[lpetrich]

The quaternion versions of the 3D rotation groups are very interesting-looking.

The cyclic one:
QC = all {cos(a), 0, 0, sin(a)}
where a finite one can have an odd number of a's as well as an even number.

QC(2m) -> C(m)
QC(2m+1) -> C(2m+1)

The dihedral one:
QD = QC + all {0, cos(a), sin(a), 0}
where a finite one must have an even number of a's.

QD(m) -> D(m)

QD(2) is {+-1, 0, 0, 0}, {0, +- 1, 0, 0}, {0, 0, +-1, 0}, {0, 0, 0, +-1}

Tetrahedral: QT = QD(2) + all sign combinations of {+-1, +-1, +-1, +-1}/2

Octahedral: QO = QT + all permutations and sign combinations of {+-1, +-1, 0, 0}/sqrt(2)

Icosahedral: QI = QT + all *even* permutations and all sign combinations of {+- (sqrt(5)+1)/4, +- 1/2, +- (sqrt(5)-1)/4, 0}