# Simplicial Approach to Fractal Structures

Authors: Carlo Cattani; Ettore Laserra; Ivana Bochicchio
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

A fractal lattice is defined by iterative maps on a simplex. In particular, Sierpinski gasket and von Koch flake are explicitly obtained by simplex transformations.


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## Body

## 1. Introduction

Simplicial calculus [1–3] has been since the beginning a suitable tool for investigating discrete models in many physical problems such as discrete models in space-time [4–9] complex networks [10–13], molecular crystals, aggregates and diamond lattices [14–17], computer graphics [18, 19], and more recently signal processing and computer vision, such as stereo matching and image segmentation [20, 21].In some recent papers [22–25], fractals [26–29] generated by simplexes, also called fractal lattices, were proposed for the analysis of nonconventional materials as some kind of polymers [24, 25] or nanocomposites [22, 23, 30, 31] having extreme physical and chemical properties. Moreover, the analysis of complex traffic on networks [32, 33] and image analysis [20, 21] based on fractal geometry and simplicial lattices has focussed on the importance of these methods in handling modern challenging problems.However, only a few attempts were made in order to define the fractal lattice (structure) by an iterated system of functions on simplexes [34, 35]. The main scheme for affine contraction has been given in [35], whereas some generation of fractals by simplicial maps can be found in [34].In this paper, we define a method based on simple algorithms for the generation of fractal-like structures by continuously deforming a simplex. This algorithm is based on a well-defined analytical map, which can be used to finitely describe fractals. Instead of recursive law, or nested maps (see, e.g., [1, 2, 15]), we propose a method which can be more easily implemented.In the following, we will study anm-dimensional fractal structure defined by the transformation group of a simplicial complex. Starting from a simplex, it will define the group of transformation on it, so that the intrinsic (affine) metric remains scale invariant. The group of transformations (isometries and homotheties) will be characterized by matrices acting on the skeleton of the simplex. We will derive the basic properties of the fractal lattice and give a suitable definition of self-similarity on lattices. The concept of self-similarity is shown to be fulfilled by some classical transformation on simplices (homothety) and, simplicial based, fractals as the Sierpinski tessellations and the von Koch flake.


## 2. Euclidean Simplexes

In the ordinary Euclidean spaceℝn, we assume that there exists a triangulation of ℝn, in the sense that there is at least a finite set of n+1 points geometrically independent (simplexes). A simplex will be considered both as a set of points and as the convex subspace of ℝn, defined by the geometrical support of the simplex. Union of n-adjacent simplexes is an n-polyhedron 𝒫 [4, 18, 19].The euclideanm-simplex σm, of independent vertices V0,V1,…,Vm, is defined [1–3] as the subset of ℝn,(2.1)σm=def{P∈Rn∣P∑i=0mλiViwith∑i=0mλi=1,0≤λi≤1}.
Let us denote with [σm]=[V0,V1,…,Vm] the set of points which form the skeleton of σm, and let #σm=m+1 be the cardinality of the set of points. The p-face of σm, with p≤m, is any simplex σp such that [σp]∩[σm]≠∅, and we write σp⪯σm.The number ofp-faces of σm is (m+1p+1).Them-dimensional simplicial complexΣm is defined as the finite set of p simplexes (p≤m) such that(1) for allσk∈Σm if σh⪯σk, then σh∈Σm,(2) for allσk,σh∈Σm, then either [σh]∩[σk]=∅ or [σh]∩[σk]=[σj] with σj∈Σm.The set of pointsP such that P∈σp, p≤m, and σp∈Σm is the geometric support of Σm also called m-polyhedron ℳm. The p-skeleton of Σm is [Σm]p=def[σp]for allσp∈Σm. The boundary∂Σm of Σm is the complex Σm-1 such that each σm-1∈Σm-1 is face of only one m-simplex of Σm. A finite set of simplexes is also called lattice (or tessellation).


### 2.1. Barycentric Coordinates and Barycentric Bases

In each simplex, it is possible to define thebarycentric basis as follows: given the m-simplex σm with vertices V0,…,Vm,  the barycentric basis is the set of (m+1) vectors(2.2)ei=defVi-Gm,
based on the barycenter(2.3)Gm=defG(σm)=∑i=0m1m+1Vi.These vectorsei belong to the n-dimensional vector space E isomorphic to ℝn. Moreover, they are linearly dependent, since according to their definition, it is(2.4)∑i=0mei=0.Each pointP∈σm can be characterized by a set of barycentric coordinates(λ0,…,λm) such that(2.5)0≤λi≤1,∑i=0mλi=1,i=0,…,m,
and P-𝒢m=∑i=0mλiei⇒(2.2),(2.4)P=∑i=0mλiVi. Therefore, each point of σm can be formally expressed as a linear combination of the skeleton [σm].The dual space is defined as the linear map of the vector spaceE into ℝ as(2.6)〈ei,ek〉=δ̃ki,
with [14](2.7)δ̃ki=defδki-1m+1={-1m+1,i≠k,+mm+1,i=k,δki being the Kroneker symbol. According to the definition  (2.7), it is(2.8)∑i=0mδ̃ki=∑k=0mδ̃ki=0.In addition, the metric tensor inσm is defined as [5](2.9)g̃ij=def-12δ̃ihδ̃jklhk2,(i,j,h,k=0,1,…,m)
being ℓhk2=def(Vk-Vh)2=(ek-eh)2.


