Math 348 (2006): Assignment #4
due Monday, July 31, 2006
1. Show that any orientation reversing isometry of
E
3
can be written as the composition of a reﬂection
and a halfturn.
2. Consider a dilative rotation in
E
3
. This is the composition of a dilation
O
λ
about a point
O
with
a rotation
R
~
`,θ
about a line
`
passing through
O
. This similarity always has an invariant plane.
Which plane is it? If
θ
= 180, there are more invariant planes. Which planes are they?
3. Recall that a regular
p
sided polygon has interior angles
θ
= 180

360
p
. In a regular polyhedron,
we have
q
such polygons meeting at each vertex. The sum of the angles at each vertex is thus
q
(180

360
p
). The amount by which this sum falls short of 360 is thus 360

q
(180

360
p
). Show that
this shortfall is exactly equal to
720
V
, where
V
is the number of vertices of the regular polyhedron.
That is, prove that
360

q
±
180

360
p
²
=
720
V
4. Recall that ellipses and hyperbolas are called
central conics
. Express the eccentricity of a central
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 Summer '06
 Karigiannis
 Math, Geometry, Euclidean geometry, Regular polyhedron, Real projective plane

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