Symmetry makes groups
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Ayuda a traducir →La pregunta
A square can be turned or flipped and still fit exactly into its outline. How many different moves like that are there? And what do you get if you do one move and then another?
Prueba el experimento antes de seguir leyendo. No hay una forma equivocada de explorar.
Experimento
Explorador de figuras
Elige una figura y activa sus características: lados, vértices, ángulos, simetría y diagonales.
CuadradoCuadrilátero
- Ejes de simetría
- 4
- Simetría de rotación
- Orden 4
Cuatro lados iguales y cuatro ángulos rectos. Un cuadrado es un rectángulo especial y también un rombo especial.
Cada línea discontinua es un eje de simetría: si doblas por ella, las dos mitades coinciden exactamente.
Pistas, si las quieres
Qué está pasando
Qué está pasando
A square has exactly eight symmetries: turns through , , and , and flips in its two diagonals and its two mid-lines. These moves have four properties:
- doing one symmetry and then another always gives a symmetry;
- "do nothing" is a symmetry;
- every symmetry can be undone by another symmetry;
- for three moves, it does not matter how you group them: equals .
Any collection of moves with these four properties is called a group. The symmetries of the square form the dihedral group of order . Order matters: a quarter turn followed by a flip is usually not the same as the flip followed by the quarter turn, so this group is not commutative. A regular polygon with sides has symmetries; a circle has infinitely many, one for every angle of turning.
Groups are the mathematics of symmetry everywhere. There are exactly space groups describing the possible symmetries of crystals; Évariste Galois used groups to explain why there is no formula for the roots of a general polynomial of degree five; and physicists classify elementary particles by the symmetry groups of the laws they obey. Groups of symmetries that vary continuously, like turning by any angle, are Lie groups, named after Sophus Lie.
Adónde lleva esto
- Universidad avanzada · Lección disponibleGroupsGroups and subgroups, cyclic, dihedral and symmetric groups, cosets and Lagrange's theorem, homomorphisms and normal subgroups, and group actions with the orbit–stabiliser theorem.
- Universidad avanzada · Lección disponibleMatrix groups and continuous symmetryGroups of matrices as both algebraic and geometric objects: O(n), SO(n), U(n), SU(2), SL(2, ℝ) and the Euclidean group; rotations and quaternions, the double cover SU(2) → SO(3), the matrix exponential, one-parameter subgroups and the tangent space at the identity.
- Primer año de posgrado · Lección disponibleLie groups and Lie algebrasGroups that are also smooth manifolds: Lie groups and their Lie algebras, the exponential map, the correspondence between Lie subgroups and subalgebras, the adjoint representation, compact groups and Haar measure, representations of sl(2, ℂ) and a first look at root systems.