Mark Dow

Geek art

Simple recursive systems and fractal patterns

Brigid's cross tessellations and patterns

Brigid's octomino Thue-Morse tesselations link Brigid's cross similarity tilings, link to Brigid's cross similarity tiling thumbnail Brigid's pyramids articulated thumbnail Brigid's pyramid spiral face thumbnail Octomino f-tiling Robert-W-Fathauer link

Brigid's octomino and cross
Brigid's octomino Thue-Morse tessellations
Brigid's octomino rule systems
Brigid's squares
Brigid's cross similarity tiling
Brigid's pyramids
Brigid's number sequences
Fractal dissection of the octomino

Brigid's octomino and cross

    An octomino with a chiral (2-D) cross symmetry, and its mirror image:
Brigid's 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).

Brigids octomino balanced plain weave
A  two color balanced plain weave that tesselates this pattern.
[ To Do: A good quality tiled rendering. Describe as a periodic recursive system. ]

Saint Brigid's cross
center of a Saint Brigid's cross
Brigids cross center pattern
    A "Brigid's cross, or Saint Brigid's cross" (left) has the same symmetry, but usually refers to a pattern woven with reeds. The pattern at the center of a woven cross (right)  is itself an interesting system, a variation of a similarity tiling. The length of the rectangular elements (with no gaps) follow an arithmetic progression (not a geometric progression) while the widths are constant. This "arithmetic similarity" pattern is also chiral and tiles the plane. In the woven version, due to a weaving method that starts with a single straight reed, the central elements' rotational symmetry is broken.

    The Brigid's octomino tessellates the plane:

Brigid's octomino tiling patch Brigid's octomino horizontal tile
Horizontal tile (repeat unit)

Tessellation with this tile
Brigid's octomino diagonal tile
Diagonal tile (repeat unit)

Tessellation with this tile
    This tesselation has the property that all 16 possible 2x2 sub-arrays (of black and white square sub-units) occur with equal frequency. See minimal arrays containing all sub-array combinations  for details.

Brigid's octomino Thue-Morse tessellations

    These patterns use Brigid's octominos, along with mirrors and inversions, as motifs in Thue-Morse patterns.


Brigid's octomino 2-symbol mirrored horizontal motif

Thue-Morse algorithm graphic
Brigid's octomino 2-symbol mirrored horizontal Thue-Morse tessellation
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]  );




Brigid's octomino 2-symbol inverted horizontal motif

Thue-Morse algorithm graphic
Brigid's octomino 2-symbol inverted horizontal Thue-Morse tessellation
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]  );


Thue-Morse algorithm graphic

Thue-Morse algorithm graphic
Brigid's octomino 2-symbol inverted horizontal motif

Thue-Morse algorithm graphic
Brigid's octomino 2-symbol inverted horizontal Thue-Morse tessellation
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]  );




Brigid's octomino 2-symbol inverted and mirrored horizontal motif

Thue-Morse algorithm graphic
Brigid's octomino 2-symbol inverted and mirrored horizontal Thue-Morse tessellation
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]  );





Brigid's octomino 4-symbo inversions and mirrors horizontal motif

Thue-Morse algorithm graphic
4-symbol by generation graphic
Brigid's octomino 2-symbol inversions and mirrors horizontal Thue-Morse tessellation
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]  );





Brigid's octomino 4-symbo inversions and mirrors horizontal motif

4-symbol cyclic algorithm graphic
4-symbol cyclic by generation graphic
Brigid's octomino 2-symbol inversions and mirrors horizontal Thue-Morse tessellation
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]  );

Brigid's octomino rule systems

4x4 2-symbol algorithm graphic
6x6 2-symbol by generation graphic
4x4 2-symbol Brigid's octomino and mirror rules

Integral resample x2 with offset (1,1)
4x4 2-symbol Brigid's octomino and mirror rules
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 ]  );

Brigid's squares

    The octomino can be generalize to other shapes, Brigid's squares, with more sides:
Brigid's cross by composition, generations 2 and 3
By composition, in the same way that squares can be formed by composition from small squares.
Brigid's squares by accretion, generations 2 and 3
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?

Brigid's cross similarity tilings

    Brigid's octomino has a discrete similarity tiling with two alternating rules that differ only by a pi/4 rotation. Sequential generations are shown below. In this example the four "corner" elements are colored differently than the four central elements to show the nested construction. The interior boundaries in the full images are weighted to delineate the hierarchical construction.
    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);



Brigids cross replacement system algorithm graphic    ...or, represented as a recursive
      replacement of  lattice points:
Brigids cross lattice replacement system algorithm graphic

    The interior boundaries in the full images are weighted to emphasize the hierarchical construction.
Brigid's octomino similarity tiling, generation 1 Brigid's octomino similarity tiling, generation 2 Brigid's octomino similarity tiling, generation 3
Brigid's octomino similarity tiling, generation 4 Brigid's octomino similarity tiling, generation 5 Brigid's octomino similarity tiling, generation 6




    The sequential generations of this similarity tiling alternates between the octomino tiling rules (above) and Brigid's squares, by composition and accretion.
Brigid's cross similarity tiling element 7 Brigid's cross similarity tiling element 6 Brigid's cross similarity tiling element 5 Brigid's cross similarity tiling element 4
Brigid's cross similarity tiling element 3 Brigid's cross similarity tiling element 2 Brigid's cross similarity tiling element 1 Brigid's cross similarity tiling



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


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

Brigid's pyramids

    Brigid's squares with unit thickness can be stacked to formed stepped pyramids, with adjacent planes differing by four units in width.

Brigid's pyramid articulated link

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
Brigid's pyramid spiral face link

The geometry of this arrangement and coloring was inspired by Leanord Claggett's "Square Reduction 1".


    With respect to discrete similarity tilings, also see Robert W. Fathauer's Fractal Knots Created by Iterative Substitution. From the abstract:

 "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."

Brigid's number sequences

[To Do: Finish this.]
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:
48 x 48 x 25

Area/volume sequence:
 [ To Do ]


Fractal dissection of the octomino

    Robert W. Fathauer generated and described the image below, a fractal dissection of Brigid's octomino, in Fractal Tilings Based on Dissections of Polyominoes:

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.
Octomino f-tiling by Robert W. Fathauer


Quadratic Koch island

A 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:

Brigid's octomino algorithm graphic
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 
 rule: F -> F+F-F-FF+F+F-F


      
Octomino2x2 7-symbol 9th generation


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