Enzymind Labs

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 "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:

Horizontal tile (repeat unit) Tessellation with this tile

Diagonal tile (repeat unit) Tessellation with this tile

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 rule systems

Integral resample x2 with offset (1,1)

Example coloring showing contiguity of integral resample x2.

Brigid's squares

By composition, in the same way that squares can be formed by composition from small squares.

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 ].

    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);



    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

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

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

Brigid's number sequences

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.

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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