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12
1. ALGEBRAIC THEMES
map vertices to vertices!) Now there is exactly one point in
R
that is
equidistant from the four vertices; this is the centroid of the ﬁgure, which
is the intersection of the two diagonals of
R
. Denote the centroid
C
. What
is
±.C/
? Since
±
is an isometry and maps the set of vertices to itself,
±.C/
is still equidistant from the four vertices, so
±.C/
D
C
. We can assume
without loss of generality that the ﬁgure is located with its centroid at
0
,
the origin of coordinates. It follows from the results quoted in the previous
paragraph that
±
extends to a
linear
isometry of
R
3
.
The same argument and the same conclusion are valid for many other
geometric ﬁgures (for example, polygons in the plane, or polyhedra in
space). For such ﬁgures, there is (at least) one point that is mapped to
itself by every symmetry of the ﬁgure. If we place such a point at the
origin of coordinates, then every symmetry of the ﬁgure extends to a linear
isometry of
R
3
.
Let’s summarize with a proposition:
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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