Weaving Quasicrystals

"Weaving Quasicrystals" is a series of wall works and large-scale installations that explore the technical connections between weaving, mathematics, and color theory.

Created in collaboration with mathemetician, Phil Engel and produced with support from the National Science Foundation grant DMS-2201221, this ongoing series of woven patterns is based on the following:

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To understand “why quasicrystals”, it is first important to understand crystals.

Why does a snowflake always have 6-fold symmetry? It is because ice molecules pack into a regular hexagonal grid, or crystal, just as hexagonal tiles might tile a bathroom floor. Thus, the microscopic structure of ice informs the macroscopic shapes which are produced. The same phenomenon happens for precious gemstones, which is why emerald grows in a hexagonal prism, or why pyrite forms a perfect cube (see below).

It has long been known to mathematicians that a crystal can only have 1-, 2-, 3-, 4-, or 6-fold rotational symmetry. This is called the crystallographic restriction.

Quasicrystals were actually first discovered theoretically, by mathematicians! Like crystals, they are made of some fixed set of tiny constituent parts, or tiles. But unlike crystals, they never repeat periodically. A mathematician Hao Wang and his student Robert Berger were the first to construct a set of tiles which can be used to cover the entire two-dimensional plane, but in a way that never repeats.

Their work was simplified and popularized by physicist Roger Penrose, who came up with what are now called “Penrose tilings” (see above). The pattern can be infinitely extended but in a manner which is always non-repeating, or “new”. As you can see, the figure has a 5-fold symmetry, violating the crystallographic restriction!

These patterns, which one could imagine growing like a crystal but never precisely repeating, are called quasicrystals. Their platonic existence was, by the 1960’s, a mathematically established fact. But quasicrystals had never been observed in nature, and it became a pursuit of chemists to see whether these amazing structures existed in some physical material. Amazingly, in 1984 Dan Schechtman discovered a material whose x-ray crystallography showed 5-fold symmetry! Due to the crystallographic restriction, it could not be a crystal.

The discovery was shocking to the chemistry community, leading to a bitter debate between Schechtman and Linus Pauling (a famous crystallographer and scientist who is the only individual to have two unshared Nobel Prizes). Pauling couldn’t believe such a material existed in nature and famously said, “There is no such thing as quasicrystals, only quasi-scientists.” Pauling continued to disparage Schechtman’s work until his death. But Schechtman’s research was ultimately proven valid, receiving a Nobel Prize in 2011.

There are a lot of parallels between crystals, quasicrystals, and weaving, especially the idea that the microscopic structure of a material guides its overall shape. We are particularly fascinated by the way that the binary code of a Jacquard loom can be used to emphasize or de-emphasize certain strands of color, by hiding those strands “under” other strands. Also amazing to us is how color blending can be achieved, as though on a computer screen, via RGB mixing of color strands. Weavings look so different up close versus far away. It is amazing how sensitive the macroscopic properties of the weave are, to tiny changes in the weave pattern. Just like an inclusion error in a crystal structure leads to a visible flaw in the gem, being off by 1 register on the Jacquard weaving program leads to wildly different results.

“Weaving Quasicrystals” explores how structures emerge at multiple length scales, and in multiple dimensions. To mathematicians, a “dimension” is simply an aspect of some object. There is no special “fourth dimension” which is time, and the “three dimensions” need not be spatial! For example, the three dimensions of RGB color are “amount of red”, “amount of green”, and “amount of blue” and so each color mixture has a total of 3 dimensions, or aspects. Each point of the weaving corresponds to a point in a 3-dimensional space of color, corresponding to the ratio of the 3 different strands of colored yarn.

The tilings patterns themselves are made by curving a 2-dimensional surface (which ultimately forms the weaving surface) in a 5-dimensional space. As this surface moves in the 5-dimensional space, the tilings become regular and periodic, like a crystal, or irregular and aperiodic, like a quasicrystal. Thus, in a concrete mathematical sense, we are exploring a 5-dimensional space by deforming the weaving surface between the crystalline and the quasicrystalline.

On the smallest scale, weaving is bound by a rectilinear form: The warp is vertical, and the weft is horizontal. This makes it difficult in weaving to escape the underlying structure of periodic, rectangular, grid lines. But the technology of the Jacquard loom gives us the ability to defy a grid, just as a computer monitor, consisting of a square grid of pixels, can still depict a curved shape, at least to the human eye. The pixel size on our weavings are much larger than on a computer, allowing the viewer to see how the illusions, both of color and shape, are created.