Golomb Rep-Tiles and Fractals
Year: 2003 Authors: Imameddin Amiraslanov
Core claim
Repeated fragments from Golomb rep-tiles can be colored and arranged into complementary nonperiodic ornaments that exhibit fractal structure.
Topics
rep-tiles, nonperiodic tiling, fractal ornament, plane coverage
Domains
geometry, tiling theory, fractals, self-similarity, ornament, decorative pattern, church and cathedral decoration
Methods
fragment repetition, coloring, visual inspection, plane tiling
Media
tile fragments, colored figures, expanded view illustration
Paper text
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ISAMA The International Society of the Arts, Mathematics, and Architecture BRIDGES Mathematical Connections in Art, Music, and Science
Golomb Rep-Tiles and Fractals
Imameddin Amiraslanov Institute of Inorganic and Physical Chemistry National Academy of Science, 370143 Husein Javid avenue 29, Baku, AZERBAIJAN Email: imam@gate.sinica.edu.tw
In 1964 Solomon W. Golomb suggested an unusual type of tile: nonperiodic rep-tiles. Unlike other kinds of tiling, rep-tiles one obtained by grouping individual tiles together to form larger replicas of themselves. One of the Golomb rep-tiles, nameli Rep-4 (L-triomino) is shown in Figure.1(a,b) [1,2].
Figure 1 a
b
c
The multiple repetitive nature of the fragments, shown in Figure.1b, can be used to cover a plane completely, without leaving gaps or overlapping (Figure.2a). Through coloring the individual fragments shown in Figure.1c one can distinguish more clearly the organization of the system. As a result two kinds of mutually complementary and cross-tree like nonperiodic ornaments are obtained (Figure.2b). By careful examination of the final picture it is easy to see the fractal character of both kinds of cross-tree.
Figure 2 a
b
575
Figure.3 an expanded view of the image shown in Figure.2b.
Figure 3
This ornament can have many applications, such as in decorating churches and cathedrals.
References
- Golomb, Solomon W. “Replicating Figures in the Plane.” Mathematical Gazette 48 (December 1964): 403-12.
- Gardner, Martin. The Unexpected Hanging and Other Mathematical Diversions. Chicago: University of Chicago Press, 1991.