m348-assignment4 - Math 348(2006 Assignment#4 due Monday 1...

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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 half-turn. 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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