Math 331 - April 5, 1999

Lattices in synthetic n-dimensional affine space

• What is synthetic n-dimensional affine space? Intuitively it is R^{n} without coordinate axes and without an origin. (A precise definition will follow at a later date.)

• Observation. The question of whether or not an affine transformation of R^{n} is a translation is independent of the choice of an affine coordinate system in R^{n} since

  1. In the representation of a given affine transformation as x -> A x + b the similarity class of the matrix A is not changed by a change of coordinates.

  2. The only matrix similar to the identity matrix is the identity matrix.

  3. A translation is represented in any coordinate system by x -> x + b.

• Proposition. The set of translations of synthetic n-dimensional affine space forms a vector space, independent of the choice of a coordinate system, in which

  1. Vector addition, independent of the choice of a coordinate system, is composition of translations.

  2. The meaning of multiplying the translation tau by the scalar c is given by the formula

     (c tau) (x) = (1 - c) x + c tau(x) ,

     which is independent of the choice of coordinates since the right-hand side is a barycentric combination of x and tau(x).

  Note that if, in some coordinate system, one has tau(x) = x + a, for each point x, then x + c a = x - c x + c x + c a = (1 - c) x + c tau(x) = (c tau) (x) .

  That is, in every affine coordinate system the vector a that gives tau as x -> x + a is multiplied by the scalar c when tau is multiplied by c. Beware of the incorrect formula: (c tau) (x) = c (tau(x)) where the right hand side expands to c x + c a instead of x + c a.

• Definition. A lattice in synthetic n-dimensional affine space is a set of points in that space that consists of all barycentric combinations

  m_{0} P_{0} + m_{1} P_{1} + ... m_{n} P_{n}

  of n + 1 barycentrically independent points where the coefficients m_{0}, m_{1}, ..., m_{n} are integers that sum to 1. Note that if one of these points is designated as ``origin'', then this is the same thing as a vector lattice, as previously defined: the set of all integer linear combinations of the members of a vector space basis.

  Proposition. If Lambda is a given lattice in synthetic n-dimensional affine space, lambda_{0} is a given point of Lambda, and for each lambda in the lattice tau_{lambda} denotes the unique translation of the affine space for which tau(lambda_{0}) = lambda, then

  1. L is a lattice in the vector space of all translations of the given affine space.

  2. Each element tau of L leaves Lambda invariant.

  3. Every translation of the affine space that leaves Lambda invariant is in L.

Short Assignment for Wednesday, April 7

Prepare for a quiz.

Let Lambda denote the lattice in R^{2} that consists of all points (n_{1}, n_{2}) with integer coordinates n_{1} and n_{2}.

1. Convince yourself that Lambda is a lattice in the synthetic Euclidean plane.

2. What is the vector lattice of all translations between members of Lambda ?

3. Find all isometries of the plane that fix the origin and that leave Lambda invariant.

4. What is the group of all isometries of the plane that leave Lambda invariant?