Daina Taimina and David Henderson
are mathematicians at Cornell University. They are coauthors of
Experiencing Geometry a classic text on euclidean and noneuclidean
space. In 1997 Daina worked out how “hyperbolic” space
could be modeled by crochet. Since then, she and David have used
her woolen models to further explore this peculiar topology. Here,
David and Daina will talk about crocheting the hyperbolic plane,
the geometry of lettuce, and the architecture of the universe.
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Fluted and crenelated, the leaf of a lettuce curves away from itself.
From the modest beginning of the stubby stalk where it abuts the
stem, to the flared and scalloped edge, the space literally expands,
each point seeking to maximize its domain. Lettuces, cabbages and
certain types of kelp all embody hyperbolic geometry  the geometric
opposite of the sphere.
Used to model the World Wide Web and the structure of the human
brain, hyperbolic space deviates from rectilinear norms. In “flat”
or Euclidean space parallel lines remain equidistant, but on curved
surfaces a more complex topology reigns. Think of the surface of
the earth: here, lines of Longitude are “parallel” at
the equator yet intersect at the poles. In hyperbolic space, parallel
lines curve away from each other  the further one travels from
any point the more room there is.
For Isaac Newton and his followers, physical space was Euclidean
 endless, formless and flat. But in 1919 measurements of starlight
bending around the sun showed that space is intrinsically curved.
In one recent model proposed by physicists, our universe is shaped
like soccer ball; in another it resembles a trumpet. The WMAP satellite
currently taking pictures of the distant cosmos may at last determine
which, if any, of the proposed models describes the global geometry
of our world.
