Simple recursive systems and fractal patterns

Brigid's
squares

Brigid's cross similarity tiling

Brigid's pyramids

Brigid's number sequences

Fractal dissection of the octomino

Brigid's cross similarity tiling

Brigid's pyramids

Brigid's number sequences

Fractal dissection of the octomino

I don't know of, or if there is, a standard name for this shape so I refer to it as a Brigid's octomino. It seems likely that various weave and/or quilting communities have named this pattern as its shape is prototypical of these common symmetries (chiral with the 4-fold rotational symmetry of a square).

A two color balanced plain weave that tesselates this pattern.

[ To Do: A good quality tiled rendering. Describe as a periodic recursive system. ]

[ To Do: A good quality tiled rendering. Describe as a periodic recursive system. ]

The Brigid's octomino tessellates the plane:

Horizontal tile (repeat unit) Tessellation with this tile |
Diagonal tile (repeat unit) Tessellation with this tile |

A 2-D Thue-Morse
tessellation with the octomino
horizontal tile and it's mirror as
motifs.

Matlab
command:

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_m_horizontal_motif.png', 0, [0 1; 1 0], [1 0; 0 1] );

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_m_horizontal_motif.png', 0, [0 1; 1 0], [1 0; 0 1] );

A 2-D Thue-Morse tesselation with the octomino horizontal tile and it's
inversion as motifs.

Matlab command:

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_i_horizontal_motif.png', 0, [0 1; 1 0], [1 0; 0 1] );

Matlab command:

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_i_horizontal_motif.png', 0, [0 1; 1 0], [1 0; 0 1] );

Logical difference (modulo 2) of the mirrored and inverted motif
Thue-Morse
tessellations. This is equivalent to the Thue-Morse tesselation of the
logical difference of these motifs (center).

Matlab
command:

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_i-m_horizontal_motif.png', 0, [0 1; 1 0], [1 0; 0 1] );

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_i-m_horizontal_motif.png', 0, [0 1; 1 0], [1 0; 0 1] );

A 2-D Thue-Morse tesselation with the octomino horizontal tile and it's
inversion as motifs.

Matlab command:

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_im_horizontal_motif.png', 0, [0 1; 1 0], [1 0; 0 1] );

Matlab command:

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_im_horizontal_motif.png', 0, [0 1; 1 0], [1 0; 0 1] );

A 4-symbol system tesselation with the octomino horizontal tile and
it's inversions and mirrors as motifs. The columns of rules 0, 2 and 1,
3 are
independent Thue-Morse systems, so there are only two unique rows.

Matlab command:

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_im_horizontal_motif.png', 0, [0 1; 2 3], [1 2; 3 0], [2 3; 0 1], [3 0; 1 2] );

Matlab command:

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_im_horizontal_motif.png', 0, [0 1; 2 3], [1 2; 3 0], [2 3; 0 1], [3 0; 1 2] );

A 4-symbol system tesselation with the octomino horizontal tile and
it's inversions and mirrors as motifs.

Matlab command:

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_im_horizontal_motif.png', 0, [0 1; 3 2], [1 2; 0 3], [2 3; 1 0], [3 0; 2 1] );

Matlab command:

>> L_system_tiling( 'TM', 5, 2, 1, 0, 'Brigids_octomino_2s_im_horizontal_motif.png', 0, [0 1; 3 2], [1 2; 0 3], [2 3; 1 0], [3 0; 2 1] );

Integral resample x2 with offset (1,1) |
Example coloring showing contiguity of integral resample x2. |

A 4-symbol system tesselation with the octomino horizontal tile and
it's inversions and mirrors.

