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 advanced_tools:group_theory [2018/04/13 07:42]bogumilvidovic [Why is it interesting?] advanced_tools:group_theory [2020/04/02 13:01] (current)130.246.243.49 Fix degree symbols Both sides previous revision Previous revision 2020/04/02 13:01 Fix degree symbols2018/04/13 07:42 bogumilvidovic [Why is it interesting?] 2018/04/08 15:15 georgefarr ↷ Links adapted because of a move operation2018/03/28 14:25 jakobadmin [Abstract] 2018/03/28 14:24 jakobadmin 2018/03/28 14:23 jakobadmin [Intuitive] 2018/03/28 14:23 jakobadmin [Intuitive] 2018/03/28 14:22 jakobadmin [Intuitive] 2018/03/28 14:18 jakobadmin [Concrete] 2018/03/28 14:18 jakobadmin [Concrete] 2018/03/28 14:18 jakobadmin [Concrete] 2018/03/28 14:07 jakobadmin [Concrete] 2018/03/28 14:07 jakobadmin 2018/03/28 14:06 jakobadmin [Student] 2018/03/28 11:19 jakobadmin [Overview] 2018/03/28 10:07 jakobadmin [Researcher] 2018/03/27 05:18 jakobadmin [Overview] 2018/03/26 11:33 jakobadmin [Overview] 2018/03/26 11:30 jakobadmin [Overview] 2018/03/26 09:36 jakobadmin [Overview] 2018/03/26 09:33 jakobadmin [Overview] 2018/03/26 09:32 jakobadmin [Overview] 2018/03/26 09:25 jakobadmin [Overview] 2018/03/24 13:40 jakobadmin [Why is it interesting?] 2018/03/24 13:40 jakobadmin [Why is it interesting?] 2018/03/24 13:02 jakobadmin 2018/03/24 13:00 jakobadmin 2020/04/02 13:01 Fix degree symbols2018/04/13 07:42 bogumilvidovic [Why is it interesting?] 2018/04/08 15:15 georgefarr ↷ Links adapted because of a move operation2018/03/28 14:25 jakobadmin [Abstract] 2018/03/28 14:24 jakobadmin 2018/03/28 14:23 jakobadmin [Intuitive] 2018/03/28 14:23 jakobadmin [Intuitive] 2018/03/28 14:22 jakobadmin [Intuitive] 2018/03/28 14:18 jakobadmin [Concrete] 2018/03/28 14:18 jakobadmin [Concrete] 2018/03/28 14:18 jakobadmin [Concrete] 2018/03/28 14:07 jakobadmin [Concrete] 2018/03/28 14:07 jakobadmin 2018/03/28 14:06 jakobadmin [Student] 2018/03/28 11:19 jakobadmin [Overview] 2018/03/28 10:07 jakobadmin [Researcher] 2018/03/27 05:18 jakobadmin [Overview] 2018/03/26 11:33 jakobadmin [Overview] 2018/03/26 11:30 jakobadmin [Overview] 2018/03/26 09:36 jakobadmin [Overview] 2018/03/26 09:33 jakobadmin [Overview] 2018/03/26 09:32 jakobadmin [Overview] 2018/03/26 09:25 jakobadmin [Overview] 2018/03/24 13:40 jakobadmin [Why is it interesting?] 2018/03/24 13:40 jakobadmin [Why is it interesting?] 2018/03/24 13:02 jakobadmin 2018/03/24 13:00 jakobadmin 2018/03/24 12:59 jakobadmin 2018/03/24 12:58 jakobadmin [Overview] 2018/03/24 12:58 jakobadmin [Overview] 2018/03/24 12:57 jakobadmin [Overview] 2018/03/24 12:57 jakobadmin [Overview] 2018/03/24 12:49 jakobadmin [Overview] 2018/03/24 12:47 jakobadmin [Overview] 2018/03/24 12:46 jakobadmin [Overview] 2018/03/24 10:49 jakobadmin [Overview] 2018/03/24 10:44 jakobadmin [Overview] 2018/03/24 10:43 jakobadmin [Overview] 2018/03/24 10:43 jakobadmin [Overview] 2018/03/24 10:42 jakobadmin [Overview] 2018/03/24 10:39 jakobadmin 2018/03/24 10:38 jakobadmin [Why is it interesting?] 2018/03/24 10:37 jakobadmin [Why is it interesting?] 2018/03/24 10:37 jakobadmin [Why is it interesting?] 2018/03/24 10:00 jakobadmin [Why is it interesting?] 2018/03/24 10:00 jakobadmin [Why is it interesting?] 2018/03/21 12:25 jakobadmin [Why is it interesting?] 2018/03/21 11:08 jakobadmin [Why is it interesting?] 2018/03/21 11:08 jakobadmin [Why is it interesting?] 2018/03/21 11:01 jakobadmin [Why is it interesting?] 2018/03/21 10:57 jakobadmin [Why is it interesting?] Line 10: Line 10: A square is defined mathematically as a set of points. A symmetry of the square is a transformation that maps this set of points into itself. This means concretely that by the transformation,​ no point is mapped to a point outside of the set that defines the square. A square is defined mathematically as a set of points. A symmetry of the square is a transformation that maps this set of points into itself. This means concretely that by the transformation,​ no point is mapped to a point outside of the set that defines the square. - Obvious examples of such transformations are rotations, by $90^\circ$, $180^\circ$,​ $270^\circ$,​ and of course $0^\circ$. + Obvious examples of such transformations are rotations, by $90^{\circ}$, $180^{\circ}$, $270^{\circ}$, and of course $0^{\circ}$. {{ :​advanced_tools:​einheitsquadrat-gedreht22-150x150.png?​nolink|}} {{ :​advanced_tools:​einheitsquadrat-gedreht22-150x150.png?​nolink|}} - A counter-example is a rotation by, say $5^\circ$. The upper-right corner point $A$ of the square is obviously mapped to a point $A'$ outside of the initial set. Of course, a square still looks like a square after a rotation by $5^\circ$, but, by definition, this is a different square, mathematically a different set of points. Hence, a rotation by $5^\circ$ is no symmetry of the square. + A counter-example is a rotation by, say $5^{\circ}$. The upper-right corner point $A$ of the square is obviously mapped to a point $A'$ outside of the initial set. Of course, a square still looks like a square after a rotation by $5^\circ$, but, by definition, this is a different square, mathematically a different set of points. Hence, a rotation by $5^{\circ}$ is no symmetry of the square. - A characteristic property of the symmetries of the square is that the combination of two transformations that leave the square invariant is again a symmetry. For example, combining a rotation by $90^\circ$ and $180^\circ$ is equivalent to a rotation of $270^\circ$,​ which is again a symmetry of the square. We will elaborate on this in the next post. In fact, the basic axioms of group theory can be derived from such an easy example. + A characteristic property of the symmetries of the square is that the combination of two transformations that leave the square invariant is again a symmetry. For example, combining a rotation by $90^{\circ}$ and $180^{\circ}$ is equivalent to a rotation of $270^{\circ}$, which is again a symmetry of the square. We will elaborate on this in the next post. In fact, the basic axioms of group theory can be derived from such an easy example. ---- ----