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Unformatted text preview: Math Biophysics, Fall 2011 Homework 2 (1) The face centered cubic (fcc) lattice is generated by the basis vectors (0 , 1 , 1) , (1 , 1 , 0) , (1 , , 1) , which means it is the set of all vectors of the form a (0 , 1 , 1) + b (1 , 1 , 0) + c (1 , , 1) where a,b, and c are integers. Show that this lattice is also the set of vectors ( p,q,r ) where p,q , and r are integers and p + q + r is even. (2) Show that the rotations 1 1 1 0 and 0 0 1 1 0 0 0 1 0 are symmetries of the fcc lattice. (3) Let u = (1 , 1 , 1) / √ 3. Use the formula at the end of the lecture on rota tions to find the matrix for R ( u , 2 π/ 3). (4) Use Maple and the Frenet Formula F t d F ds =  κ κ τ τ to find the curvature and torsion of the curve (3 t t 3 , 3 t 2 , 3 t + t 3 ). You can use the procedure presented in the lecture for computing the Frenet Frame F ....
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 Fall '11
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 arg, Differential geometry of curves, Frenet Frame

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