# 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:

**i**^{2} = **j**^{2} =
**k**^{2} = -**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

*Q*_{n+1} = Q_{n}^{2} + Qc,
Q_{0} = 0, Qc = (c_{r} , c_{i} ,
c_{j} , c_{k} ).

As since (*a + b ***i*** + c ***j*** +
d ***k**)^{2} = a^{2} - (b^{2} +
c^{2} + d^{2}) + 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 = a*^{2} - (b^{2} +
c^{2} + d^{2}) + c_{r}

b = 2ab + c_{i}

c = 2ac + c_{j}

d = 2ad + c_{k}

a^{2} + b^{2} +
c^{2} + d^{2} < BailOut^{2}

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 = a*^{2} - |**r**|^{2}
+ c_{r}

**r** = *2a***r** + **c**

*a*^{2} + |**r**|^{2} <
BailOut^{2}

where **r** = (*b, c, d*) and **c** = (*c*_{i} ,
c_{j} , c_{k}) 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, γ**^{0},
γ^{1},
γ^{2},
γ^{3}) matrixes for which

**γ**^{i} γ^{j} =
- γ^{j} γ^{i},
(γ^{0})^{2} = 1,
(γ^{1})^{2} =
(γ^{2})^{2} =
(γ^{3})^{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