Matlab command:

>> L_system_tiling( 'BC', 5, 2, 1, 0, '', 0, [ 0 1 0 0; 0 1 1 1; 1 1 1 0; 0 0 1 0 ], [ 0 0 1 0; 1 1 1 0; 0 1 1 1; 0 1 0 0 ] );

Matlab command:

>> L_system_tiling( 'BC', 5, 2, 1, 0, '', 0, [ 0 1 0 0; 0 1 1 1; 1 1 1 0; 0 0 1 0 ], [ 0 0 1 0; 1 1 1 0; 0 1 1 1; 0 1 0 0 ] );

The octomino can
be generalize to other shapes, Brigid's squares, with more sides:

By composition, in the same way that squares can be formed by composition from small squares. |
These shapes can also be formed by accretion, adding to every 8-neighbor and add a new "corner". |

These Brigid's squares are also chiral and can tile (cover without overlap) the plane. The family is a discrete similarity tiling, but the algorithm must be equivalent to a context sensitive cellular automata:

successive generations requires rules that depend on the [ i + 1, j - 1 ] through [ i + 1, j + 3 ] neighbors, with 3 possible cell states [Null, a, b]. Compare this with the Brigid's cross similarity tiling below, which is context free (a D0L-system).

Is there a 3-D generalization, "Brigid's cubes", and can they tile space?

The system description that results in this fractal can be summarized by the replacement rules (squares are replaced with octominos in this alternating spatial pattern, initialized with the square at left);

...or, represented as a recursive replacement of lattice points: |

The interior
boundaries in the full images are weighted to emphasize the
hierarchical construction.

The sequential generations of this similarity tiling alternates between the octomino tiling rules (above) and Brigid's squares, by composition and accretion.

From A Gallery of L-Systems, a recursive edge replacement system:

Using
the turtle
graphics L-system, whos
rules are composed of direction and position changes:

```
axiom:
F+F+F+F
```

rule: F -> FF+F+F+F+F+F-F

An axiom-rule representation. |
Fixed point of the system's evolution. |

Composite of renderings of the thirteenth generation pyramids (in Space Software volume format): Brigids_alt_a_13_half_x12.vol.gz Mag. = 2, off window Smoothing scale = 1.0 theta = +- 30, phi = +- 30 no shadow, no fade with depth, no back surface lighting: -.7, -1.0, 1.0 Specularity = 4 |
The geometry of this arrangement and coloring was inspired by Leanord Claggett's "Square Reduction 1". |

"A
widely-applicable method for
iterating
knots is described. This method relies on substitution of portions of a
knot with smaller copies of the entire knot. A starting knot is first
arranged as a patch of tiles that contains individual tiles similar in
shape to the overall patch. Iterative substitution leads to the
creation of complex knots that are often esthetically pleasing,
particularly for knots possessing a high degree of symmetry. The
iteration process is designed to allow repetition ad infinitum; i.e.,
an infinite number of iterations leads to a unicursal fractal that is,
therefore, a (wild) knot."

Pyramid
width sequence, ( w x w x h,
tip-to-tip):

4
x 4 x 1

8 x 8 x 3

12 x 12 x 5

...

= ( 4n ) x ( 4n ) x ( 2n + 1 )

n = 12:

8 x 8 x 3

12 x 12 x 5

...

= ( 4n ) x ( 4n ) x ( 2n + 1 )

n = 12:

48
x 48 x 25

Area/volume
sequence:

[
To Do ]

From the
abstract:
"Polyominoes, shapes made up of squares connected edge-to-edge, provide
a rich source of prototiles for edge-to-edge fractal tilings. We give
examples of fractal tilings with 2-fold and 4-fold rotational symmetry
based on prototiles derived by dissecting polyominoes with 2-fold and 4-fold rotational symmetry, respectively. A systematic analysis is made of candidate prototiles based on lower-order polyominoes. In some of these fractal tilings, polyomino-shaped holes occur repeatedly with each new generation. We also give an example of a fractal knot created by marking such tiles with Celtic-knot-like graphics. |

## Quadratic Koch islandA boundary similar to Fathauer's f-tiling based on an octomino (see above) above can be formed from a recursive replacement system. The 7-symbol system can be describe by these replacement rules:Non-black symbols form the boundary. This system was designed by finding a boundary system that bisected a square. Notice that the light boundary crosses each quadrant from corner to corner. [To Do: Illustrate.] The quadrants that are diagonally opposite each other are identical; note the symmetry of the initial condition. See Fractals and graphic interpretation of strings for an equivalent specification of this boundary in terms of a turtle graphics L-system whos rules are composed of direction and position changes: axiom: F+F+F+F |

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