Mark
Dow
Geek art
Simple recursive systems and fractal patterns
Brigid's cross tessellations and patterns
Brigid's octomino and cross
An octomino with a chiral cross symmetry, and its mirror image:
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. ]
A "Brigid's cross, or Saint Brigid's cross" (right) has the same symmetry, but usually refers to a pattern
woven with reeds. The pattern at the center of a woven cross (left) 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.
This shape tessellates the plane:
Brigid's octomino Thue-Morse tessellations
These patterns use Brigid's octominos, along with mirrors and inversions, as motifs in Thue-Morse patterns.
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] );
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] );
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] );
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] );
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] );
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
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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 ] );
Brigid's squares
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.
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These shapes can also be formed by accretion, by adding to every
8-neighbor and add a new "corner". This procedure requires a context sensitive cellular automata.
A cellular automata that generates successive generations from a
Brigid's octomino requires rules that depend on the [ i + 1, j - 1 ]
through [ i + 1, j + 3 ] neighbors, with 3 possible cell states [ Null, a, b ].
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These Brigid's squares are also chiral and can tile (cover without
overlap) the plane. The family is a discrete similarity tiling.
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 in reverse reading order. 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 (crosses indicate
replacement with the result of the previous rule, initialized with the
square at left);
The interior boundaries in the full images are weighted to emphasize the hierarchical construction.
This similarity tiling is a combination of the octomino tiling (above) and Brigid's squares, by composition and accretion. Sequential generations, with different rules, are
shown in reverse reading order.
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.
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 |
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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
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. |
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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:
Non-black symbols from the boundary.
This system was designed by finding a boundary system that bisected a
square. Notice that the light boundary crossed each quadrant from
corner to corner. 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 who's rules are composed of direction and distance changes.
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Comments are welcome (dow[at]uoregon.edu).