### 2.2. Measures of them-Simplex

Let(2.10)Lij=defVj-Vi(=ej-ei),lij=def〈Lij,Lij〉,by using the ordinary wedge product of the vectorsej1,…,ejp, we can define the p-form ω,(2.11)ω=1p!∑j1,…,jmωj1…jpej1∧⋯∧ejp,whose affine components areωj1…jp=def〈ω,ej1∧ej2∧⋯∧ejp〉 [14].The euclidean measure of them-simplex σ (volume) is [14] (2.12)ɛΩ2=def1m!|L01∧⋯∧L0m|,from where, it follows that(2.13)Ω2=(1m!)3∑k1,…,kmj1,…,jmɛj1…jmɛk1…km∏a=1mljaka2,
being(2.14)ɛj1…jm=def±1,
according to the even/odd permutation j0,j1,…,jm of the indices 0,1,…,m.In particular, the volume of eachp-face σi1…im-p (see also [9]) is(2.15)Ωi1…im-p2=(-12)p(1p!)3∑k1,…,kpj1,…,jpɛj1…jpɛk1…kp∏a=1pljaka2(0<p≤m),
where j1,…,jp,k1,…,kp≠i1,…,im-p.


## 2.1. Barycentric Coordinates and Barycentric Bases

In each simplex, it is possible to define thebarycentric basis as follows: given the m-simplex σm with vertices V0,…,Vm,  the barycentric basis is the set of (m+1) vectors(2.2)ei=defVi-Gm,
based on the barycenter(2.3)Gm=defG(σm)=∑i=0m1m+1Vi.These vectorsei belong to the n-dimensional vector space E isomorphic to ℝn. Moreover, they are linearly dependent, since according to their definition, it is(2.4)∑i=0mei=0.Each pointP∈σm can be characterized by a set of barycentric coordinates(λ0,…,λm) such that(2.5)0≤λi≤1,∑i=0mλi=1,i=0,…,m,
and P-𝒢m=∑i=0mλiei⇒(2.2),(2.4)P=∑i=0mλiVi. Therefore, each point of σm can be formally expressed as a linear combination of the skeleton [σm].The dual space is defined as the linear map of the vector spaceE into ℝ as(2.6)〈ei,ek〉=δ̃ki,
with [14](2.7)δ̃ki=defδki-1m+1={-1m+1,i≠k,+mm+1,i=k,δki being the Kroneker symbol. According to the definition  (2.7), it is(2.8)∑i=0mδ̃ki=∑k=0mδ̃ki=0.In addition, the metric tensor inσm is defined as [5](2.9)g̃ij=def-12δ̃ihδ̃jklhk2,(i,j,h,k=0,1,…,m)
being ℓhk2=def(Vk-Vh)2=(ek-eh)2.


## 2.2. Measures of them-Simplex

Let(2.10)Lij=defVj-Vi(=ej-ei),lij=def〈Lij,Lij〉,by using the ordinary wedge product of the vectorsej1,…,ejp, we can define the p-form ω,(2.11)ω=1p!∑j1,…,jmωj1…jpej1∧⋯∧ejp,whose affine components areωj1…jp=def〈ω,ej1∧ej2∧⋯∧ejp〉 [14].The euclidean measure of them-simplex σ (volume) is [14] (2.12)ɛΩ2=def1m!|L01∧⋯∧L0m|,from where, it follows that(2.13)Ω2=(1m!)3∑k1,…,kmj1,…,jmɛj1…jmɛk1…km∏a=1mljaka2,
being(2.14)ɛj1…jm=def±1,
according to the even/odd permutation j0,j1,…,jm of the indices 0,1,…,m.In particular, the volume of eachp-face σi1…im-p (see also [9]) is(2.15)Ωi1…im-p2=(-12)p(1p!)3∑k1,…,kpj1,…,jpɛj1…jpɛk1…kp∏a=1pljaka2(0<p≤m),
where j1,…,jp,k1,…,kp≠i1,…,im-p.


## 3.m-Dimensional Homothety

Letℐ(σi) be the subspace of ℝm to which σi belongs; it can be easily proved that [14] (3.1)∀v∈I(σi)〈ni,v〉=0(ifixed),
where the normal vector ni is defined as(3.2)ni=def-mΩΩiei,hi=defmΩΩi.The above definition of vector orthogonal to a(m-1)-face allows us to characterize the m-parallelism of simplexes as follows. Let σ,σ⌢ be two simplexes in ℝm; let σi,σ⌢i be the ith (m-1)-faces of σ and σ⌢, respectively, and let ni,n⌢i be their normal vectors, then we say that σ is m-parallel to σ⌢ ( σ∥mσ⌢) if and if only σi∥σ⌢i, that is,ni=n⌢i(i=0,…,m).Letφ be a map (3.3)φ:Rm⟶Rm,σ⟼φσ⌢
such that(1)φ is a bijective simplicial map on σ,(2)  thes-adjacent faces of σ correspond (under the map φ) to s-adjacent faces of σ⌢,(3)σ and σ⌢ are m-parallel.We also assume that this transformation depends on the edge vectors and in particular on the edge lengths,so that  any quantity, defined on the simplex, transforming under the action ofφ, is a function of the edge lengths. Furthermore, we assume the following conditions:(4) there exists a fixed point under the action ofφ: (3.4)∃O∈Rm∣φ(O)≡O,(5) each(m-1)-face σi translates of an amount t∈[0,∞).Let us choose as a fixed point one of the vertices, for example,V0. We define this bijective simplicial map applying any P∈σ into P⌢∈σ⌢ (t∈[0,∞)) as(3.5)P⌢=defP+tΩ0mΩ∑i=0mλiL0i;
in particular, this function acts on any vertex Vi as(3.6)V⌢i=Vi+tΩ0mΩL0i,(V⌢0=V0),
so that we can easily prove that all the previous conditions are easily satisfied [14]. According to the above equations, each edge transforms as(3.7)L⌢ij=(1+tΩ0mΩ)Lij,
where L⌢ij=V⌢j-V⌢i.


