1 April 2011

Weird geometry: Art enters the hyperbolic realm

Hyperbolic space is a Pringle-like alternative to flat, Euclidean geometry where the normal rules don’t apply: angles of a triangle add up to less than 180 degrees and Euclid’s parallel postulate, governing the properties of parallel lines, breaks down. That fascinates mathematical artist Vi Hart, who creates hyperbolic “tilings” from a range of different materials. View our gallery to get to grips with this alternative geometry.

Read more: “Mathematical artist: Why hyperbolic space is awesome”

Ghostly skirt




Rare balloon tiling

Hart is not the first person to come up with unique ways to visualise the hyperbolic plane: crocheting has been used before, for example. But she says twisted balloons provide unique insights.

"When crocheting, you are building it outward, curlier and curlier as you go out. That gets across one part of it, but with balloons, I want you to see the geometry precisely," she says.

"Very few hyperbolic tilings have been made physically," she adds. "Finding a tiling isn't very hard, but making the twisted balloons is the challenge."

Beading the plane

Hart has also experimented with stringing together hyperbolic planes from beads. "I don't have very much experience in beading, but it's not too hard to make a hyperbolic plane!" she writes on her blog.

Hyperbolic planes take off

Mathematics can be used to define precisely how to combine balloons or beads to form a hyperbolic plane. But painting what one might look like if it sailed past is a very different challenge. "Drawing a hyperbolic plane out of my head is a completely different experience," says Hart. "In order to draw it, I have to understand it. If I draw it, and it looks right, I know I was visualising it right."


Pringles are often used as the example of edible hyperbolic planes. But Hart is fascinated by how fruit slices start to look hyperbolic as they dry out and shrivel up. "This is something that everyone has seen and which surrounds us in the natural world," she says. She is now working on a mathematical model to explain why fruit takes on hyperbolic qualities as it dries.