Geometric Topology
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.
- Acme Klein Bottle.
A topologist's delight, handcrafted in glass.
- Are most manifolds hyperbolic? From Dave Rusin's known math pages.
- 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.
- Boy's surface:
Wikipedia,
MathWorld,
Geometry
Center,
and an
asymmetric animated gif from the Harvard zoo.
- Constructing Boy's surface out of paper and tape.
- 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.
- Figure eight knot / horoball diagram.
Research of A. Edmonds into the symmetries of knots,
relating them to something that looks
like a packing of spheres.
The MSRI Computing Group uses
another horoball
diagram as their logo.
- The
flat torus in the three-sphere. Thomas Banchoff animates the
Hopf fibration.
- 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?
- Geometric topology preprint server.
- Hilbert's
3rd Problem and Dehn Invariants.
How to tell whether two polyhedra can be dissected into each other.
See also Walter
Neumann's paper connecting these ideas with problems of
classifying manifolds.
- Hopf fibration.
R. Kreminski,
the U.
Sheffield maths dept., and
MathWorld
explain and animate the partition of a 3-sphere
into circles.
- 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.
- Mathematics
in John Robinson's symbolic sculptures. Borromean rings, torus
knots, fiber bundles, and unorientable geometries.
- Mathenautics. Visualization of 3-manifold geometry at the Univ. of Illinois.
- A
minimal winter's tale. Macalester College's snow sculpture of
Enneper's surface wins second place at Breckenridge.
- Möbius
at the Shopping Mall. Topological sculpture as public seating. From MathTrek.
- Nested
Klein bottles. From the London Science Museum gallery, by way of Boing
Boing. Topological glassware by Alan Bennett.
- 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).
- Pseudospherical surfaces.
These surfaces are equally "saddle-shaped" at each point.
- 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.
- Morwen Thistlethwait,
sphere packing, computational topology, symmetric knots,
and giant ray-traced floating letters.
- 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.
- UMass Gang
library of knots, surfaces, surface deformation movies, and
minimal surface meshing software.
- Matthias
Weber's gallery of ray-traced mathematical objects, such as minimal
surfaces floating in ponds.
- A zoo of surfaces.
From the Geometry Junkyard,
computational
and recreational geometry pointers.
Send email if you
know of an appropriate page not listed here.
David Eppstein,
Theory Group,
ICS,
UC Irvine.
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from a common source file.
