Recall that a vector lattice is the set of all integer linear combinations of the members of some basis of a vector space.
Recall that a lattice in an affine space is the set of all barycentric combinations with integer coefficients of the members of a barycentric basis of that affine space.
The last proposition in the handout of April 5 was missing a
clause. It should read:
Proposition. If Lambda is a given lattice in synthetic
n-dimensional affine space, lambda_{0} is a given point of
Lambda, tau_{a} denotes the unique translation of the
affine space carrying lambda_{0} to the point a, and L
denotes the set of members tau_{lambda} of the vector space of
translations as lambda varies in Lambda, then
L is a lattice in the vector space of all translations of the given affine space.
Each element tau of L leaves Lambda invariant.
Every translation of the affine space that leaves Lambda invariant is in L. In fact, every translation of the affine space that carries a single member of Lambda to a member of Lambda is in L.
The theorem in the last discussion classified the affine transformations that leave a lattice invariant. In the case where the vector space is Euclidean, and, therefore, it is meaningful to speak of distance, among these transformations the only transformations that are obviously isometries are the translations by members of the lattice and the involutions x -> lambda_{1} + lambda_{2} - x (a barycentric combination), where the lambda_{j} are members of the lattice. (An involution is a transformation that is its own inverse.)
Given an equilateral triangle show that the group of its symmetries, i.e., the group of plane isometries leaving invariant the set of its vertices (which we know ``to be'' the dihedral group D_{3}) is the smallest group of isometries containing the three reflections in the three medians of the triangle.
Explain why every lattice in synthetic n-dimensional affine space is invariant under each involution x -> lambda_{1} + lambda_{2} - x where the lambda_{j} are any points (vertices) in the lattice. Show that the midpoint of the line segment connecting the vertices lambda_{1} and lambda_{2} (which may or may not be a point of the lattice) is the unique fixed point of this involution. Note that this involution is a half turn when n = 2.
Explain why most lattices in R^{2} are invariant under no rotation about a vertex other than the identity and the half turn. Given such a vertex of a lattice, how many lines can there be through the vertex for which the reflection in the line leaves the lattice invariant?
Given an equilateral triangle identify a plane lattice with the property that the group of isometries leaving the lattice invariant is the smallest group of isometries of the plane that contains the following six reflections: the three reflections in the sides of the triangle and the three reflections in its medians.
Revisit exercise 7 of the assignment due March 26 (on the handout dated March 24): What subgroup of the group leaving invariant the lattice of the previous exercise is the smallest group containing the three reflections in the sides of the given equilateral triangle?
Propose definitions for the terms rotation and reflection in the context of affine transformations of R^{n} that are consistent with those already in this course when n = 2 and that correspond to common usage when n = 3.