This area of mathematics is about the assignment of geometric structures
to topological spaces, so that they "look like"
geometric spaces. For instance, compact two dimensional surfaces
can have a local geometry based on the sphere
(the sphere itself, and the projective plane), based
on the Euclidean plane (the torus and the Klein bottle), or based
on the hyperbolic plane (all other surfaces).
Similar questions in three dimensions have more complicated answers;
Thurston showed that there are eight possible geometries,
and conjectured that all 3-manifolds can be split into pieces having
these geometries.
Computer solution of these questions by programs like SnapPea
has proved very useful in the study of knot theory
and other topological problems.
Bending a
soccer ball mathematically. Michael Trott animates morphs between a
torus and a double-covered sphere, to illustrate their topological
equivalence, together with several related animations.
Crystallographic
topology. C. Johnson and M. Burnett of Oak Ridge National Lab use
topological methods to understand and classify the symmetries of the
lattice structures formed by crystals. (Somewhat technical.)
Double
bubbles. Joel Hass investigates shapes formed by soap films
enclosing two separate regions of space.
Gallery of interactive on-line geometry.
The Geometry Center's collection includes programs for generating
Penrose tilings, making periodic drawings a la Escher in the Euclidean
and hyperbolic planes, playing pinball in negatively curved spaces,
viewing 3d objects, exploring the space of angle geometries, and
visualizing Riemann surfaces.
Geometric probability question.
What is the probability that the shortest paths between three random
points on a projective plane form a contractible loop?
Laying
Track. The combinatorics and topology of Brio train layouts. From
Ivars Peterson's MathTrek.
3-Manifolds from regular solids.
Brent Everitt lists the finite volume orientable hyperbolic and
spherical 3-manifolds obtained by identifying the faces of regular solids.
Mathematically
correct breakfast. George Hart describes how to cut a single bagel
into two linked Möbius strips. As a bonus, you get more surface
area for your cream cheese than a standard sliced bagel.
The Optiverse.
An amazing 6-minute video on how to turn spheres inside out.
The
Pretzel Page. Eric Sedgwick uses animated movies of twisting pretzel knots
to visualize a theorem about Heegard splittings
(ways of dividing a complex topological space into two simple pieces).
Riemann Surfaces and the Geometrization of 3-Manifolds,
C. McMullen, Bull. AMS 27 (1992).
This expository (but very technical) article outlines Thurston's
technique for finding geometric structures in 3-dimensional topology.
SnapPea, powerful software for computing geometric properties of
knot complements and other 3-manifolds.
The Thurston Project: experimental differential geometry, uniformization and quantum field theory.
Steve Braham hopes to prove Thurston's uniformization conjecture
by computing flows that iron the wrinkles out of manifolds.
Tiling dynamical systems.
Chris Hillman describes his research
on topological spaces in which each point represents a tiling.
Lun-Yi Tsai paints fine
art of foliatied 3-manifolds, differentiable atlases, and other
topological structures.
