Consider the set D of vectors (a1, a2, a3 , a4) 6 R4 such that all a,- are integers, and the sum a1 + a2 + a3 + a4 is even. 1. Find the nonzero elements of D which have minimal distance from the origin. 2. Find inequalities defining the set of elements which are closer to the origin than they are to any non-zero element of D. (Hint: By the Pythagorean theorem, a vector 35 is on the hyperplane midway between 33 and 2 if)? - (i Z) = %(l lil |2 IIZI |2)). This set AD is a 4d polytope you are familiar with. Explain which one it is and find its vertices. 3. Translating this polytope by each element of D, we obtain the cells in a tesselation of R4 (with a cell for each d E D given by the points closer to d than to any other element of D). Give an algebraic description of the set C of vertices of the polyhedra which appear, that is all the translates of the vertices of your original polytope by elements of D. Just as before, for each element c E C, there is a polytope AC given by the set of x E R4 which are closer to c than any other element of C. Again is it is one you are familiar with. Explain which one it is. (Hint: it might be easier to do the calculations if you change coordinates so that c is at the origin.) 4. The translates of AC by elements of D also give a tesselation of R4, which is dual to the first tesselation we found (so their Schlafli symbols are related like those of dual polytopes). Which Schlafli symbols do these two tesselations show cannot correspond to a regular polytope