Quaternions & the Mandelbrot set

Quaternions have 4 real components Q = (a + b i + c j + d k) or (a 1 + b i + c j + d k) compared to the two of complex numbers. (1, i, j, k) make a group and satisfy the rules:
    i2 = j2 = k2 = -1,
    i j = - j i = k,     j k = - k j = i,     k i = - i k = j.
Operations such as addition and multiplication can be performed on quaternions, but multiplication is not commutative.

The Mandelbrot set may be generalized by quaternion numbers
    Qn+1 = Qn2 + Qc,   Q0 = 0,  Qc = (cr , ci , cj , ck ).
As since (a + b i + c j + d k)2 = a2 - (b2 + c2 + d2) + 2ab i + 2ac j + 2ad k
due to terms like bc i j and cb j i cancel each other (bc i j = -cb j i). We get
    a = a2 - (b2 + c2 + d2) + cr
    b = 2ab + ci
    c = 2ac + cj
    d = 2ad + ck
    a2 + b2 + c2 + d2 < BailOut2

The Quaternions Mandelbrot set is a 4 dimensional object. For Qc = (a, b, 0, 0 ) all points belongs to 2D (a, b ) plane and the set coincides with the classical M-set. For Qc =(a, b, c, 0 ) we get the 3D M-set.
We can rewrite the last equations as
    a = a2 - |r|2 + cr
    r = 2ar + c
    a2 + |r|2 < BailOut2
where r = (b, c, d) and c = (ci , cj , ck) are 3 dimensional vectors. By rotations of coordinate system any 3D vector c may be put into (|c|, 0, 0) or Qc = (a, |c|, 0, 0) - the classical Mandelbrot plane. Rotations conserve |r|. As since dynamic of the a - component and exiting condition depend on |r| only, therefore the QM-set is spherical symmetric in (b, c, d) subspace.

Due to the spherical symmetry we can get (a, b, c) 3D QM-set rotating the classical M-set relatively the real axis. The 3D QMandelbrot cactus consists of spheres and toruses so its cross-sections are circles and ellipses. Here you see the QM-set cross sections by (Re, Im, c, 0 ) planes for different c.

A few more questions:
Ouaternions generators (1, i, j, k) group has tight connection with the group of 3D space rotations. Is the QM-set symmetry consequence of this connection? Will we get similar symmetry for other generators groups? E.g. hyperbolic rotation symmetry for Dirac's (1, γ0, γ1, γ2, γ3) matrixes for which
    γi γj = - γj γi,    (γ0)2 = 1,    (γ1)2 = (γ2)2 = (γ3)2 = -1

Sorry that quaternions are a bit "alien" here. Further we will return to the complex quadratic mappings.


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updated 2 April 2000