The Group of Translations of The Plane
A translation of the plane is a rigid movement of every point of the plane for a certain distance in a certain direction. For instance "move in the North-East direction for 2 miles" is a translation of the plane. If you have two such translations a and b, they can be composed to form a new translation a ∘ b as follows: first follow the prescription of b, then that of a. For instance, if
- a = "move North-East for 3 miles"
and
- b = "move South-East for 4 miles"
then
- a ∘ b = "move East for 5 miles"
(see Pythagorean theorem for why this is so, geometrically).
The set of all translations of the plane with composition as operation forms a group:
- If a and b are translations, then a ∘ b is also a translation.
- Composition of translations is associative: (a ∘ b) ∘ c = a ∘ (b ∘ c).
- The identity element for this group is the translation with prescription "move zero miles in whatever direction you like".
- The inverse of a translation is given by walking in the opposite direction for the same distance.
This is an Abelian group and our first (nondiscrete) example of a Lie group: a group which is also a manifold.
Read more about this topic: Examples Of Groups
Famous quotes containing the words group, translations and/or plane:
“Unless a group of workers know their work is under surveillance, that they are being rated as fairly as human beings, with the fallibility that goes with human judgment, can rate them, and that at least an attempt is made to measure their worth to an organization in relative terms, they are likely to sink back on length of service as the sole reason for retention and promotion.”
—Mary Barnett Gilson (1877?)
“Woe to the world because of stumbling blocks! Occasions for stumbling are bound to come, but woe to the one by whom the stumbling block comes!”
—Bible: New Testament, Matthew 18:7.
Other translations use temptations.
“As for the dispute about solitude and society, any comparison is impertinent. It is an idling down on the plane at the base of a mountain, instead of climbing steadily to its top.”
—Henry David Thoreau (18171862)