From Sierpinski’s Carpet to Fractal Tapestries (Joshua Holden, DMV Mathart 2022)

Описание: Weaving is possibly the earliest artistic medium to divide pictures up into pixels. Many weaving techniques involve a rectangular grid of horizontal and vertical threads, where each intersection can be thought of as a pixel. The color of the pixel is determined by whether the horizontal or the vertical thread is on top. Modern hobbyist weaving looms generally use frames called shafts to select which threads are going to be raised and lowered in the weaving process. Each shaft controls a set of vertical threads, and therefore the color pattern of each column of pixels is determined by a specific set of shafts. The number of distinct vertical color patterns, known as blocks, in a design is thus bounded by a function of the number of shafts. Because of constraints necessary to keep the fabric from falling apart, it is not normally possible to weave n blocks with only n shafts. Rather, between n + 2 to 4n or more shafts are required, depending on the exact structure of the weaving. The author recently purchased through his institution a 32-shaft computer-controlled loom, and started looking for mathematical patterns which could be woven on it. Fractals seemed like the obvious answer, leading to the question of how complicated a fractal could be produced with 30 blocks? A variation on the Sierpinski carpet can be designed using a number of blocks which is a power of three, which so far appears to be the fractal leaving the fewest number of unused shafts.

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Talk by Joshua Holden (Rose-Hulman Institute of Technology)
https://wordpress.rose-hulman.edu/hol...

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Talk given at: Minisymposium "Mathematics and Arts" at the annual meeting of the German Mathematical Society, 12. - 16. September 2022 (https://ms-math-computer.science/proj...)
Organized by: Milena Damrau and Martin Skrodzki (https://ms-math-computer.science/)
Twitter:   / msmathcomp  

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See also:

Minisymposium "Mathematics and Arts" at the annual meeting of the German Mathematical Society, 14-17 September 2020 (https://ms-math-computer.science/proj...)
Playlist:    • DMV Minisymposium "Mathematics and Arts" 2020  

Minisymposium "Mathematics and Arts" at the annual meeting of the German Mathematical Society, 27 September - 1 October 2021 (https://ms-math-computer.science/proj...)
Playlist:    • Jigsaw puzzles, bell ringing, and Hamilton...