Feb 13, 2019 07:03 AM
http://nautil.us/issue/69/patterns/impos...se-tile-rp
EXCERPT: In 1974, Roger Penrose, a British mathematician, created a revolutionary set of tiles that could be used to cover an infinite plane in a pattern that never repeats. [...] Penrose garnered public renown on a scale rarely seen in mathematics ... defied human intuition and changed our basic understanding of nature’s design, revealing how infinite variation could emerge within a highly ordered environment.
At the heart of their breakthroughs is “forbidden symmetry,” so-called because it flies in the face of a deeply ingrained association between symmetry and repetition. Symmetry is based on axes of reflection—whatever appears on one side of a line is duplicated on the other. In math, that relationship is reflected in tiling patterns. [...] Repeated patterns are called “periodic” and are said to have “translational symmetry.” If you move a pattern from place to place, it looks the same.
Penrose, a bold, ambitious scientist, was interested less in identical patterns and repetition, and more in infinite variation. To be precise, he was interested in “aperiodic” tiling, or sets of tiles that can cover an infinite plane with neither gap nor overlap, without the tiling pattern ever repeating itself.
[...] Edmund Harriss, an assistant clinical professor in mathematical studies at the University of Arkansas, who wrote his Ph.D. thesis on Penrose tiles, offers a comparison. “Imagine you’re on a world that is just made up of squares,” Harriss says. “You start walking, and when you get to the edge of the square, and the next square is exactly the same, you know what you’re going to see if you walk forever.” Penrose tiling has the exact opposite nature. “No matter how much information you have, how much you’ve seen of the tiling, you’ll never be able to predict what happens next. It will be something that you’ve never seen before.”
One of the curious aspects of aperiodic division of the plane is that information about positioning is somehow communicated across great distances—a Penrose tile placed in one position prevents the placement of other pieces hundreds (and thousands and millions) of tiles away. “Somehow a local constraint imposes a global constraint,” says Harriss. “You impose that at no scale will these tiles give you something that is periodic.” [...]
The tiles, which form an infinite non-repeating pattern, express the Fibonacci ratio, also known as “the golden ratio.”
MORE (practical uses): http://nautil.us/issue/69/patterns/impos...se-tile-rp
EXCERPT: In 1974, Roger Penrose, a British mathematician, created a revolutionary set of tiles that could be used to cover an infinite plane in a pattern that never repeats. [...] Penrose garnered public renown on a scale rarely seen in mathematics ... defied human intuition and changed our basic understanding of nature’s design, revealing how infinite variation could emerge within a highly ordered environment.
At the heart of their breakthroughs is “forbidden symmetry,” so-called because it flies in the face of a deeply ingrained association between symmetry and repetition. Symmetry is based on axes of reflection—whatever appears on one side of a line is duplicated on the other. In math, that relationship is reflected in tiling patterns. [...] Repeated patterns are called “periodic” and are said to have “translational symmetry.” If you move a pattern from place to place, it looks the same.
Penrose, a bold, ambitious scientist, was interested less in identical patterns and repetition, and more in infinite variation. To be precise, he was interested in “aperiodic” tiling, or sets of tiles that can cover an infinite plane with neither gap nor overlap, without the tiling pattern ever repeating itself.
[...] Edmund Harriss, an assistant clinical professor in mathematical studies at the University of Arkansas, who wrote his Ph.D. thesis on Penrose tiles, offers a comparison. “Imagine you’re on a world that is just made up of squares,” Harriss says. “You start walking, and when you get to the edge of the square, and the next square is exactly the same, you know what you’re going to see if you walk forever.” Penrose tiling has the exact opposite nature. “No matter how much information you have, how much you’ve seen of the tiling, you’ll never be able to predict what happens next. It will be something that you’ve never seen before.”
One of the curious aspects of aperiodic division of the plane is that information about positioning is somehow communicated across great distances—a Penrose tile placed in one position prevents the placement of other pieces hundreds (and thousands and millions) of tiles away. “Somehow a local constraint imposes a global constraint,” says Harriss. “You impose that at no scale will these tiles give you something that is periodic.” [...]
The tiles, which form an infinite non-repeating pattern, express the Fibonacci ratio, also known as “the golden ratio.”
MORE (practical uses): http://nautil.us/issue/69/patterns/impos...se-tile-rp