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 (2-D) 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" (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:
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.
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
|
|
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.
|
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);
 |
...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. |
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 |
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. |
 |
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:
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
|
|
 |
-----------------------------------
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Domain.
There
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Comments are welcome
(dow[at]uoregon.edu).