### 3.1. Variation Law of thep-Faces of σ

The variation law of the edge lengths, resulting from (3.7), is given by the formula(3.8)l⌢ij=(1+tΩ0mΩ)lij,
where lij is the length of the edge Lij, and l⌢ij is the length of the edge L⌢ij.According to (2.13), the volume Ω is a homogeneous function of degree m of the m(m+1)/2 variable {lij2}i<j, so that its variation law is(3.9)Ω⌢=(1+tΩ0mΩ)mΩ,
and for any p-face,(3.10)Ω⌢i1…im-p=(1+tΩ0mΩ)pΩi1…im-p(0<p<m);
analogously, taking into account the definition (5.5)2, we have the transformation law of hi:(3.11)h⌢i=m(1+tΩ0mΩ)hi.
There follows, for the fundamental vectors of σ⌢, that  (3.12){e⌢i=defV⌢i-G⌢=(1+tΩ0mΩ)ei,n⌢i=ni,e⌢i=Ω⌢i/ΩiΩ⌢/Ωei=(1+tΩ0mΩ)-1ei.


## 3.1. Variation Law of thep-Faces of σ

The variation law of the edge lengths, resulting from (3.7), is given by the formula(3.8)l⌢ij=(1+tΩ0mΩ)lij,
where lij is the length of the edge Lij, and l⌢ij is the length of the edge L⌢ij.According to (2.13), the volume Ω is a homogeneous function of degree m of the m(m+1)/2 variable {lij2}i<j, so that its variation law is(3.9)Ω⌢=(1+tΩ0mΩ)mΩ,
and for any p-face,(3.10)Ω⌢i1…im-p=(1+tΩ0mΩ)pΩi1…im-p(0<p<m);
analogously, taking into account the definition (5.5)2, we have the transformation law of hi:(3.11)h⌢i=m(1+tΩ0mΩ)hi.
There follows, for the fundamental vectors of σ⌢, that  (3.12){e⌢i=defV⌢i-G⌢=(1+tΩ0mΩ)ei,n⌢i=ni,e⌢i=Ω⌢i/ΩiΩ⌢/Ωei=(1+tΩ0mΩ)-1ei.


## 4. Self-Similar Structure

Let(ℝn,d) be the complete metric space with the standard Euclidean metric d, and let K(ℝn) be the set(4.1)K(Rn)={K⊆Rn:Kis  a  nonempty  compact  set}.The iterated function system (IFS)(4.2){wi}=(Rn,d,w1,w2,…,wn)is the finite set of contractionswi on the complete metric space (ℝn,d), being the contraction w defined as (4.3)d(w(x),w(y))≤cd(x,y),∀x,y∈Rn,withc contraction coefficient.For eachA∈K(ℝn), the (IFS) contracting mapping is (4.4)w:A∈K(Rn)⟶w1(A)⋃⋯⋃wn(A)∈K(Rn),with contraction coefficientc=max{c1,…,cn}. Each function wi usually is linear, or more generally an affine transformation, but sometimes it can be nonlinear, including projective and Möbius transformations [27].According to the Banach fixed-point theorem (see, e.g., [36]), every contraction mapping on a nonempty complete metric space has a unique fixed point, so that there exists a unique compact (i.e., closed and bounded) fixed set A such that A=w(A). The set A is also known as the fixed set of the Hutchinson operator [28].One way of constructing such fixed set is to start with an initial setA and by iterating the actions of w. Hence, (4.5)A=⋃i1,…,ih=1,…,nwi1∘⋯∘wih(A),so thatA is a self-similar set, expressed as the finite union of its conformal copies, each one reduced by a factor ch.The attractorA of IFS is characterized by a similarity dimension as follows.Definition 4.1.
Given an IFS ofn contraction mappings with the same contraction coefficient c, the similarity dimension is defined as
(4.6)s=log⁡nlog⁡1/c(=-log⁡nlog⁡c).Sets having noninteger similarity dimensions are called fractal sets, or simply fractals. There follows that the iterated function systems are a method of constructing fractals; the resulting constructions are always self-similar such thatw(μx)=μHw(x). Hence, each map  w  is also called a self-similar map [27].


