This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 348 (2006): Assignment #4: Solutions 1. We know that there are three types of orientationreversing isometries of E 3 : reflection, glide, and rotatary reflection. We also know that a halfturn H ` about a line ` can be written as the composition r M 2 r M 1 of two reflections across any two perpendicular planes M 1 and M 2 intersecting in the line ` . Consider a reflection r M about a plane M . Let ` be any line contained in M and let M be the plane through ` perpendicular to M . Then H ` = r M r M and thus r M = r M r M r M = H ` r M . Consider a glide g . It can be written as g = r M 3 r M 2 r M 1 where M 1 k M 3 and M 2 is perpendicular to both M 1 and M 3 . But then r M 3 r M 2 = H ` where ` is the line of intersection of M 2 and M 3 , and hence g = H ` r M 1 . Finally, consider a rotary reflection f . It can be written as f = r M 3 r M 2 r M 1 where M 1 and M 2 intersect in some line and are both perpendicular to M 3 . But then r M 3 r M 2 = H ` where ` is the line of intersection of M 2 and M 3 , and hence f = H ` r M 1 ....
View Full
Document
 Summer '06
 Karigiannis
 Math, Geometry

Click to edit the document details