Symmetry makes groups
Ukurasa huu bado haujatafsiriwa kwa Kiswahili, kwa hiyo unaonyeshwa kwa English.
Saidia kutafsiri →Swali
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?
Fanya jaribio kabla ya kuendelea kusoma. Hakuna njia mbaya ya kuchunguza.
Jaribio
Mchunguzi wa maumbo
Chagua umbo na uwashe sifa zake: pande, vipeo, pembe, ulinganifu na milalo.
MrabaPembenne
- Mistari ya ulinganifu
- 4
- Ulinganifu wa mzunguko
- Daraja 4
Pande nne sawa na pembe nne mraba. Mraba ni mstatili maalum na pia ni rombasi maalum.
Kila mstari wa vistari ni mstari wa ulinganifu: ukikunja kwenye mstari huo, nusu mbili hulingana kabisa.
Vidokezo, ukivitaka
Kinachoendelea
Kinachoendelea
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.
Hii inaelekea wapi
- Shahada ya kwanza (juu) · Somo linapatikanaGroupsGroups and subgroups, cyclic, dihedral and symmetric groups, cosets and Lagrange's theorem, homomorphisms and normal subgroups, and group actions with the orbit–stabiliser theorem.
- Shahada ya kwanza (juu) · Somo linapatikanaMatrix 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.
- Mwaka wa kwanza wa uzamili · Somo linapatikanaLie 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.