Definition. A crystallographic group is a group of isometries of an n-dimensional Euclidean affine space for which the translation subgroup is a lattice. When n is 2, the term wallpaper group is used.
Proposition. When G is a crystallographic group of isometries of R^{n}, and Lambda is the point lattice containing the origin in R^{n} whose associated translation lattice is the translation subgroup of G, the following statements hold:
If f is in G and f(x) = U x + b, then U(Lambda) = Lambda. Note. This does not say that f leaves Lambda invariant, nor does it say that the isometry x -> U x is in G.
If f_{1} and f_{2} are two elements of G that share the same ``linear part'' U, i.e., if f_{1}(x) = U x + b_{1} and f_{2}(x) = U x + b_{2}, then f_{2} must be the composite tau \circ f_{1} where tau is a translation in G, i.e., translation by a point of Lambda. It follows that b_{2} - b_{1} must be in Lambda.
Proof. If f(x) = U x + b is in G, then for each lambda in Lambda it must be true that f tau_{lambda} f^{-1} is in G. Moreover the conjugate of a translation by any isometry must be a translation. Hence, the conjugate of tau_{lambda} by f is a translation in the group G and, therefore, is the translation by some member of Lambda since G is crystallographic. A standard calculation shows that the conjugate of tau_{lambda} by f is tau_{U(lambda)}. Hence, U(lambda) must be in Lambda for each lambda, which means that the image of Lambda under U is contained in Lambda. Applying the same argument to f^{-1} instead of f shows that the image of Lambda under U^{-1} is also a subset of Lambda, and this latter statement is equivalent to the statement Lambda is contained in U(Lambda). Since Lambda and U(Lambda) are contained in each other, they must be equal, as asserted.
The second assertion follows from the simple calculation
This calculation shows that f_{2} \circ f_{1}^{-1} must be both an element of G and a translation and, therefore, since G is crystallographic, the translation by a member of Lambda.
Theorem. If a wallpaper group G contains a rotation, then the angle of the rotation must be an integer multiple of one of the angles pi/3 or pi/2.
Proof. The use of the term ``wallpaper'' means that the ambient space has dimension 2. One may assume that G is a wallpaper group in the group of all isometries of R^{2}. Let Lambda be the point lattice in R^{2} containing the origin whose associated translation lattice is the translation subgroup of G. If f(x) = U x + b is a rotation in G, then the angle theta of rotation is apparent from the rotation matrix
| cos theta | - sin theta |
| sin theta | cos theta |
) .
Since U(Lambda) = Lambda by the preceding proposition, there must be an integer matrix M (the matrix of x -> U x relative to a basis of Lambda) that is similar to M. Consequently, inasmuch as similar matrices have the same trace, 2 cos theta, the trace of U, must be an integer, and, therefore, cos theta must be half an integer. Hence, cos theta = 0, {+/-} 1/2, or {+/-} 1.
Up to isomorphism determine all crystallographic groups in the group of isometries of the real line R^{1}.
What is the translation subgroup of the smallest group of isometries of the Euclidean plane that contains the four reflections in the sides of a given square?
What is the translation subgroup of the group of all isometries of the Euclidean plane that leaves invariant the smallest lattice containing the three vertices of a given equilateral triangle?
What is the translation subgroup of the smallest group of isometries of the Euclidean plane that contains the reflections in the three sides of a given equilateral triangle?
Let G be the smallest group of isometries that contains the reflections in the three sides of the triangle in R^{2} with vertices at the points (0, 0), (1, 0), and ( cos (pi/5), sin (pi/5)).
Show that G contains the rotation about the origin through the angle 2pi/5.
Explain why G cannot possibly be a wallpaper group.
How are the polynomial t^{5} - 1 and the number SQRT{5} related to the geometry of this exercise?
How many non-parallel lines can you find for which G contains translations parallel to the line with length no more than twice the length of the longest of the translational components of the six glide reflections formed by composing the three given reflections?