The Geometry Center was a

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

research and education center at the

University of Minnesota The University of Minnesota, formally the University of Minnesota, Twin Cities, (UMN Twin Cities, the U of M, or Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Tw ...

. It was established by the

National Science Foundation The National Science Foundation (NSF) is an independent agency of the United States government that supports fundamental research and education in all the non-medical fields of science and engineering. Its medical counterpart is the National I ...

in the late 1980s and closed in 1998. The focus of the center's work was the use of

computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...

and

visualization Visualization or visualisation may refer to: *Visualization (graphics), the physical or imagining creation of images, diagrams, or animations to communicate a message * Data visualization, the graphic representation of data * Information visualiz ...

for research and education in

pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...

and

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

. 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 Paul Dobkin is an American computer scientist and the Phillip Y. Goldman '86 Professor of Computer Science at Princeton University. His research has concerned computational geometry and computer graphics. Early life and education Dobkin wa ...

."Post-mortem on the Geometry Center" ''Math in the Media'' (AMS)
/ref>

Geomview

Much of the work done at the center was for the development of Geomview, a three-dimensional interactive geometry program. This focused on mathematical visualization with options to allow

hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. Th ...

to be visualised. It was originally written for

Silicon Graphics Silicon Graphics, Inc. (stylized as SiliconGraphics before 1999, later rebranded SGI, historically known as Silicon Graphics Computer Systems or SGCS) was an American high-performance computing manufacturer, producing computer hardware and soft ...

workstations, and has been ported to run on

Linux Linux ( or ) is a family of open-source Unix-like operating systems based on the Linux kernel, an operating system kernel first released on September 17, 1991, by Linus Torvalds. Linux is typically packaged as a Linux distribution, which ...

systems; it is available for installation in most Linux distributions through the package management system. ''Geomview'' can run under

Windows Windows is a group of several proprietary graphical operating system families developed and marketed by Microsoft. Each family caters to a certain sector of the computing industry. For example, Windows NT for consumers, Windows Server for serv ...

using

Cygwin Cygwin ( ) is a POSIX-compatible programming and runtime environment that runs natively on Microsoft Windows. Under Cygwin, source code designed for Unix-like operating systems may be compiled with minimal modification and executed. The Cygwin in ...

and under

Mac OS X macOS (; previously OS X and originally Mac OS X) is a Unix operating system developed and marketed by Apple Inc. since 2001. It is the primary operating system for Apple's Mac (computer), Mac computers. Within the market of ...

. Geomview has a web site at . Geomview is built on 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 Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...

Videos

''Geomview'' was used in the construction of several mathematical movies including: *''

Not Knot ''Not Knot'' is a 16-minute film on the mathematics of knot theory and low-dimensional topology, centered on and titled after the concept of a knot complement. It was produced in 1991 by mathematicians at the Geometry Center at the University of ...

'', exploring hyperbolic space rendering of

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

complements

*''Outside In'', a movie about

sphere eversion In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word '' eversion'' means "turning inside out"). Remarkably, it is possible to smoothly and continuously turn a sphere in ...

*''The shape of space'', exploring possible three dimensional spaces

Other software

Other programs developed at the Center included: *'' WebEQ'', a

web browser A web browser is application software for accessing websites. When a user requests a web page from a particular website, the browser retrieves its files from a web server and then displays the page on the user's screen. Browsers are used on ...

plugin allowing mathematical equations to be viewed and edited

*''Kali'', to explore plane

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...

s

*''The Orrery'', a

Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...

visualizer

*''SaVi'', a satellite visualisation tool for examining the orbits and coverage of

satellite constellation A satellite constellation is a group of artificial satellites working together as a system. Unlike a single satellite, a constellation can provide permanent global or near-global coverage, such that at any time everywhere on Earth at least one sa ...

s

*''Crafter'', for structural design of spacecraft

*''

Surface Evolver Surface Evolver is an interactive program for the study of surfaces shaped by surface tension and other energies, and subject to various constraints. A surface is implemented as a simplicial complex. The user defines an initial surface in a datafi ...

'', to explore

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

s

*''

SnapPea SnapPea 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 ...

'', a

hyperbolic 3-manifold In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It ...

analyzer

*''qhull'', to explore

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

s

*''KaleidoTile'', to explore tessellations of the sphere, Euclidean plane, and hyperbolic plane

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 a

as an archive of a wide range of geometric topics, including: *''Geometry and the Imagination'' handouts for a two-week course by

John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston ...

and others

*''Science U'', a collection of interactive exhibits

*''The Geometry Forum'', an electronic community focused on geometry and math education

*''Preprints'', 99 preprints from the center

*''The Topological Zoo'', a collection of curves and surfaces

''Geomview'' is supported through the dedicate
''Geomview'' website

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 Calabi Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...

Donald E. Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...

David Mumford David Bryant Mumford (born 11 June 1937) is an American mathematician known for his 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 t ...

, and Jeff Weeks. There were over thirty postdocs, apprentices and graduate students.

References

{{DEFAULTSORT:Geometry Center, The Mathematics education in the United States National Science Foundation mathematical sciences institutes Research institutes in Minnesota University of Minnesota Visualization (research) Geometry education