# On the convergence of multicomplex M-J sets to the Steinmetz hypersolids

### Andrzej Katunin

Journal of Applied Mathematics and Computational Mechanics |
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ON THE CONVERGENCE OF MULTICOMPLEX M-J SETS TO THE STEINMETZ HYPERSOLIDS

Andrzej Katunin

Institute of Fundamentals of Machinery
Design, Silesian University of Technology

Gliwice, Poland

andrzej.katunin@polsl.pl

**Abstract.** In
this paper, the analysis of generalized multicomplex Mandelbrot-Julia
(henceforth abbrev. M-J) sets is performed in terms of their shape when a
degree of an iterated polynomial tends to infinity. Since the multicomplex
algebras result from a tensor product of complex algebras, the dynamics of
multicomplex systems described by iterated polynomials is different with
respect to their complex and hypercomplex analogues. When the
degree of an iterated polynomial tends to infinity the M-J sets tend to the
higher dimensional generalization of the Steinmetz solid, depending on the
dimension of a vector space, where a given generalization of M-J sets is
constructed. The paper describes a case of
bicomplex M-J sets with appropriate visualizations as well as a tricomplex one,
and
the most general case - the muticomplex M-J sets, and their corresponding
geometrical convergents.

*Keywords: **generalized M-J sets,
multicomplex M-J sets, convergence, Steinmetz hypersolid*

1. Introduction

The Mandelbrot set and corresponding Julia
sets were intensively studied
during the 80s of the XX century and many fascinating properties of these sets
were observed to date. Considering that M-J sets are defined on a complex plane
**C** in the form of a quadratic polynomial:

(1) |

nothing stands in a way of generalization of
these sets both in terms of degree
of the polynomial as well as in terms of a vector space in which it is
constructed. The first generalization of complex M-J sets to quaternionic **H**
ones was defined
by Holbrook [1], and then further developed and analysed by several authors
[2-6]. Then, Wang and Sun [7] proposed a generalization of quaternionic M-J
sets
in terms of a degree of an iterated polynomial *p*:

(2) |

However, these studies show that the dynamics of
quaternionic maps is trivial
with respect to their complex analogues. The same can be observed for the
further generalization - the M-J sets in octonions **O**, introduced and
studied by Griffin
and Joshi in the early 90s of the XX century [8-10].

An alternative to the above-mentioned
generalization was proposed by Rochon and his team. In 2000, Rochon introduced
a generalization of M-J sets to a bicomplex vector space [11], further several
studies on properties of this type of M-J sets [12-14] as well as their
generalization to tricomplex [15] and multicomplex [16] analogues were
proposed. The generalized version of bicomplex M-J sets with
respect to a degree of an iterated polynomial was also studied by Zireh [17],
and Wang and Song [18]. The performed studies of the above-mentioned teams
show, in general, the different dynamics with respect to hypercomplex M-J sets
which
reflect, in particular, in varying character of changing a shape of these sets
for varying values of a constant *c* in an iterated polynomial of type
(2).

In the following study, the convergents of
the multicomplex M-J sets, i.e. the sets for , are analysed. The study starts with
the simplest case - the bicom-
plex M-J sets, through the tricomplex ones, and ends with the most general case
*n*-complex or multicomplex M-J sets. In each of the considered cases it
was shown that the shapes of the convergents of M-J sets in multicomplex vector
spaces tend to higher-dimensional generalizations of a Steinmetz solid.

2. Preliminaries to multicomplex algebras

Let us begin with the preliminaries of
multicomplex vector spaces from the sim-
plest case - a 4-algebra of bicomplex numbers . For simplicity, the following notation
is introduced: **C**_{2},
where the lower index denotes a number of tensor product operations on complex
algebras (thus **C**_{2} denotes
an algebra of bicomplex numbers,** C**_{3} denotes an algebra of tricomplex numbers, etc.). The
bicomplex
numbers can be expressed in the symbolic representation as follows:

, | (3) |

where are the associators, , and are the imaginary units with the
following interrelations: , , .
Since **C**_{2} is
commutative and considering the existence of idempotents for **C**_{2} (which follows from the
definition of bicomplex numbers):

, | (4) |

where (or just **C**
since **C** ≡ **C**_{1}),
and are
idempotents which means that during multiplication_{ }_{ }and_{ } do not change the initial
result: , , , . **C**_{2} is closed under addition and
multiplication operations (which are necessary to perform
an iteration of a polynomial of type (2)), and these operations can be
performed element-wise. Considering that , , , ,
the addition and multiplication is defined as follows:

, | (5) |

. | (6) |

The next generalization of **C**_{2} is an 8-algebra of tricomplex
numbers **C**_{3} with
the following symbolic representation [15]:

, | (7) |

with the
following interrelations between imaginary units: , , , , . Similarly to **C**_{2}, the tri-complex numbers can be
presented as a pair of bicomplex elements:

, , | (8) |

where and are idempotents, or in a form of a quadruple of complex elements:

