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Triangular periodic tilings

Mon, 2008-11-24 10:05 — Mark Dow

Mark Dow

Geek art

Simple recursive systems and fractal patterns

Triangular periodic tilings

    By dividing the edges of a set of triangles such that there is always a matching division, regular triangular tilings result in nice (high continuity) periodic spatial patterns. These are a few examples of the simplest (smallest prime integer) divisions.

 2:2:3 tilings, link to 2:2:3 tilings, link to Mixed_symmetries tesselation, link to

2:2:3 tilings

2008-11-12 While writing the one-dimensional code for a stillanimation component (for a pinwheel "wedge" edge, EC_radial.m), this high symmetry "cone" structure came up and I thought about how it tiles:

checkerboard twist

   An overlapping square tiling with no gap, smooth square-profile "teeth" and matching the pattern on the diagonal and vertical boundaries. The result is a triangular tiling with edge matching that corresponds with various periodic colorings.

    What are the matching conditions, which correspond with periodic triangular colorings?


2:2:3 tilings

    The simplest of this class of motifs has boundaries that split two edges into two equal parts (a boundary bisects the edges) and two boundaries split the third into three parts (a bilaterally symmetric split, but not necessarily equal divisions). A simple motif of this type uses straight segments as boundaries:
 
2:2:3 motif a, link to
 2:2:3 motif b, link to  
Base motif ...and its color inverse  

    Periodic tilings with matching (color) boundaries:
2:2:3 stipe tiling, link to
 2:2:3 star tiling, link to 2:2:3 star with stripe surroung tiling, link to
Periodic stripe tiling, three possible orientations. Star tiling, using just one prototile Star with stripe surround tiling

2:2:3 key frieze 1 tiling, link to
 2:2:3 key frieze 2 tiling, link to 2:2:3 radial tiling, link to
Key frieze tiling, two unique column offsets.  Key frieze tiling, two unique column offsets.  Radial tiling (concentric triangles). Not strictly periodic, it is a  piece-wise periodic trisection of plane.

     And a few notable mixed matching and anti-matching tilings:
2:2:3 checker tiling, link to
 2:2:3 key frieze 1 tiling, link to 2:2:3 star alternating tiling, link to
Checker tiling A variation on the key frieze above. Star alternating tiling, with every boundary anti-matched.

Mixed symmetries

Mixed symmetries

    This is a 5:3 division example, that is periodic in the third dimension (time in the animation below). It is a 3-color tiling of triangular or diamond prisms. The pattern is particularly simple, but each frame consists of curved boundaries and continuities that mask the simple structure.

Mixed symmetries animation, link to
Mixed_symmetries.swf   1 MB
 Mixed symmetries frame, link to
Full resolution example frame
 

MATLAB command to generate a single triangular element (see header and hardcoding to generate the three required elements):
>> EC_radial_MS( 256, 512, 1.0 );
See Mixed symmetries notes and programs and code for construction and program details.


Programs and code

    MATLAB code used to generate the Mixed symmetries motifs.  See Mixed symmetries notes for construction and program details:
EC_radial_MS.m
    ->Edge_contrast_1D.m
    ->volume_merge_logical_MSBackground.m
    ->write_Space_volume.m

Space Software used to rotate and mirror 3-D motifs, and to assemble tilings.


    Each x/t plane of Mixed symmetries can be generated with:

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There are no restrictions on use of the images/animations on this page. Claiming to be the originator of the material, explicitly or implicitly, is bad karma. A link (if appropriate), a note to dow[at]uoregon.edu, and credit are appreciated but not required.

Comments are welcome (dow[at]uoregon.edu).
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Comments

Tue, 2008-11-25 12:52 — PSR

Plane singularities

Another excellelent bit of work Mark. The animation nicely conveys the perception of a flat plane extending outward and twisting on itselfs which evokes the idea -at least for me- of a space-time wormhole.

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