 
Summary: FACTORING EUCLIDEAN ISOMETRIES
NOEL BRADY AND JON MCCAMMOND
Abstract. Every isometry of a finite dimensional euclidean space
is a product of reflections and the minimum length of a reflection
factorization defines a metric on its full isometry group. In this
article we identify the structure of intervals in this metric space by
constructing, for each isometry, an explicit combinatorial model
encoding all of its minimal length reflection factorizations. The
model is largely independent of the isometry chosen in that it only
depends on whether or not some point is fixed and the dimension
of the space of directions that points are moved.
Every good geometry book proves that each isometry of euclidean n
space is a product of at most n+1 reflections and several moreadvanced
sources include Scherk's theorem which identifies the minimal length
of such a reflection factorization from the basic geometric attributes
of the isometry under consideration [5, 7, 8, 10]. The structure of the
full set of minimal length reflection factorizations, on the other hand,
does not appear to have been given an elementary treatment in the
literature even though the proof only requires basic geometric tools.1
In this article we construct, for each isometry, an explicit combinatorial
