![]() ![]() For each choice of L, antipodal points R and – R Left-multiplication with elementsįrom a path around the other, full copy of SU(2) gives a net rotation of Since they leave the identity vector unchanged. These are effectively three-dimensional rotations, ![]() In the second view, the half-copy of SU(2) is used to produce rotations Halves of both copies of SU(2) would be redundant. Is given by the function T( v)= L v R, and since (– L) v(– R)= L v R, using both Reveals a 4-dimensional swath of the whole 6-dimensional manifold. Path that wraps around another copy of the group (not shown). To reveal the interior – and left-multiplication with elements taken from a Taken from one half of SU(2) – drawn here as a 3-dimensional ball, with a wedge removed In theįirst view, rotations are produced by right-multiplication with elements Multiplication by a fixed element of SU(2) rotates the whole 3-sphere,Īnd any rotation can be produced by a combination of left and right multiplication.Ĭlicking on the applet alternates between two different views of SO(4). Multiplication, and comprises a group known as SU(2). Surface of the four-dimensional unit sphere in the quaternions is closed under The hypercube’s 24 faces buried in the interior of the dodecahedron.įour-dimensional Euclidean space can be equipped with a rule for multiplying vectors,įorming an algebraic structure known as the quaternions. Projects down to three dimensions as an irregular rhombic dodecahedron, with half SO(4) portrays the group of rotations in four dimensions,īy showing the effect of various rotations on a hypercube. ![]()
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