## 5. Fractal Structures from Simplicial Maps

In this section, some examples of self-similar (scale invariant) structures obtained by IFS on simplexes are given inℝ2. In particular, the IFS will be defined by affine transformations, as conformal maps of the affine metrics.In the following, we will introduce some self-similar maps defined both on 2-simplexes and 1-simplexes, so that, from (4.1), (5.1)K(R2)={σ2;σ1;σ0},w:K(R2)⟼K(R2).In particular, letσ2 be the simplex [V1,V2,V3], then it is (5.2)K(R2)={[V1,V2,V3];[V1,V2],[V1,V3],[V2,V3];[V1],[V2],[V3]},so that a mapw on K(ℝ2) could be the more general function defined on any face of σ2.Examples 5.
If the skeleton ofσ2 is the set of vertices {V1,V2,V3} with V1=(x1,y1),V2=(x2,y2), and V3=(x3,y3), the affine map w is defined by the matrix
(5.3)W=(a11a12a13a14a15a16a21a22a23a24a25a26a31a32a33a34a35a36a41a42a43a44a45a46a51a52a53a54a55a56a61a62a63a64a65a66)
and the constant vector
(5.4)U=(u1,u2,u3,u4,u5,u6).
The function w maps a 2-simplex into a 2-simplex whereas, by a matrix product, the vector
(5.5)X=(x1,y1,x2,y2,x3,y3)
is mapped into the vector
(5.6)WX+U,
so that the skeleton of w(σ2) is given by the vector WX+U. For instance, a rotation with fixed point V1 is given by the matrix
(5.7)W=(10000001000000a33a340000a43a44000000a55a560000a65a66),
with
(5.8)a33a44-a34a43=±1,a55a66-a56a65=±1,
and the vector U={0,0,0,0,0,0}.Some more special maps will be given in the following where, in particular, we consider, without restriction, some special maps on the 1-faces ofσ2 such that
(5.9)w(σ2)=w1(σ12)∪w2(σ22)∪w3(σ32),#w(σ2)=3.
In this case, the matrix W, acting on σ2, follows from the direct sum of lower-order matrices acting on σ1 simplexes, as follows:(a)  the first vertexV1 remains fixed, and the map w on σ2 is a consequence of the transformation of the simplex σ1=[V2,V3], that is, by defining
(5.10)I=(1001),W1=(a33a34a35a36a43a44a45a46a53a54a55a56a63a64a65a66),
it is
(5.11)W=I⊕W1=(10000001000000a33a34a35a3600a43a44a45a4600a53a54a55a5600a63a64a65a66),(b)  the second vertexV2 remains fixed, and the map w on σ2 is a consequence of the transformation of the simplex σ1=[V1,V3], so that
(5.12)W2=(a11a12a15a16a21a22a25a26a51a52a55a56a61a62a65a66),W=(a11a1200a15a16a21a2200a25a26001000000100a51a5200a55a56a61a6200a65a66),(c) the third vertexV3 remains fixed, and the map w on σ2 is a consequence of the transformation of the simplex σ1=[V1,V2], that is,
(5.13)W=W3⊕I=(a11a12a13a1400a21a22a23a2400a31a32a33a3400a41a42a43a4400000010000001),
being
(5.14)W3=(a11a12a13a14a21a22a23a24a31a32a33a34a41a42a43a44).In the following, we will characterize the transformation on a 2-simplex as a result of iterative maps on its boundary 1-simplexes. These maps on 1-simplexes are defined by the matricesW1,W2,andW3, applied to the vectors of coordinates of [V2,V3], [V1,V3], and [V1,V2], respectively.


### 5.1. Homothety

Let us consider the 2-simplexσ2={A,B,C} and the map (Figure 1)(5.15)σ2={A,B,C}⟹w(σ2)={A,B′,C′},Figure 1
Homothety map.such thatnC=±nC′. This map, according to (5.9), is obtained as a combination of 3 maps acting on the faces of σ2, since(5.16)w(σ2)=w1([A,B])∪w2([B,C])∪w3([A,C]).
This map is a scale invariant, since there results (5.17)lAB2=λlA′B′2,(0≤λ),ℓBC2=λℓB′C′2,andℓAC2=λℓA′C′2, as well.So that whenλ<1, we have a contraction and a dilation when λ>1.Moreover, according to  (2.9), the metric g̃′ij of the transformed simplex is given by a conformal transformation g̃′ij=λg̃ij.


### 5.2. Sierpinski Gasket

As a first example of fractal defined by IFS on simplexes, we will consider the Sierpinski gasket. To this end, let us introduce an orthogonal coordinate system0xy in ℝ2 and three homothety maps w1, w2, and w3. Each wi is uniquely and completely determined once we know as it acts on the paired points A=(xA,yA),B=(xB,yB), and C=(xC,yC), vertices of the 2-simplex [σ2]=[A,B,C].In order to define the Sierpinski gasket by IFS of maps, we consider a sequence of maps that, at each step, shrink the area ofσ2 by a factor 0.25 and move the edges by a suitable homothety (Figure 2). In particular, the 3 maps are explicitly defined as follows:(5.18)w1:(xAyAxByBxCyC)⟹M⋅(xAyAxByBxCyC),w2:(xAyAxByBxCyC)⟹M⋅(xAyAxByBxCyC)+(01/201/201/2),w3:(xAyAxByBxCyC)⟹M⋅(xAyAxByBxCyC)+(1/201/201/20),
where M is the matrix(5.19)M=(1/20000001/20000001/20000001/20000001/20000001/2).Figure 2
Fundamental maps.Once we get the vertices ofw(σ2), we can easily define the map for each point P of the σ2 convex domain.Comment
In fact, letλ1, λ2, and λ3be the barycentric coordinates of a given point P inside σ2, as given by  (2.1), then we can write the barycentric expansion of P≡(x,y) in terms of the coordinates of vertices A,B,andC as
(5.20)x=λ1xA+λ2xB+λ3xC,y=λ1yA+λ2yB+λ3yC.Substitutingλ3=1-λ1-λ2 into the above and rearranging, this linear transformation can be written as (5.21)H⋅Λ=P-C,whereΛ is the vector of barycentric coordinates, and H is the matrix (5.22)H=(xA-xCxB-xCyA-yCyB-yC).SinceH is invertible, we can easily obtain the barycentric coordinates of P=(x,y): (5.23)λ1=(yB-yC)(x-xC)+(xC-xB)(y-yC)(yB-yC)(xA-xC)+(xC-xB)(yA-yC),λ2=(yC-yA)(x-xC)+(xA-xC)(y-yC)(yC-yA)(xB-xC)+(xA-xC)(yB-yC),λ3=1-λ1-λ2.According to  (5.9) each map wi,i={1,2,3} is a contraction (dilation) of the σ2 faces, such that the union gives rise to a 2-simplex (Figure 2). Any P∈[σ2] is mapped into P⌢∈σ⌢=wi([σ2]) as(5.24)P⌢=P-12∑i=0mλiL0i.
Moreover, each vertex in the wi([σ2]),i=1,2,3 can be expressed as in  (3.6)(5.25)V⌢i=Vi-12L0i,
so that(5.26)L⌢ij=12Lij,l⌢ij=12lij,Ω⌢=14Ω.Reiterating this process for each remaining triangle, at the stepk, we will obtain the compact set Tk given by 3k triangles whose edges are contracted by (1/2k). In other words,(5.27)L⌢ij=12kLij;l⌢ij=12klij,Ω⌢=122kΩ.Finally, we note that through the three simplicial maps inℝ2, provided with the natural metric d, we are able to construct the IFS (ℝ2,d,w1,w2w3) that has the well-known Sierpinski gasket T=⋂kTk as fractal attractor. So, we have obtained the Sierpinski gasket, as the combination of homothety maps (Figure 2). The iterating function will generate the known fractal-shaped curve (Figure 3).Figure 3
Sierpinski gasket.The Sierpinski gasket supplies one of the most simple cases of construction of fractals through simplicial maps. In fact, the fractal structure is obtained acting on the 2-simplex only with homothetic transformations.  Sometimes a fractal object can be constructed not only acting on simplexes with one map, but considering the compositions of different suitable transformations. Hereafter, in order to obtain another fractal object, we will consider, in details, some more elementary maps: the translation and the rotation (which are special cases of the matrixW).


