The Pauli matrices ( after multiplication by i to make them anti-hermitian ), also generate transformations in the sense of Lie algebras: the matrices form a basis for, which exponentiates to the spin group, and for the identical Lie algebra, which exponentiates to the Lie group of rotations of 3-dimensional space.

In particular, this vector field is a Killing vector field belonging to an element of the Lie algebra so ( 3 ) of the 3-dimensional rotation group SO ( 3 ).

There is a natural 2-to-1 homomorphism from the group of unit quaternions to the 3-dimensional rotation group described at quaternions and spatial rotations.

It is also diffeomorphic to the real 3-dimensional projective space RP < sup > 3 </ sup >, so the latter can also serve as a topological model for the rotation group.

Since unit quaternions can be used to represent rotations in 3-dimensional space ( up to sign ), we have a surjective homomorphism from SU ( 2 ) to the rotation group SO ( 3 ) whose kernel is

This is the only model geometry that cannot be realized as a left invariant metric on a 3-dimensional Lie group.

The group G has 2 components, and is a semidirect product of the 3-dimensional Heisenberg group by the group O ( 2, R ) of isometries of a circle.

Moreover if the volume does not have to be finite there are an infinite number of new geometric structures with no compact models ; for example, the geometry of almost any non-unimodular 3-dimensional Lie group.

* rotations in space ( this forms the non-Abelian Lie group of 3-dimensional rotations )

This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in navigation, nautical engineering, and aerospace engineering, among many other uses.

For example, if one considers a Chern – Simons theory with gauge group G on a manifold with boundary then all of the 3-dimensional propagating degrees of freedom may be gauged away, leaving a 2-dimensional conformal field theory known as a G Wess – Zumino – Witten model on the boundary.

A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e. g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translate of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previous sense.

A prism manifold is a closed 3-dimensional manifold M whose fundamental group

For example, projective representations of the 3-dimensional rotation group, which is the special orthogonal group SO ( 3 ), are ordinary representations of the special unitary group SU ( 2 ).

The group PSL ( 2, C ) of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic space H < sup > 3 </ sup >, and as orientation preserving conformal maps of the open unit ball B < sup > 3 </ sup > in R < sup > 3 </ sup > to itself.

The theory of spherical functions for the Lorentz group, required for harmonic analysis on the 3-dimensional hyperboloid in Minkowski space, or equivalently 3-dimensional hyperbolic space, is considerably easier than the general theory.

The second part of the problem asks whether there exists a polyhedron which tiles 3-dimensional Euclidean space but is not the fundamental region of any space group ; that is, which tiles but does not admit an isohedral ( tile-transitive ) tiling.

One example is the Banach – Tarski paradox which says that it is possible to decompose (" carve up ") the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original.

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field.

That is, curvature does not depend on how the surface might be embedded in 3-dimensional space or 2-dimensional space.

Still, in the absence of naked singularities, the universe is deterministic — it's possible to predict the entire evolution of the universe ( possibly excluding some finite regions of space hidden inside event horizons of singularities ), knowing only its condition at a certain moment of time ( more precisely, everywhere on a spacelike 3-dimensional hypersurface, called the Cauchy surface ).

where E is the electric field and δ < sup > 3 </ sup > is the 3-dimensional delta function.

The square is bounded by 1-dimensional lines, the cube by 2-dimensional areas, and the tesseract by 3-dimensional volumes.

In practice, this allows for true 3-dimensional level design that was previously impossible, although the engine is still 2D.

where is the 3-dimensional delta function.

A hypercube is a generalization of a 2-dimensional square, a 3-dimensional cube, and so on to n dimensions.

The parameterization of the 3-dimensional immersion of the bottle itself is much more complicated.

The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.

The first view is the functional view: the input is transformed into a 3-dimensional vector, which is then transformed into a 2-dimensional vector, which is finally transformed into.

In mathematics, with 2-or 3-dimensional vectors with real-valued entries, the idea of the " length " of a vector is intuitive and can easily be extended to any real vector space R < sup > n </ sup >.

The Poincaré conjecture asserts that the same is true for 3-dimensional spaces.

A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces.

Cross-linking between amino acids in different linear amino sugar chains occurs with the help of the enzyme transpeptidase and results in a 3-dimensional structure that is strong and rigid.

Moreover, the algebra generated by the three matrices is isomorphic to the 3-dimensional Euclidean real Clifford Algebra.

Photon mapping is another method that uses both light-based and eye-based ray tracing ; in an initial pass, energetic photons are traced along rays from the light source so as to compute an estimate of radiant flux as a function of 3-dimensional space ( the eponymous photon map itself ).

Note that the ordinary sphere is a 2-sphere, because it is a 2-dimensional surface ( which is embedded in 3-dimensional space ).

A pentacube puzzle or 3D pentomino puzzle, amounts to filling a 3-dimensional box with these 1-layer pentacubes, i. e. cover it without overlap and without gaps.

Parallel transport of polarization vectors along such sphere gives rise to Thomas precession, which is analogous to the rotation of the swing plane of Foucault pendulum due to parallel transport along a sphere S < sup > 2 </ sup > in 3-dimensional Euclidean space.

* 3-dimensional Flat Euclidean geometry, generally notated as E < sup > 3 </ sup >

* 3-dimensional spherical geometry with a small curvature, often notated as S < sup > 3 </ sup >

In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R < sup > 3 </ sup >.

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R < sup > 3 </ sup >, considered up to continuous deformations ( isotopies ).

In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n x n x n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is equal to a single number, the so-called magic constant of the cube, denoted M < sub > 3 </ sub >( n ).

Consider a smooth surface S in 3-dimensional Euclidean space R < sup > 3 </ sup >.

The superspace R < sup > 1 | 2 </ sup > is a 3-dimensional vector space.

For instance, in a 3-dimensional space, the magnitude of 5, 6 is √( 4 < sup > 2 </ sup > + 5 < sup > 2 </ sup > + 6 < sup > 2 </ sup >) = √ 77 or about 8. 775.

* The 3-dimensional sphere S < sup > 3 </ sup >

Now consider Alexander's horned sphere as an embedding into the 3-sphere, considered as the one-point compactification of the 3-dimensional Euclidean space R < sup > 3 </ sup >.

For algebraically closed fields of characteristic p > 0 Lie's theorem holds provided the dimension of the representation is less than p, but can fail for representations of dimension p. An example is given by the 3-dimensional nilpotent Lie algebra spanned by 1, x, and d / dx acting on the p-dimensional vector space k /( x < sup > p </ sup >), which has no eigenvectors.

* The complement to any knot in a 3-dimensional sphere S < sup > 3 </ sup > is of type K ( G, 1 ); this is called the " asphericity of knots ", and is a 1957 theorem of Christos Papakyriakopoulos.

it is a model of 3-dimensional hyperbolic space H < sup > 3 </ sup >.

* 3-dimensional boron frameworks that include boron polyhedra, example NaB < sub > 15 </ sub > with boron icosahedra

In physics, the algebra of physical space ( APS ) is the use of the Clifford or geometric algebra Cℓ < sub > 3 </ sub > of the three-dimensional Euclidean space as a model for ( 3 + 1 )- dimensional space-time, representing a point in space-time via a paravector ( 3-dimensional vector plus a 1-dimensional scalar ).