By Ronnie Brown, Professor Emeritus, University of Wales Bangor.
Royal Institution Proceedings, Volume 64 pp207-243. This paper is an account of given in 1992 as a Royal Institution Friday Evening Discourse, and to the British Association for the Advancement of Science as the Presidential Address to the Mathematics section, in Southampton. It is addressed to a general audience. It was an honour to be asked to be the speaker at this prestigious and traditional event in 1992. The title reflects the kernel of the subject matter: pushing the boundaries of what is known by questioning the status quo, my own research being the case in point. The main part of the talk was to describe the nature and scope of the area of research - Higher Dimensional Algebra (pertaining to a branch of mathematics known as algebraic topology) - in visual, engaging terms for a general audience.
This revised (colour graphics and some additions/corrections/clarifications to text) web version of the paper includes images from the popularisation web site (click on the image) and accompanying CD. Contact the author for more information and to obtain a copy of the CD.
For mathematicians: "From groups to groupoids: a brief survey" (ref ) also by the author.
BOOK: Topology and Groupoids. R. Brown,
Full paper PDF 27pp, 1MB OPEN/DOWNLOAD
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III. Symmetry and Abstract Groups
|Formula on blocks.
2D representation vs in a line formula.
Examples for algebraic interpretation.
Symmetry operation x4
Symmetry operation xyxy = 1.
Symmetry space of rectangles
Symmetry space of rhombuses
Symmetry space of parallellograms
Illustrations from video "Pivoted Lines and the Möbius Band".
Brehm Möbius band.
`Journeys` by John Robinson.
Dirac string trick
Representing positions of a pendulum
Positions of a double pendulum.
Movement of a double pendulum (by Eric Weisstein).
Trefoil on a torus.
|`Rhythm of Life` by John Robinson.
Pentoil Knot with path.
Loops around a knot.
Product of loops.
Inverse of loops
Pentoil with labels and arrows.
Passing an overpass.
Tying on a string.
Commutativity of 2D groups.
Arrow from source to target.
Product of arrows.
Compositions of squares.
Identities for squares.
Boundary of a cube.
Composition of a cube.
Representing a product xy.
: Table of Contents.
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