## 5.1. Homothety

Let us consider the 2-simplexσ2={A,B,C} and the map (Figure 1)(5.15)σ2={A,B,C}⟹w(σ2)={A,B′,C′},Figure 1
Homothety map.such thatnC=±nC′. This map, according to (5.9), is obtained as a combination of 3 maps acting on the faces of σ2, since(5.16)w(σ2)=w1([A,B])∪w2([B,C])∪w3([A,C]).
This map is a scale invariant, since there results (5.17)lAB2=λlA′B′2,(0≤λ),ℓBC2=λℓB′C′2,andℓAC2=λℓA′C′2, as well.So that whenλ<1, we have a contraction and a dilation when λ>1.Moreover, according to  (2.9), the metric g̃′ij of the transformed simplex is given by a conformal transformation g̃′ij=λg̃ij.


## 5.2. Sierpinski Gasket

As a first example of fractal defined by IFS on simplexes, we will consider the Sierpinski gasket. To this end, let us introduce an orthogonal coordinate system0xy in ℝ2 and three homothety maps w1, w2, and w3. Each wi is uniquely and completely determined once we know as it acts on the paired points A=(xA,yA),B=(xB,yB), and C=(xC,yC), vertices of the 2-simplex [σ2]=[A,B,C].In order to define the Sierpinski gasket by IFS of maps, we consider a sequence of maps that, at each step, shrink the area ofσ2 by a factor 0.25 and move the edges by a suitable homothety (Figure 2). In particular, the 3 maps are explicitly defined as follows:(5.18)w1:(xAyAxByBxCyC)⟹M⋅(xAyAxByBxCyC),w2:(xAyAxByBxCyC)⟹M⋅(xAyAxByBxCyC)+(01/201/201/2),w3:(xAyAxByBxCyC)⟹M⋅(xAyAxByBxCyC)+(1/201/201/20),
where M is the matrix(5.19)M=(1/20000001/20000001/20000001/20000001/20000001/2).Figure 2
Fundamental maps.Once we get the vertices ofw(σ2), we can easily define the map for each point P of the σ2 convex domain.Comment
In fact, letλ1, λ2, and λ3be the barycentric coordinates of a given point P inside σ2, as given by  (2.1), then we can write the barycentric expansion of P≡(x,y) in terms of the coordinates of vertices A,B,andC as
(5.20)x=λ1xA+λ2xB+λ3xC,y=λ1yA+λ2yB+λ3yC.Substitutingλ3=1-λ1-λ2 into the above and rearranging, this linear transformation can be written as (5.21)H⋅Λ=P-C,whereΛ is the vector of barycentric coordinates, and H is the matrix (5.22)H=(xA-xCxB-xCyA-yCyB-yC).SinceH is invertible, we can easily obtain the barycentric coordinates of P=(x,y): (5.23)λ1=(yB-yC)(x-xC)+(xC-xB)(y-yC)(yB-yC)(xA-xC)+(xC-xB)(yA-yC),λ2=(yC-yA)(x-xC)+(xA-xC)(y-yC)(yC-yA)(xB-xC)+(xA-xC)(yB-yC),λ3=1-λ1-λ2.According to  (5.9) each map wi,i={1,2,3} is a contraction (dilation) of the σ2 faces, such that the union gives rise to a 2-simplex (Figure 2). Any P∈[σ2] is mapped into P⌢∈σ⌢=wi([σ2]) as(5.24)P⌢=P-12∑i=0mλiL0i.
Moreover, each vertex in the wi([σ2]),i=1,2,3 can be expressed as in  (3.6)(5.25)V⌢i=Vi-12L0i,
so that(5.26)L⌢ij=12Lij,l⌢ij=12lij,Ω⌢=14Ω.Reiterating this process for each remaining triangle, at the stepk, we will obtain the compact set Tk given by 3k triangles whose edges are contracted by (1/2k). In other words,(5.27)L⌢ij=12kLij;l⌢ij=12klij,Ω⌢=122kΩ.Finally, we note that through the three simplicial maps inℝ2, provided with the natural metric d, we are able to construct the IFS (ℝ2,d,w1,w2w3) that has the well-known Sierpinski gasket T=⋂kTk as fractal attractor. So, we have obtained the Sierpinski gasket, as the combination of homothety maps (Figure 2). The iterating function will generate the known fractal-shaped curve (Figure 3).Figure 3
Sierpinski gasket.The Sierpinski gasket supplies one of the most simple cases of construction of fractals through simplicial maps. In fact, the fractal structure is obtained acting on the 2-simplex only with homothetic transformations.  Sometimes a fractal object can be constructed not only acting on simplexes with one map, but considering the compositions of different suitable transformations. Hereafter, in order to obtain another fractal object, we will consider, in details, some more elementary maps: the translation and the rotation (which are special cases of the matrixW).


