More Advanced Geometry Exercises

1. Show that any two different bases of a lattice L differ by an element of the group SL(2, Z) of 2 × 2 matrices with integer entries and determinant 1.
   If a lattice L has a basis {e₁ , e₂} show that one can change its basis so that the first element of the new basis is any vector of the form pe₁ + qe₂ with p, q coprime integers. [Hint: the Euclidean algorithm]

2. For each of the 17 plane crystallographic groups identify their lattices and point groups. Identify the stabilizer subgroups (subgroups which leave a point fixed) of points in a design with each of these as groups of symmetries. For which examples is the point group not a stabilizer of some point ?

   Solution to question 3

3. If G is a discrete subgroup of I(R²) with lattice L and point group P, then we have L G P with the first map the inclusion as a subgroup and the second map onto the quotient. For which of the 17 2-dimensional crystallographic groups is the group G the direct product L × P ?

   Solution to question 4

4. Let G be the crystallographic group generated by a glide reflection in the y-direction and a translation in the x-direction. Draw a figure which has G as a symmetry group and identify it from the list of 17.

   Solution to question 5

5. Which of the 17 groups have patterns which can be obtained from the seven Frieze groups by "laying down strips"?

   Solution to question 6

6. The diagram below represents part of a design with symmetry group pgg.
   Identify the lattice of this group, its point group P and the shift vectors of each element of P.
   
   
   
   Draw two other designs whose symmetry groups have the same lattice and point group as the above, but whose symmetry groups are different from the above and from each other. In each case indicate the shift vectors of elements of the point groups.

   Solution to question 7
   

   SOLUTIONS TO WHOLE SET

   ---

   Previous page (Advanced Geometry Exercises)

   Contents

   Next page (Some more practice)

   ---

   JOC March 2003