Xmax = 2^level - 1

    Ymin = - (4^floor(level/2) - 1) * 2 / 3
         = binary 1010...10
    Ymax = (4^ceil(level/2) - 1) / 3
         = binary 10101...01

    height = Ymax - Ymin = 2^level - 1

The Y range increases alternately above and below by a power of 2, so the
result for Ymin and Ymax is a 1 bit going alternately to Ymax and Ymin,
starting with Ymax for level 1.

    level     Ymin    binary       Ymax   binary
    -----     --------------       -------------
      0         0                    0
      1         0          0         1 =       1
      2        -2 =      -10         1 =      01
      3        -2 =     -010         5 =     101
      4       -10 =    -1010         5 =    0101
      5       -10 =   -01010        21 =   10101
      6       -42 =  -101010        21 =  010101
      7       -42 = -0101010        85 = 1010101

The power of 4 divided by 3 formulas above for Ymin/Ymax have the effect of
producing alternating bit patterns like this.

For odd levels -Ymin/height approaches 1/3 and Ymax/height approaches 2/3,
ie. the start point is about 1/3 up the total extent.  For even levels it's
the other way around, with -Ymin/height approaching 2/3 and Ymax/height
approaching 1/3.

=head2 Closed Curve

Wierum's idea for the curve is a closed square made from four betas,

    *---*      *---*
    |   |      |   |
    *   *--  --*   *
    |              |

    |              |
    *   *--  --*   *
    |   |      |   |
    *---*      *---*

And at the next expansion level

    *---*---*---*       *---*---*---*
    |           |       |           |
    *---*   *---*       *---*   *---*
        |   |               |   |
    *---*   *   *--   --*   *   *---*
    |       |   |       |   |       |
    *---*   *---*       *---*   *---*
        |                       |

        |                       |
    *---*   *---*       *---*   *---*
    |       |   |       |   |       |
    *---*   *   *--   --*   *   *---*
        |   |               |   |
    *---*   *---*       *---*   *---*
    |           |       |           |
    *---*---*---*       *---*---*---*

The code here could be used for that by choosing a level and applying four
copies of the path suitably mirrored and offset in X and Y.

For an odd level, the path N=0 to N=4^level-1 here is the top-right quarter,
entering on the left and exiting downwards.  For an even level it's the
bottom-right shape instead, exiting upwards.  The difference arises because
when taking successively greater detail sub-parts the initial direction
alternates up or down, but in the code here it's kept fixed (as noted
above).

The start point here is also fixed at Y=0, so an offset Ymin must be applied
if say the centre of the sections is to be Y=0 instead of the side entry
point.

=head1 FUNCTIONS

See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.

=over 4