Course MT3818 Topics in Geometry

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## Summary of Results

A Geometry consists of a set S and a subgroup G of the group Bij(S) of all bijections from S to itself.

For Euclidean geometry, S = Rn and the group is the set I(Rn) of all length preserving maps or isometries.

Every element of I(Rn) is of the form TaL with Ta a translation xa + x and L a length preserving linear map ∈ O(n).
Then det(L) = ±1. If det(L) = +1 the isometry is direct, otherwise it is opposite.

If f is a symmetry of the line: fI(R), then f(x) = a + x or f(x) = a - x and f is either a translation or reflection (in a point).

Direct isometries of the plane: I(R2) are either rotations (about a point) or translations.
Opposite symmetries of R2 are either reflections (in a line) or glides (a reflection in a line followed by translation parallel to the line).

Direct isometries of R3 are either rotations (about an axis in R3) or translations or screws (a rotation about a line followed by translation parallel to the line).
Opposite symmetries of R3 are either reflections (in a plane) or glides (a reflection in a plane followed by translation parallel to the plane) or rotatory reflections (rotaion about a line followed by reflection in a plane perpendicular to the line).

A symmetry group of a figure F ⊆ Rn is the set of all symmetries f in I(Rn) for which f(F) = F.

Finite subgroups of I(R2) are isomorphic either to Cn: a cyclic group of order n (generated by a rotation by 2π /n about some point) or Dn: a dihedral group of order 2n (generated either by a similar rotation and one reflection in a line through the point, or by two such reflections) and this classifies the subgroups up to conjugacy inside the group I(R2).

Finite subgroups of I(R3) which consist only of direct symmetries are isomorphic either to Cn (generated by a rotation by 2π /n about some axis) or Dn (a dihedral group of order 2n consisting of rotations as in the Cn case and rotations by π) or to the rotational symmetry groups of one of the Platonic solids (A4 , S4 or A5).

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JOC March 2003