The Geometry Center
Encyclopedia
The Geometry Center was a mathematics
research and education center at the University of Minnesota
. It was established by the National Science Foundation
in the late 1980s and closed in 1998. The focus of the Center's work was the use of computer graphics
and visualization
for research and education in pure mathematics
and geometry
.
The Center's founding director was Al Marden. Richard McGehee directed the Center during its final years. The Center's governing board was chaired by David P. Dobkin
.
. This focused on mathematical visualization with options to allow hyperbolic space
to be visualised. This originally ran on Silicon Graphics
machines, and has been ported to run under X11 on unix
machines. Geomview can run under Windows
using Cygwin
and under Mac OS X
. Geomview is still being supported today.http://www.geomview.org/
Geomview is built around the Object Oriented Graphics Library (OOGL). The displayed scene and the attributes of the objects in it may be manipulated by the graphical command language (GCL) of Geomview. Geomview may be set as a default 3-D viewer for Mathematica
.
Despite the Center being closed, its website is still online at http://www.geom.uiuc.edu/ as an archive of a wide range of geometric topics, including:
Geomview is supported through the dedicated Geomview website.
Support for software developed at the Geometry Center is available through Geometry Technologies.
, John Horton Conway
, Donald E. Knuth, David Mumford
, William Thurston
, and Jeff Weeks
. There were over thirty postdocs, apprentices and graduate students.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
research and education center at the University of Minnesota
University of Minnesota
The University of Minnesota, Twin Cities is a public research university located in Minneapolis and St. Paul, Minnesota, United States. It is the oldest and largest part of the University of Minnesota system and has the fourth-largest main campus student body in the United States, with 52,557...
. It was established by the National Science Foundation
National Science Foundation
The National Science Foundation is a United States government agency that supports fundamental research and education in all the non-medical fields of science and engineering. Its medical counterpart is the National Institutes of Health...
in the late 1980s and closed in 1998. The focus of the Center's work was the use of computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
and visualization
Visualization
The term visualization or visualisation may refer to:* Creative visualization* Flow visualization* Geovisualization* Illustration* Information graphics, visual representations of information, data, or knowledge* Information visualization...
for research and education in pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
.
The Center's founding director was Al Marden. Richard McGehee directed the Center during its final years. The Center's governing board was chaired by David P. Dobkin
David P. Dobkin
David Paul Dobkin is the Dean of the Faculty and Phillip Y. Goldman '86 Professor of Computer Science at Princeton University.Dobkin was born February 29, 1948, in Pittsburgh, Pennsylvania. After receiving a B.S. from the Massachusetts Institute of Technology in 1970, he moved to Harvard University...
.
Geomview
Much of the work at the Center was of the development of Geomview, a three-dimensional interactive geometry programInteractive geometry software
Interactive geometry software are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry. In most IGS, one starts construction by putting a few points and using them to define new objects such as lines, circles or other points...
. This focused on mathematical visualization with options to allow hyperbolic space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
to be visualised. This originally ran on Silicon Graphics
Silicon Graphics
Silicon Graphics, Inc. was a manufacturer of high-performance computing solutions, including computer hardware and software, founded in 1981 by Jim Clark...
machines, and has been ported to run under X11 on unix
Unix
Unix is a multitasking, multi-user computer operating system originally developed in 1969 by a group of AT&T employees at Bell Labs, including Ken Thompson, Dennis Ritchie, Brian Kernighan, Douglas McIlroy, and Joe Ossanna...
machines. Geomview can run under Windows
Microsoft Windows
Microsoft Windows is a series of operating systems produced by Microsoft.Microsoft introduced an operating environment named Windows on November 20, 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces . Microsoft Windows came to dominate the world's personal...
using Cygwin
Cygwin
Cygwin is a Unix-like environment and command-line interface for Microsoft Windows. Cygwin provides native integration of Windows-based applications, data, and other system resources with applications, software tools, and data of the Unix-like environment...
and under Mac OS X
Mac OS X
Mac OS X is a series of Unix-based operating systems and graphical user interfaces developed, marketed, and sold by Apple Inc. Since 2002, has been included with all new Macintosh computer systems...
