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Pictural calculus of isometries
by
Oleg Viro
Stony Brook University
In many homogeneous spaces any isometry is a composition of two involutions which are defined by
their fixed point sets. Hence, an isometry is presented by an ordered pair of subspaces, the
fixed point sets of the involutions. In low-dimensional spaces compositions of isometries can be
easily expressed in terms this presentation. We will discuss multiplication rules similar to
the head to tail addition of vectors for all isometries of plane, 3-space, 2-sphere, projective
plane, hyperbolic plane. This will be compared to Hamilton's presentation of quaternions as
fractions of vectors.
Date received: December 20, 2014
Copyright © 2014 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbjz-65.