, , | (9) |

, , , , the addition and multiplication of two tricomplex numbers and can be performed element-wise:

, | (10) |

. | (11) |

The above expressions can be generalized to a
hypercomplex number space **C*** _{n}*, which is

*n*-tensor product 2

*-algebra with a following symbolic representation [19]:*

^{n}, | (12) |

where , or alternatively:

, | (13) |

where , . Thus, every multicomplex number in **C*** _{n}* contains 2

*elements with the associators defined in*

^{n}**R**, or equivalently 2

*elements defined in*

^{n–m}**C**

*for .*

_{m}The multicomplex algebra is commutative and idempotent representation of multicomplex number has the form [16]:

, | (14) |

where and are
idempotents. Extending (5)-(6) and (10)-(11) to the case of **C*** _{n}* the addition and
multiplication operations of two multi-complex numbers and can be performed element-wise
as:

, | (15) |

. | (16) |

3. Multicomplex M-J sets

Having defined multicomplex algebras and basic operations on multicomplex numbers, one can define M-J sets as follows:

for , | (17) |

and correspondingly:

for . | (18) |

The authors of [16] proved that M sets and
filled J sets defined in **C*** _{n}* are
connected and the escaping-time radius (known also as a bailout values) equals
2. Moreover, the connectedness of J sets defined in

**C**

*are of three types: connected, when , totally disconnected (i.e. homeomorphic to the Cantor dust) when and , where is a strong basin of attraction at infinity (see details in [16]), and disconnected, but not totally in all other cases. More details on connectedness and other properties of M-J sets defined in*

_{n}**C**

*can be found in [16].*

_{n}4. Convergence analysis

When changing a degree *p* of an
iterated polynomial of type (2) significant changes in a shape of the resulting
M-J sets can be observed for the small values
of *p*. However, when , a shape of these sets tends to a specific geometrical shape.
An example of such behavior for is presented in
Figure 1.

(a) ; (b) ; (c) ;

(d) ; (d) ; (f) ;

Fig. 1. 3D
projections of bicomplex J sets for *c* =
(–0.5,0.5,0,0) and various values of *p*

**Theorem 1.** The
generalized bicomplex M-J sets of type (2) tend to a 4-dimen-sional Steinmetz
solid when_{ }.

Before proceeding
with the proof of the Theorem 1, it is necessary to define
the generalized *n*-dimensional Steinmetz solid or the Steinmetz
hypersolid.

**Definition 1. **We
say that the Steinmetz hypersolid is
an *n*-dimensional solid resulting from an intersection of *Q* *n*-cylinders
(*Q* ≥ 2) of equal radii *r* denoted as ,
where , is a
*q*-th coordinate axis and is a unitary basis
in **R*** ^{n}* unique for each

*q*, and being their common:

. | (19) |

*Proof.* Suppose *c* = 0
in (2) defined in bicomplex numbers which is equivalent to the case when . Then (2) takes
a form:

, , . | (20) |

Recalling representation of a product of two bicomplex numbers by pairs of complex numbers (6), one can present (20) in the following form [20]:

, | (21) |

where , *x*_{1,…,4} are the coordinates in **C**_{2}. Following this, we can
rewrite (20) in the form:

, | (22) |

and
considering that **C**_{2} results
from a tensor product , the
boundary of sets of prisoner points, i.e. the M and “filled” J sets are given
by a common of intersection of two 3-cylinders:

(23) |

where *r*
= 1. In the limit case (when_{ }) the resulting set does not have fractal properties. The system (23) describes_{ }.

**Corollary 1. **One
can extend Theorem 1 as follows: The generalized tricomplex M-J sets of type
(2) tend to 8-dimensional Steinmetz solid when_{ }.

*Proof.*
Considering Definition 1 and a recursive equation (20) defined in **C**_{3},
one can express such an equation in terms of quadruple of complex numbers
as follows:

. | (24) |

By analogy
to the proof of Theorem 1, the tricomplex M-J sets tend to_{ }_{ }given
by:

(25) |

when_{ }.

**Corollary 2. **One
can also extend Theorem 1 to the most general form: The generalized
multicomplex M-J sets of type (2) tend to an *n*-dimensional Steinmetz
solid when_{ }.

*Proof.*
Considering Definition 1 and a recursive equation (20) defined in **C*** _{n}* one can express such
equation in terms of

*n*/2-tuple of complex numbers. By analogy to the proofs of Theorem 1 and Corollary 1, the multicomplex M-J sets tend to

_{ }given by:

(26) |

when .

5. Conclusions

In the presented study, the convergents of
multicomplex M-J sets were investigated. Starting from the simplest case - the
bicomplex algebra, it was shown that
in contrast to hypercomplex generalizations of M-J sets [21], the multicomplex ones tend to higher dimensional Steinmetz
solids (with_{ }) which, in a general case, results from the intersection of *n*/2
*n*-cylinders in **C**_{n}_{ }.

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