. Geomview is still being supported today.http://www.geomview.org/
Geomview is built around the Object Oriented Graphics Library (OOGL). The displayed scene and the attributes of the objects in it may be manipulated by the graphical command language (GCL) of Geomview. Geomview may be set as a default 3-D viewer for Mathematica
Mathematica
Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
.
Videos
Geomview was used in the construction of several mathematical movies including:- Not Knot, exploring hyperbolic space rendering of knotKnot theoryIn topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
complements. http://www.geom.uiuc.edu/video/NotKnot/ - Outside In, an award winning movie about the sphere eversionSmale's paradoxIn differential topology, Smale's paradox states that it is possible to turn a sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, a process often called sphere eversion...
. http://www.geom.uiuc.edu/docs/outreach/oi/ - The shape of space, exploring possible three dimensional spaces.http://www.geom.uiuc.edu/video/sos/
Other software
Other programs developed at the Center included:- WebEQ, a web browserWeb browserA web browser is a software application for retrieving, presenting, and traversing information resources on the World Wide Web. An information resource is identified by a Uniform Resource Identifier and may be a web page, image, video, or other piece of content...
plugin allowing mathematical equations to be viewed and edited. http://www.geomtech.com/portfolio - Kali, to explore plane symmetry groupSymmetry groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
s. http://geometrygames.org/Kali/index.html - The Orrery, a Solar systemSolar SystemThe Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
visualizer. - SaVi, a satellite visualisation tool for examining the orbits and coverage of satellite constellationSatellite constellationA group of artificial satellites working in concert is known as a satellite constellation. Such a constellation can be considered to be a number of satellites with coordinated ground coverage, operating together under shared control, synchronised so that they overlap well in coverage and...
s. http://savi.sourceforge.net/ - Crafter, for structural design of spacecraft. http://www.geom.uiuc.edu/software/crafter/
- Surface Evolver, to explore minimal surfaceMinimal surfaceIn mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
s. http://www.geom.uiuc.edu/software/download/evolver.html http://www.susqu.edu/facstaff/b/brakke/evolver/evolver.html - SnapPeaSnapPeaSnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds. The primary developer is Jeffrey Weeks, who created the first version as part of his doctoral thesis, supervised by William Thurston. The latest version is 3.0d3...
, a hyperbolic 3-manifoldHyperbolic 3-manifoldA hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
analyzer.http://www.geometrygames.org/SnapPea/index.html - qhull, to explore convex hullConvex hullIn mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
s. http://www.qhull.org/ - KaleidoTile, to explore tessellations of the sphere, Euclidean plane, and hyperbolic plane. http://www.geometrygames.org/KaleidoTile/index.html
Website
Richard McGehee, the center's director, has stated that the website was one of the first one hundred websites ever published.Despite the Center being closed, its website is still online at http://www.geom.uiuc.edu/ as an archive of a wide range of geometric topics, including:
- Geometry and the Imagination handouts for a two week course by John Horton ConwayJohn Horton ConwayJohn Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
, William ThurstonWilliam ThurstonWilliam Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
and others. http://www.geom.uiuc.edu/docs/education/institute91/handouts/handouts.html - Science U, a collection of interactive exhibits. http://www.scienceu.com/
- The Geometry Forum, an electronic community focused on geometry and math education.http://www.geom.uiuc.edu/docs/forum/
- Preprints, 99 preprints from the center.http://www.geom.uiuc.edu/docs/preprints/
- The Topological Zoo, a collection of curves and surfaces.http://www.geom.uiuc.edu/zoo/
Geomview is supported through the dedicated Geomview website.
Support for software developed at the Geometry Center is available through Geometry Technologies.
Research
During its time of operation, a large number of mathematical workshops were held at the Center. Many well-known mathematicians visited the Center, including Eugenio CalabiEugenio Calabi
Eugenio Calabi is a Italian American mathematician and professor emeritus at the University of Pennsylvania, specializing in differential geometry, partial differential equations and their applications....
, John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
, Donald E. Knuth, David Mumford
David Mumford
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
, William Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
, and Jeff Weeks
Jeffrey Weeks (mathematician)
Jeffrey Renwick Weeks is an American mathematician, a geometric topologist and cosmologist.-Biography:Weeks received his B.A. from Dartmouth College in 1978, and his Ph.D. in mathematics from Princeton University in 1985, under the supervision of William Thurston...
. There were over thirty postdocs, apprentices and graduate students.