## 6. Von Koch Curve

The von Koch curve [27, 28] can be obtained as a combination of homothety, translation, and rotation maps, so that the von Koch snowflake is obtained by their iteration.


### 6.1. Translation

Let the translation operator be defined as the operator(6.1)T:Rm⟶Rm,σ⟼Tσ⌢such that(6.2)T(P)=P+v,wherev=(v1,v2,…,vm) is a given vector of ℝm, then the image of a simplex σ under the function T is the translation of σ by T so that any vertex Vi is transformed into(6.3)V⌢i=Vi+v.
Since in a Euclidean space, any translation is an isometry, we have no variation of the edge lengths of σ.According to the definitions  (5.3), (5.4) in ℝ2, it is (6.4)U=(v1,v2,v1,v2,v1,v2),v1=Cnst.,v2=Cnst.,beingW the zero matrix.


### 6.2. Rotation

Rotation is characterized by having a fixed point, however, like the translation which is an isometry. This is like the previous maps on simplexes that can be defined by a suitable matrix (5.3). ℝ2 rotation is defined by (5.7), which however can be expressed by a single parameter (rotation angle). Hence, in two dimensions, a rotation with fixed point V0 is the operator (6.5)R:R2⟶R2,σ⟼Rσ⌢such that(6.6)R(P)=V0+Rθ(P-V0),whereRθ is the matrix(6.7)Rθ=(cos⁡θ-sin⁡θsin⁡θcos⁡θ),
so that  (5.7), when applied to the simplex σ2=[V0,V1,V2] with one fixed vertex, becomes(6.8)W=(10000001000000cos⁡θ-sin⁡θ0000sin⁡θcos⁡θ000000cos⁡θ-sin⁡θ0000sin⁡θcos⁡θ).With respect to an orthogonal coordinate system with originO, for any P∈σ, we define the rotation as the bijective simplicial map which applies P∈σ into P⌢∈σ⌢,(6.9)P⌢=def[Rθ(P-v)]+v,
where v=V0-O; in particular, the vertices V0,V1,andV2 are transformed into(6.10)V⌢0=V0,V⌢i=[Rθ(L0i+O)]+(V0-O),(i=1,2).


### 6.3. Von Koch Snowflake

Let0xy be an orthogonal coordinate system for ℝ2, and let σ2=[A,B,C] be a two simplex under the homothety map. According to (5.16), this map can be realized by a composition of maps on the 1-simplexes σ1=[A,B], σ1=[B,C], and σ1=[A,C]. The coordinates of vertices are A=(xA,yA), B=(xB,yB), and C=(xC,yC), respectively. In the following we will give both the construction of the Koch curve as IFS on σ1 and the construction of the Koch snowflake as IFS on σ2.


#### 6.3.1. Von Koch Curve

Koch curve can be classically constructed by starting with a line segment, then recursively altering the shape as follows: divide the line segment into three segments of equal length; draw an equilateral triangle that has the middle segment from step 1 as its base and points outward; remove the line segment that is the base of the triangle from step 2 (see Figure5).Following the classical construction, we consider the following maps on 1-simplexes:(6.11)w1:(xAyAxByB)⟹M⋅(xAyAxByB),w2:(xAyAxByB)⟹M⋅(xAyAxByB)+(2/302/30),
where M is the matrix(6.12)M=(1/300001/300001/300001/3).
Hence, w1 is a factor of the homothety w having A as a fixed vertex, while w2 leaves B unchanged: (6.13)w1(P)=P-23λiL0i,L0i=Vi-A,w2(P)=P-23λiL0i,L0i=Vi-B.Moreover, each vertex in thewi([σ1]),i=1,2, can be expressed as in(3.6)(6.14)V⌢i=Vi-23L0i.Let us now consider the transformation, on two steps, which first rotatesw1([A,B]) of an angle θ=60° around the fixed point A, and then it translates the rotated simplex by a vector v=(1/3,0).So that we obtain(6.15)w3:(xAyAxByB)⟹M′⋅(xAyAxByB)+(1/301/30),
where M′ is the matrix(6.16)M′=(1/6-3/6003/61/600001/6-3/6003/61/6).Finally, let us apply the transformation which first rotatesw1([A,B]) of an angle θ=120° around the fixed point A, and then it translates the rotated simplex by the vector v=(2/3,0). Accordingly, it is(6.17)w4:(xAyAxByB)⟹M′′⋅(xAyAxByB)+(2/302/30),
where M′′ is the matrix(6.18)M′′=(-1/6-3/6003/6-1/60000-1/6-3/6003/6-1/6).Since, as previously shown, rotation and translation are isometries, for eachwi([σ1]),i=1,2,3,4, we obtain(6.19)L⌢ij=13Lij,l⌢ij=13lij,Ω⌢=13Ω.In order to visualize the von Koch pattern, let us consider the 1-simplex{A,B}={{0,0},{1,0}}; since the point A has been chosen as the origin of the reference system, and w3 and w4 are obtained as rotation leaving fixed the origin, the transformed instances can be easily computed so that, at the first step, the IFS maps on 1-simplexes can be drawn (Figure 4).Figure 4
Image ofw([σ1])=⋃i=1,…,4wi([σ1]),where the 1-simplex σ1 is the unitary interval.Figure 5
Koch curve.Reiterating this process for each remaining segment, at the stepk we will obtain the compact set Tk made of 22k segments whose sides are contracted by a factor (1/3)k. The IFS map (ℝ1,d,w1,w2,w3,w4) gives us the Koch curve L=⋂kLk with similarity dimension equal to (6.20)s=log⁡4log⁡1/(1/3)=log⁡4log⁡3.This is also the similarity dimension of the Koch snowflake [27, 28]. So, the Koch curve (Figure 5) is obtained as a combination of IFS simplicial maps generating the known fractal-shaped curve.


