By Ronald Brown, Emeritus Professor, Bangor University,FLSW.
Royal Institution Proceedings, Volume 64 pp207243. This paper is an account of lectures 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. The lectures were addresed to a general audience. 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.
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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 [4]) also by the author.
BOOK: Topology and Groupoids. R. Brown,
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III. Symmetry and Abstract
Groups 
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V.
Motion 
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IX.Implications. References. 
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Fig 1 Fig 2 Fig 3 Fig 4 Figs 57 Figs 89 Fig 10 Fig 11 Fig 12 Fig 13 Fig 14 Figs: 1526 Fig 27 Fig 28 Fig 29 Fig 30 Fig 31 Fig 32 Fig 33 Fig 34 Fig 35 
Formula on blocks. Formula rearranged. 2D representation vs in a line formula. Labelled arrows. Examples for algebraic interpretation. Symmetry operations. Symmetry operation x^{4} 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". Möbius Band Brehm Model. 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. 
Fig 36 Fig 37 Fig 38 Fig 39 Fig 40 Fig 41 Fig 42 Fig 43 Fig 44 Fig 45 Fig 46 Fig 47 Fig 48 Figs 4950 Fig 51 Fig 52 Figs 534 Fig 55 Fig 56 Fig 57 Fig 58 
`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. Trefoil knot. Commutativity of 2D groups. Arrow from source to target. Product of arrows. Labelled square. Compositions of squares. Associativity. Identities for squares. Connections. Boundary of a cube. Composition of a cube. Permutahedra. Representing a product xy. 
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