 Math::PlanePath::QuadricCurve(3) eight segment zig-zag

## SYNOPSIS

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

## DESCRIPTION

This is a self-similar zig-zag of eight segments,

```                  18-19                                       5
|  |
16-17 20 23-24                                 4
|     |  |  |
15-14 21-22 25-26                              3
|           |
11-12-13    29-28-27                              2
|           |
2--3 10--9       30-31             58-59    ...        1
|  |     |           |              |  |     |
0--1  4  7--8          32          56-57 60 63-64     <- Y=0
|  |              |           |     |  |
5--6             33-34       55-54 61-62           -1
|           |
37-36-35    51-52-53                 -2
|           |
38-39 42-43 50-49                    -3
|  |  |     |
40-41 44 47-48                    -4
|  |
45-46                       -5
^
X=0 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
```

The base figure is the initial N=0 to N=8,

```          2---3
|   |
0---1   4   7---8
|   |
5---6
```

It then repeats, turned to follow edge directions, so N=8 to N=16 is the same shape going upwards, then N=16 to N=24 across, N=24 to N=32 downwards, etc.

The result is the base at ever greater scale extending to the right and with wiggly lines making up the segments. The wiggles don't overlap.

The name "QuadricCurve" here is a slight mistake. Mandelbrot (``Fractal Geometry of Nature'' 1982 page 50) calls any islands initiated from a square ``quadric'', only one of which is with sides by this eight segment expansion. This curve expansion also appears (unnamed) in Mandelbrot's ``How Long is the Coast of Britain'', 1967.

## Level Ranges

A given replication extends to

```    Nlevel = 8^level
X = 4^level
Y = 0
Ymax = 4^0 + 4^1 + ... + 4^level   # 11...11 in base 4
= (4^(level+1) - 1) / 3
Ymin = - Ymax
```

## Turn

The sequence of turns made by the curve is straightforward. In the base 8 (octal) representation of N, the lowest non-zero digit gives the turn

```   low digit   turn (degrees)
---------   --------------
1            +90
2            -90
3            -90
4              0
5            +90
6            +90
7            -90
```

When the least significant digit is non-zero it determines the turn, to make the base N=0 to N=8 shape. When the low digit is zero it's instead the next level up, the N=0,8,16,24,etc shape which is in control, applying a turn for the subsequent base part. So for example at N=16 = 20 octal 20 is a turn -90 degrees.

## FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"\$path = Math::PlanePath::QuadricCurve->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.

## Level Methods

"(\$n_lo, \$n_hi) = \$path->level_to_n_range(\$level)"
Return "(0, 8**\$level)".

## OEIS

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

<http://oeis.org/A133851> (etc)

```    A133851    Y at N=2^k, being successive powers 2^j at k=1mod4
```

## HOME PAGE

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

## LICENSE

Copyright 2011, 2012, 2013, 2014, 2015, 2016 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/>.