First consider the dual to Minkowski spacetime. Recall that the invariant interval of Minkowski spacetime has one dimension of time with three dimensions of space:

(*ds*)² = (*c dt*)² – (*dx _{1}*)² – (

*dx*)² – (

_{2}*dx*)² = (

_{3}*c dt*)² – (

*dr*)²

where *t* is the time coordinate and *x _{1}, x_{2}, *and

*x*are space coordinates of

_{3}*r*.

This could be called *temporal spacetime* since speeds and other ratios are referenced to duration, i.e., they have units of *time* in the denominator. The dual could be called *spatial spacetime* since measures of movement are referenced to distance, i.e., they have units of *length* in the denominator. In that case the dual invariant interval is:

(*ds*)² = (*dr*)² – (*c dt _{1}*)² – (

*c dt*)² – (

_{2}*c dt*)² = (

_{3}*dr*)² – (

*c dt*)²

where *r* is the space coordinate and *t _{1}*,

*t*, and

_{2}*t*are time coordinates of

_{3}*t*.

Here’s how one might put together the two four-dimension spacetimes into one six-dimension spacetime invariant interval:

(*ds*)² = (*c dt _{1}*)² + (

*c dt*)² + (

_{2}*c dt*)² – (

_{3}*dx*)² – (

_{1}*dx*)² – (

_{2}*dx*)² = (

_{3}*c dt*)² – (

*dr*)²

= (*c dt _{1}*)² – (

*dx*)² + (

_{1}*c dt*)² – (

_{2}*dx*)² + (

_{2}*c dt*)² – (

_{3}*dx*)²

_{3}where the three dimensions of direction are the same for space and time. The six dimensions are in two groups of three dimensions, i.e., there are 2 × 3 dimensions or three *complex* *dimensions*.

Minkowski and dual Minkowski spacetime have 10 symmetries each. Six-dimension spacetime has 6 translations (one for each dimension), 6 rotations (along the *x _{1}-x_{2}, x_{2}-x_{3}, x_{3}-x_{1}, t_{1}-t_{2}, t_{2}-t_{3}*, and

*t*planes), and 3 Lorentz boosts (about the

_{3}-t_{1}*t*and

_{1}-x_{1}, t_{2}-x_{2},*t*planes) for a total of 15 symmetries.

_{3}-x_{3}