- #1
mnb96
- 715
- 5
Hello,
I read somewhere that the set of unit quaternions identifies the [tex]\mathcal{S}^3[/tex] sphere.
This makes sense; however, what happens if we consider instead a quaternion as an element of the even-grade subalgebra [tex]\mathcal{C}\ell^+_{3,0}[/tex] ?
Now a unit quaternion is represented as a scalar-plus-bivector [tex]p+\mathbf{B}q[/tex] which can be written in the form [tex]cos(\alpha)+\mathbf{B}sin(\alpha)[/tex] where [itex]\alpha[/itex] is an angle on the plane B.
So why can´t we consider a quaternion as an element of [tex]\mathcal{S}^2[/tex] instead?
I read somewhere that the set of unit quaternions identifies the [tex]\mathcal{S}^3[/tex] sphere.
This makes sense; however, what happens if we consider instead a quaternion as an element of the even-grade subalgebra [tex]\mathcal{C}\ell^+_{3,0}[/tex] ?
Now a unit quaternion is represented as a scalar-plus-bivector [tex]p+\mathbf{B}q[/tex] which can be written in the form [tex]cos(\alpha)+\mathbf{B}sin(\alpha)[/tex] where [itex]\alpha[/itex] is an angle on the plane B.
So why can´t we consider a quaternion as an element of [tex]\mathcal{S}^2[/tex] instead?