#### 6.3.2. Koch Flake

According to (5.16) and to the examples previously given, Koch flake (snowflake) can be constructed in a non-classical approach as IFS of maps on a 2-simplex. Koch snowflake can be seen as the image of a suitable system of iterated homotheties acting on a 2-simplex, given by suitable translations of the boundary 1-simplexes.In this process, the total length of each side of a triangle increases by one-third, and thus, the total length at thekth step will be (4/3)k of the original triangle perimeter.


## 6.1. Translation

Let the translation operator be defined as the operator(6.1)T:Rm⟶Rm,σ⟼Tσ⌢such that(6.2)T(P)=P+v,wherev=(v1,v2,…,vm) is a given vector of ℝm, then the image of a simplex σ under the function T is the translation of σ by T so that any vertex Vi is transformed into(6.3)V⌢i=Vi+v.
Since in a Euclidean space, any translation is an isometry, we have no variation of the edge lengths of σ.According to the definitions  (5.3), (5.4) in ℝ2, it is (6.4)U=(v1,v2,v1,v2,v1,v2),v1=Cnst.,v2=Cnst.,beingW the zero matrix.


## 6.2. Rotation

Rotation is characterized by having a fixed point, however, like the translation which is an isometry. This is like the previous maps on simplexes that can be defined by a suitable matrix (5.3). ℝ2 rotation is defined by (5.7), which however can be expressed by a single parameter (rotation angle). Hence, in two dimensions, a rotation with fixed point V0 is the operator (6.5)R:R2⟶R2,σ⟼Rσ⌢such that(6.6)R(P)=V0+Rθ(P-V0),whereRθ is the matrix(6.7)Rθ=(cos⁡θ-sin⁡θsin⁡θcos⁡θ),
so that  (5.7), when applied to the simplex σ2=[V0,V1,V2] with one fixed vertex, becomes(6.8)W=(10000001000000cos⁡θ-sin⁡θ0000sin⁡θcos⁡θ000000cos⁡θ-sin⁡θ0000sin⁡θcos⁡θ).With respect to an orthogonal coordinate system with originO, for any P∈σ, we define the rotation as the bijective simplicial map which applies P∈σ into P⌢∈σ⌢,(6.9)P⌢=def[Rθ(P-v)]+v,
where v=V0-O; in particular, the vertices V0,V1,andV2 are transformed into(6.10)V⌢0=V0,V⌢i=[Rθ(L0i+O)]+(V0-O),(i=1,2).


## 6.3. Von Koch Snowflake

Let0xy be an orthogonal coordinate system for ℝ2, and let σ2=[A,B,C] be a two simplex under the homothety map. According to (5.16), this map can be realized by a composition of maps on the 1-simplexes σ1=[A,B], σ1=[B,C], and σ1=[A,C]. The coordinates of vertices are A=(xA,yA), B=(xB,yB), and C=(xC,yC), respectively. In the following we will give both the construction of the Koch curve as IFS on σ1 and the construction of the Koch snowflake as IFS on σ2.


### 6.3.1. Von Koch Curve

Koch curve can be classically constructed by starting with a line segment, then recursively altering the shape as follows: divide the line segment into three segments of equal length; draw an equilateral triangle that has the middle segment from step 1 as its base and points outward; remove the line segment that is the base of the triangle from step 2 (see Figure5).Following the classical construction, we consider the following maps on 1-simplexes:(6.11)w1:(xAyAxByB)⟹M⋅(xAyAxByB),w2:(xAyAxByB)⟹M⋅(xAyAxByB)+(2/302/30),
where M is the matrix(6.12)M=(1/300001/300001/300001/3).
Hence, w1 is a factor of the homothety w having A as a fixed vertex, while w2 leaves B unchanged: (6.13)w1(P)=P-23λiL0i,L0i=Vi-A,w2(P)=P-23λiL0i,L0i=Vi-B.Moreover, each vertex in thewi([σ1]),i=1,2, can be expressed as in(3.6)(6.14)V⌢i=Vi-23L0i.Let us now consider the transformation, on two steps, which first rotatesw1([A,B]) of an angle θ=60° around the fixed point A, and then it translates the rotated simplex by a vector v=(1/3,0).So that we obtain(6.15)w3:(xAyAxByB)⟹M′⋅(xAyAxByB)+(1/301/30),
where M′ is the matrix(6.16)M′=(1/6-3/6003/61/600001/6-3/6003/61/6).Finally, let us apply the transformation which first rotatesw1([A,B]) of an angle θ=120° around the fixed point A, and then it translates the rotated simplex by the vector v=(2/3,0). Accordingly, it is(6.17)w4:(xAyAxByB)⟹M′′⋅(xAyAxByB)+(2/302/30),
where M′′ is the matrix(6.18)M′′=(-1/6-3/6003/6-1/60000-1/6-3/6003/6-1/6).Since, as previously shown, rotation and translation are isometries, for eachwi([σ1]),i=1,2,3,4, we obtain(6.19)L⌢ij=13Lij,l⌢ij=13lij,Ω⌢=13Ω.In order to visualize the von Koch pattern, let us consider the 1-simplex{A,B}={{0,0},{1,0}}; since the point A has been chosen as the origin of the reference system, and w3 and w4 are obtained as rotation leaving fixed the origin, the transformed instances can be easily computed so that, at the first step, the IFS maps on 1-simplexes can be drawn (Figure 4).Figure 4
Image ofw([σ1])=⋃i=1,…,4wi([σ1]),where the 1-simplex σ1 is the unitary interval.Figure 5
Koch curve.Reiterating this process for each remaining segment, at the stepk we will obtain the compact set Tk made of 22k segments whose sides are contracted by a factor (1/3)k. The IFS map (ℝ1,d,w1,w2,w3,w4) gives us the Koch curve L=⋂kLk with similarity dimension equal to (6.20)s=log⁡4log⁡1/(1/3)=log⁡4log⁡3.This is also the similarity dimension of the Koch snowflake [27, 28]. So, the Koch curve (Figure 5) is obtained as a combination of IFS simplicial maps generating the known fractal-shaped curve.


