We know that the 3-sphere can be expressed with two different sets of parametric equations. There's the standard one:

sin a sin b cos c

sin a sin b sin c

sin a cos b

cos a,

and one based on the Hopf fibration:

cos a cos b

cos a sin b

sin a cos c

sin a sin c

Clearly a similar thing can be done with higher dimensional spheres, as well as rotatopes and toratopes that involve these spheres. The Hopf coordinates can be viewed as a special case of the tiger with both minor radii equal to zero.

Here's a conjecture. There is a unique set of Hopf spherical coordinates for each nD toratope that does not contain a lonely '1'. So for example, in 6D we have Hopf coordinates based on the following toratopes:

(IIIIII)

((IIII)(II))

(((II)(II))(II))

((III)(III))

((II)(II)(II))

The (((II)(II))(II)) Hopf coordinates in 6D are:

cos a cos b cos c

cos a cos b sin c

cos a sin b cos d

cos a sin b sin d

sin a cos e

sin a sin e