Week 5
Escher's world:
All motions on Euclidean two- and threespace
After last week's excursus into affine space, we now return to Euclidean space.
You have already learned enough to understand classification
theorems for Euclidean motions in two- and threedimensional Euclidean
space!
I will prove that there are only three different types of Euclidean motions
in two dimensions:
Translations, rotations, and glides (also called glide reflections), where
the latter include reflections as a special case.
If time allows, we will take a closer look at the consequences of our
classification theorem for Euclidean motions on E2. I will
explain what a crystallographic group is and show that in
Euclidean twospace there exist seventeen different crystallographic
groups. If you take a close look at
Escher's periodic drawings, then
you will find that he has used all of them in his artwork and that
you can now name and distinguish them!
For motions on Euclidean threespace you will notice that you can
generalize the technique
used in the twodimensional case without great difficulty. It is only
a little more tedious than in the lowerdimensional analog to see that
on E3, every Euclidean motion is either a twist,
a glide, or a rotary reflection, where translations and rotations
are understood as special cases of twists, and reflections in a plane
can be understood as special cases of either glides or rotary relfections.
Helpful literature:
-
M. Reid, Geometry and Topology, chapters of forthcoming
textbook by M. Reid and B. Szendröi, available from General Office:
Sections 2.11 - 2.13 and 3.3.
- H.S.M. Coxeter,
Introduction
to Geometry, John Wiley & Sons:
Sections 4.1-4.5, and table 1 on page 413.
- V.V. Nikulin and I.R. Shafarevich,
Geometries and Groups,
Springer:
Section 12.2.
Note:
Only topics covered in the lectures and exercises
will be asked in the exam.
The list of helpful literature is meant as an aid, also to raise further
interest, but not as an obligation.