### 6.3.2. Koch Flake

According to (5.16) and to the examples previously given, Koch flake (snowflake) can be constructed in a non-classical approach as IFS of maps on a 2-simplex. Koch snowflake can be seen as the image of a suitable system of iterated homotheties acting on a 2-simplex, given by suitable translations of the boundary 1-simplexes.In this process, the total length of each side of a triangle increases by one-third, and thus, the total length at thekth step will be (4/3)k of the original triangle perimeter.


## 6.3.1. Von Koch Curve

Koch curve can be classically constructed by starting with a line segment, then recursively altering the shape as follows: divide the line segment into three segments of equal length; draw an equilateral triangle that has the middle segment from step 1 as its base and points outward; remove the line segment that is the base of the triangle from step 2 (see Figure5).Following the classical construction, we consider the following maps on 1-simplexes:(6.11)w1:(xAyAxByB)⟹M⋅(xAyAxByB),w2:(xAyAxByB)⟹M⋅(xAyAxByB)+(2/302/30),
where M is the matrix(6.12)M=(1/300001/300001/300001/3).
Hence, w1 is a factor of the homothety w having A as a fixed vertex, while w2 leaves B unchanged: (6.13)w1(P)=P-23λiL0i,L0i=Vi-A,w2(P)=P-23λiL0i,L0i=Vi-B.Moreover, each vertex in thewi([σ1]),i=1,2, can be expressed as in(3.6)(6.14)V⌢i=Vi-23L0i.Let us now consider the transformation, on two steps, which first rotatesw1([A,B]) of an angle θ=60° around the fixed point A, and then it translates the rotated simplex by a vector v=(1/3,0).So that we obtain(6.15)w3:(xAyAxByB)⟹M′⋅(xAyAxByB)+(1/301/30),
where M′ is the matrix(6.16)M′=(1/6-3/6003/61/600001/6-3/6003/61/6).Finally, let us apply the transformation which first rotatesw1([A,B]) of an angle θ=120° around the fixed point A, and then it translates the rotated simplex by the vector v=(2/3,0). Accordingly, it is(6.17)w4:(xAyAxByB)⟹M′′⋅(xAyAxByB)+(2/302/30),
where M′′ is the matrix(6.18)M′′=(-1/6-3/6003/6-1/60000-1/6-3/6003/6-1/6).Since, as previously shown, rotation and translation are isometries, for eachwi([σ1]),i=1,2,3,4, we obtain(6.19)L⌢ij=13Lij,l⌢ij=13lij,Ω⌢=13Ω.In order to visualize the von Koch pattern, let us consider the 1-simplex{A,B}={{0,0},{1,0}}; since the point A has been chosen as the origin of the reference system, and w3 and w4 are obtained as rotation leaving fixed the origin, the transformed instances can be easily computed so that, at the first step, the IFS maps on 1-simplexes can be drawn (Figure 4).Figure 4
Image ofw([σ1])=⋃i=1,…,4wi([σ1]),where the 1-simplex σ1 is the unitary interval.Figure 5
Koch curve.Reiterating this process for each remaining segment, at the stepk we will obtain the compact set Tk made of 22k segments whose sides are contracted by a factor (1/3)k. The IFS map (ℝ1,d,w1,w2,w3,w4) gives us the Koch curve L=⋂kLk with similarity dimension equal to (6.20)s=log⁡4log⁡1/(1/3)=log⁡4log⁡3.This is also the similarity dimension of the Koch snowflake [27, 28]. So, the Koch curve (Figure 5) is obtained as a combination of IFS simplicial maps generating the known fractal-shaped curve.


## 6.3.2. Koch Flake

According to (5.16) and to the examples previously given, Koch flake (snowflake) can be constructed in a non-classical approach as IFS of maps on a 2-simplex. Koch snowflake can be seen as the image of a suitable system of iterated homotheties acting on a 2-simplex, given by suitable translations of the boundary 1-simplexes.In this process, the total length of each side of a triangle increases by one-third, and thus, the total length at thekth step will be (4/3)k of the original triangle perimeter.


## 7. Conclusion

In this paper, a nonclassical approach to fractal generation based on IFS of maps on simplexes has been given. Some of the most popular fractals, as the Sierpinski gasket and the von Koch flake, were obtained by iterative maps on simplexes. All maps were also intrinsically defined by using the affine (barycentric) coordinates and some basic measures on simplexes. The method proposed in this paper could be used to generate some new classes of fractals in any dimension, by simply defining suitable IFS on simplexes, thus opening new perspectives in fractal lattice geometry.



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*Source: 958101-2012-03-05.xml*