Intelligent 3D Blackboards for Mathematical Research
Sha Xin Wei
Abstract
We can now reap the progress over the past decade in symbolic
computation programs, virtual reality systems and hardware,
to assemble a smoothly functioning intelligent blackboard
suitable for research in topology, differential geometry and
physics. While researchers and developers
have begun to implement intelligent blackboards for video-
conferencing and general presentation/communication work,
few have designed intelligent blackboards for use by
mathematicians or physical scientists to augment analytical work,
in part because the technology could not meet the demands of such
disciplines, but also because these theoretical disciplines have not had
the same familiarity with, or access to technology as the
engineering disciplines. This is no longer the case.
Context
Computer environments and
languages can now begin to support entities and operations meaningful to
domains as diverse and rich as theater or differential geometry. For example, given
reasonably expressive visual models of objects with some classical physical
properties from rigid and elastic mechanics, one can design languages to
mediate interaction between human users and virtual objects. In the domain
of theater, or more generally, performance, we can begin to design
high-level "director languages" to define the motions or behaviors of
interacting 3D sprites.
Such languages build on models of appearance and behavior abstracted from
the highly evolved history of human performance (choreography, staging,
classical animation, etc.). Highly evolved models of manifolds and more
general geometric and topological structuters also exist in the domains of
mathematics and theoretical physics, but to date, interactive 3D
visualization systems have not found wide use in the mathematical or
theoretical physics communities.
Within the field of differential geometry, perhaps the best known
visualization system is the Geometry Center's {\em geomview} (\cite{Geomview93},
\cite{Sha94}), which
has been coupled to a variety of numerical research applications, and to
general algebraic systems. What {\em geomview} lacks in graphics sophistication is
offset by its accomodation of structures and operators meaningful to
working geometers.
An approach toward a geometric workspace
We seek 3D interfaces,
or for that matter, any geometric workspace at all which approaches the
suggestiveness
of freehand chalkboard but couples to algebraic subsystems.
One of the goals is the synthesis of symbolic algebra languages and
numerical analysis tools into 3D interfaces, with converse feedback
from 3D manipulation to algebraic representations of geometric
structures. Broadly, there are two ways to enrich 3D interfaces
to make them more useful for geometers and topologists: by
making graphically smart manipulables, and by linking manipulables to
computational engines such as symbolic algebra systems,
numerical analysis systems, and knowledge representation
databases tuned to mathematical theories rather than relatively
fine-grained "knowledge systems."
The general goal is not verisimilitude, or a mathematical
equivalent of photorealism, which is generally infeasible anyway,
but flexibility, expressivity, and easy definition in terms familiar
to geometers, topologists or allied researchers. (see examples below)
Applications
One of the most useful benefits of such liveboards would be the opportunity to radically augment the communication among students of mathematical and physical sciences.
Physics
Computational chemistry
Mathematics -- Topology, geometry, analysis
examples available
Bibliography
Andre Linde. Proposal: Quantum Cosmology Simulations.
(1 p. PDF, HTML).
David Mumford; Goroff. Harvard experimental mathematics course. Geometry Center Bulletin 1994.
Sha Xin Wei.
Proposal for a Geometric Reasoning Laboratory, CSLI Interface Lab talk, 12 March 1996, (18 pp. HTML).
-------. Proposal: Intelligent 3D Blackboards for Geometric Research.(7pp PDF)
-------. Notes from Workshop on geometry workspace , Mathematical Sciences Research Institute December 1994.
-------. Geometer's Workbench presentation, Stanford CS Mural project
email xinwei.