7.3. Fractal Geometry and Quotient Space Analysis
7.3.1. Introduction
A famous Sierpinski carpet is shown in Fig. 7.2. It’s a typical fractal graph. From the quotient space view point, it’s a chain of quotient spaces, where 
has one element, 
has four and 
has 13 elements. Therefore, the concepts of quotient space and fractal geometry have a close relation. We will discuss them using the quotient space approximation principle in the following sections.
has one element, has four and has 13 elements. Therefore, the concepts of quotient space and fractal geometry have a close relation. We will discuss them using the quotient space approximation principle in the following sections.7.3.2. Iterated Function Systems
In order to investigate fractal graphs from quotient space theory, the concept of quotient fractals is established first. Then, the quotient fractals are used to approximate fractal graphs. Its procedure is the following. An equivalence relation is defined through a mapping. A corresponding chain of hierarchical quotient spaces is built by the equivalence relation. Then, quotient mappings are set up on the quotient spaces. Finally, quotient fractals are obtained from the quotient mappings.
Definition 7.19
A set 
of compressed mappings on a compact metric space 
is given, where compression factor 
. Let 
. It is called an iterated function system on 
, simply denoted by 
.
of compressed mappings on a compact metric space is given, where compression factor . Let . It is called an iterated function system on , simply denoted by .Definition 7.20
Assume that 
is a complete metric space. 
, 
, where H(x) is a power set of . Define 
. 
is called a distance from point 
to set 
.
is a complete metric space. , , where H(x) is a power set of . Define . is called a distance from point to set .Definition 7.21
Assume that is a complete metric space. Sets 
, define 
. 
is called a distance from set 
to set .
is a complete metric space. Sets , define . is called a distance from set to set .Definition 7.22
is a complete metric space. The Hausdorff distance between points A and B on H(x) is defined as 
.The main theorem in iterated function systems is the following.
Theorem 7.4 (Attractor Theorem)
Assume that 
is an iterated function system on X. W is a fractal mapping on X. Then, 
holds on (H(x),h(d)). Namely, there exists a unique fixed point P on H(x), i.e.,
is an iterated function system on X. W is a fractal mapping on X. Then, holds on (H(x),h(d)). Namely, there exists a unique fixed point P on H(x), i.e.,
P is a corresponding fractal graph on iterated function system IFS=W.
7.3.3. Quotient Fractals
A mapping is used to define an equivalence relation as follows.
Definition 7.23
An iterated function system 
on is given. Construct 
and 
. Letting 
, then is a quotient space of .
on is given. Construct and . Letting , then is a quotient space of .Now, is partitioned as follows, i.e., 
is partitioned as 
. Let
is partitioned as follows, i.e., is partitioned as . Let
We obtain
(7.16)
Set 
remains unchanged.
remains unchanged.Then, we have a partition of denoted by . From Formula (7.16), it’s known that is a quotient space of .
denoted by . From Formula (7.16), it’s known that is a quotient space of .By induction, assume that 
is a known quotient space. For its elements, the partition procedure is the following. Let
is a known quotient space. For its elements, the partition procedure is the following. Let
We obtain
Obviously, 
compose a chain of hierarchical quotient spaces. For simplicity, the element 
of 
is denoted by its subscript 
in the following discussion.
compose a chain of hierarchical quotient spaces. For simplicity, the element of is denoted by its subscript in the following discussion.Definition 7.24
Define a mapping 
on as 
, where 
is a nature projection and 
.
on as , where is a nature projection and .Definition 7.25
Define 
on , where 
is an invariant subset on corresponding to mapping W, i.e., a fractal graph.
on , where is an invariant subset on corresponding to mapping W, i.e., a fractal graph.Definition 7.26
is called a quotient fractal model of iterated function system 
.Theorem 7.4
If an iterated function system IFS on is given, then it corresponds to a chain of hierarchical quotient sets on .
is given, then it corresponds to a chain of hierarchical quotient sets on .Proof
Assume that mapping 
. Let 
and .
. Let and .(1) Assume that 
. Then, 
compose a partition of X. Its corresponding quotient space is denoted by . Again, let 
and 
. Then, 
compose a partition of denoted by . Obviously, is a quotient space of .
. Then, compose a partition of X. Its corresponding quotient space is denoted by . Again, let and . Then, compose a partition of denoted by . Obviously, is a quotient space of .By induction, define
compose a partition of X denoted by . It’s easy to show that is a quotient space of .
compose a partition of X denoted by . It’s easy to show that is a quotient space of .Therefore, 
is a chain of quotient sets of X or a chain of quotient spaces corresponding to an iterated function system.
is a chain of quotient sets of X or a chain of quotient spaces corresponding to an iterated function system.(2) When some and 
are overlapping, i.e., 
. An abstract space 
can be constructed, where element i corresponds to set . Define an abstract space 
, where element 
corresponds to set , i.e., 
.
and are overlapping, i.e., . An abstract space can be constructed, where element i corresponds to set . Define an abstract space , where element corresponds to set , i.e., .Generally, define 
, where element 
corresponds to 
, 
. Let 
. The point on is an infinite sequence composed by 0,1,…,n, i.e., 
.
, where element corresponds to , . Let . The point on is an infinite sequence composed by 0,1,…,n, i.e., .Similarly, we may have a chain 
of quotient sets.
of quotient sets.We have a profound relation between quotient fractals and fractal graphs as follows.
Theorem 7.5 (Quotient Fractal Approximation Theorem)
We have the following properties.
Property 1
Property 3
is an iterated function system on . Then, 
, its convergence is based on the Hausdorff distance defined on the power set of .Property 4
7.3.4. Conclusions
Property 3 in Section 7.3.3 is the quotient fractal approximation theorem of fractal graphs. It means that in fractal geometry we can still use a set of simple quotient spaces to approximate the original space so that the computational complexity is reduced. This is just the basic principle of quotient space approximation method and the multi-granular computing as well.
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