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.

Contents (To browse click a link: paper is in 3 parts for quicker downloading).

I. Introduction.



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.

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II. Formulae in a Line or Plane.


III. Symmetry and Abstract Groups
     The Importance of Symmetry.
     Symmetry of a Square.
     Abstract Groups and Structure.
     Axioms of an Abstract Group.





IV. Space.


V. Motion
     Pivoted Lines and the Möbius Band.
     The Dirac String Trick.
     The Double Pendulum.
     Paths and Knot Spaces.





VI. Towards Higher Dimensions


VII. Groupoids: a Wider Structure Than Groups


VIII. Higher Dimensional Groupoids.


IX.Implications. References.


List of illustrations.

Fig 1
Fig 2
Fig 3
Fig 4
Figs 5-7
Figs 8-9
Fig 10
Fig 11
Fig 12
Fig 13
Fig 14
Figs: 15-26
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
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 49-50
Fig 51
Fig 52
Figs 53-4
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.
Identities for squares.
Boundary of a cube.
Composition of a cube.
Representing a product

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