Does 4d exist

Answer 1

Yes! There is one real (distance) dimension r=ct and three vector (displacement) dimensions called vectors ,V=Ix + Jy + Kz. Together they make up a 4D space called Quaternions [r,V] = [r, ix +Jy + Kz].

The Cosmos consists of these Quaternions. Most variables in Physics are Quaternions consisting of a real (small letter) and a vector (Capital letter) .e.g. [a,A]

Consider two Quaternions [a,A] and [b.B]

Addition: [a,A] + [b,B] = [a +b, A +B]

Multiplication: [a,A] [b,B] = [ab -A.B, aB + Ab + AxB]

Multiplication includes Complex Number multiplication.

A.B indicates vector parallel product |AB|cos(AB) and A.B=0 if the vectors A and B are perpendicular,e.g I.J=0.

AxB indicates vector perpendicular product |AB|sin(AB) and AxB=0 if the vectors A and B are parallel, e.g IxI=0.

Complex Numbers are two dimensional and only have one vector I, C=[a,ib] where 'i' indicates the singular vector. Quaternions contain the real numbers and the Complex Numbers as subsets of the 4D Quaternions. The three dimensions of space are vector dimensions and the real dimension is r=ct.

Mathematically, the real dimension is critical to making mathematics and Physics work.