Y is for Hyperbolic Space

| March 4, 2014

SInce our discussion about non-Euclidian geometry, I’ve been thinking about hyperbolic space – as found in art, but also those found in nature.

Non-Euclidian Art: Crochet Reefs

In searching  for artistic expressions of hyperbolic space, I came across the “Hyperbolic Crochet Coral Reef” project, a production of the Institute For Figuring, a non-profit Los-Angeles based organization that pioneers creative new methods for engaging the public about scientific and environmental issues through art. The art itself is primarily the work of the Wertheim sisters,

According to their site, “hyperbolic crochet” was discovered in 1997 by Cornell University mathematician Dr. Daina Taimina. The Wertheim sisters adopted Dr Taimina’s techniques and elaborated upon them to develop a whole taxonomy of reef-life forms.

The goal of their project is to create a crochet reef that serves as an invitation to the public to learn more about the spatial mathematics of coral reefs.

Crochet Kelp: Maximum Surface Area, Minimal Space

See this 2009 Ted Talk by Margaret Wertheim: “The beautiful math of coral“.

“The basic process for making these forms is a simple pattern or algorithm, which on its own produces a mathematically pure shape, but by varying or mutating this algorithm, endless variations and permutations of shape and form can be produced.”

Besides reefs, many marine organisms embody hyperbolic geometry in their anatomies – among them kelps, corals, sponges, sea slugs and nudibranchs. Other non-marine plants embody hyperbolic geometry as well, such as cacti, mushrooms, succulents, and green leafy vegetables like lettuce and kale.

From an evolutionary perspective, the “reason” that these organisms have thrived by embodying hyperbolic geometry is that it provides the organisms with a great way to maximize surface area in a limited space, which allows them to feed more.

“New” Shapes – Archimedean Solids

Recently, I came across an Ars Technica piece describing the discovery of a “new” class of Archimedean solids, a/k/a “Goldberg solids”. The piece artfully describes the defining characteristic of the shapes: “Take a cube and blow it up like a balloon.” This made me think of hyperbolic space. A way of taking the existing space, and expanding it from within…

Goldberg Solids

These “new” shapes are actually found in nature (carbon atoms in a diamond, salt and fool’s gold form cubic crystals, and calcium fluoride forms octahedral crystals). How novel is it that after 400 years of geometry, we ‘found’ new shapes by blowing up old shapes from within?  Everything old is new again.

What does it Mean for Art?

I’ve been thinking about how explorations of hyperbolic space can be manifest in media other than crochet.  This brings me back to music. Non-Euclidean geometry deals with matter and surface area in space. I think of rhythm and pulse as the expression of space, and the melodic content as the expression of lines within that space. So, I think of different jazz musicians who play with density, like Coltrane and Monk. I also think of hypermelodicists; those who fit the maximum amount of notes and melody into a minimal amount of space, and that brings to mind the Dirty Projectors, Hermeto Pascoal, and Frank Zappa. In particular, Conlon Nancarrow came to mind.  He was an American-born composer who lived and worked in Mexico for most of his life. Because what he was hearing in his head couldn’t really be played by humans, he wrote these insane compositions for the player piano. I thought his music might best express what I meant by hyperbolic geometry in music.

Take  a listen to this…  Maximum (melodic) surface area, minimal space.

Now that’s some hyperbolic geometry in space time.

I also thought this notion of “finding” new shapes in nature echoes the artistic process: we “find” new ways of aesthetic expression using the same resources (e.g. paint/sculpture/the chromatic scale) time and again.  Art reinvents itself constantly, but does anything really ever change?