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:
“Once it was a boat, quite wooden
and with no business, no salt water under it
and in need of some paint. It was no more
than a group of boards. But you hoisted her, rigged her.
She’s been elected.”
—Anne Sexton (1928–1974)
“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.”
“At the moment when a man openly makes known his difference of opinion from a well-known party leader, the whole world thinks that he must be angry with the latter. Sometimes, however, he is just on the point of ceasing to be angry with him. He ventures to put himself on the same plane as his opponent, and is free from the tortures of suppressed envy.”
—Friedrich Nietzsche (1844–1900)