Simplicial Approach to Fractal Structures

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.


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 an m-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.

Euclidean Simplexes
In the ordinary Euclidean space R n , we assume that there exists a triangulation of R n , 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 R n , defined by the geometrical support of the simplex.Union of n-adjacent simplexes is an n-polyhedron P 4, 18, 19 .
The euclidean m-simplex σ m , of independent vertices

2.1
Let us denote with σ m V 0 , V 1 , . . ., V m the set of points which form the skeleton of σ m , and let #σ m m 1 be the cardinality of the set of points.The p-face of σ m , with p ≤ m, is any simplex σ p such that σ p ∩ σ m / ∅, and we write σ p σ m .
The number of p-faces of σ m is m 1 p 1 .The m-dimensional simplicial complex Σ m is defined as the finite set of p simplexes p ≤ m such that 1 for all A finite set of simplexes is also called lattice or tessellation .

Barycentric Coordinates and Barycentric Bases
In each simplex, it is possible to define the barycentric basis as follows: given the m-simplex σ m with vertices V 0 , . . ., V m , the barycentric basis is the set of m 1 vectors based on the barycenter These vectors e i belong to the n-dimensional vector space E isomorphic to R n .Moreover, they are linearly dependent, since according to their definition, it is

2.4
Each point P ∈ σ m can be characterized by a set of barycentric coordinates λ 0 , . . ., λ m such that Therefore, each point of σ m can be formally expressed as a linear combination of the skeleton σ m .
The dual space is defined as the linear map of the vector space E into R as e i , e k δ i k , 2.6 with 14 δ i k being the Kroneker symbol.According to the definition 2.7 , it is In addition, the metric tensor in σ m is defined as 5

Measures of the m-Simplex
Let by using the ordinary wedge product of the vectors e j 1 , . . ., e j p , we can define the p-form ω, 2.11 whose affine components are ω j 1 ...j p def ω, e j 1 ∧ e j 2 ∧ • • • ∧ e j p 14 .

m-Dimensional Homothety
Let I σ i be the subspace of R m to which σ i belongs; it can be easily proved that 14 where the normal vector n i is defined as 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 R m ; let σ i , σ i be the ith m − 1 -faces of σ and σ, respectively, and let n i , n i be their normal vectors, then we say that such that 1 ϕ is a bijective simplicial map on σ, 2 the s-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 of ϕ, is a function of the edge lengths.Furthermore, we assume the following conditions: 4 there exists a fixed point under the action of ϕ: Let us choose as a fixed point one of the vertices, for example, V 0 .We define this bijective simplicial map applying any P ∈ σ into P ∈ σ t ∈ 0, ∞ as in particular, this function acts on any vertex V i as so that we can easily prove that all the previous conditions are easily satisfied 14 .According to the above equations, each edge transforms as where

Variation Law of the p-Faces of σ
The variation law of the edge lengths, resulting from 3.7 , is given by the formula where l ij is the length of the edge L ij , 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 {l 2 ij } i<j , so that its variation law is and for any p-face, analogously, taking into account the definition 5.5 2 , we have the transformation law of h i :

3.11
There follows, for the fundamental vectors of σ, that 3.12

Self-Similar Structure
Let R n , d be the complete metric space with the standard Euclidean metric d, and let K R n be the set The iterated function system IFS is the finite set of contractions w i on the complete metric space R n , d , being the contraction w defined as with contraction coefficient c max{c 1 , . . ., c n }.Each function w i usually is linear, or more generally an affine transformation, but sometimes it can be nonlinear, including projective and M öbius 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 set A and by iterating the actions of w.Hence, so that A is a self-similar set, expressed as the finite union of its conformal copies, each one reduced by a factor c h .The attractor A of IFS is characterized by a similarity dimension as follows.
Definition 4.1.Given an IFS of n contraction mappings with the same contraction coefficient c, the similarity dimension is defined as 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 that w μx μ H w x .Hence, each map w is also called a self-similar map 27 .

Fractal Structures from Simplicial Maps
In this section, some examples of self-similar scale invariant structures obtained by IFS on simplexes are given in R 2 .In particular, the IFS will be defined by affine transformations, as conformal maps of the affine metrics.

Mathematical Problems in Engineering
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
In particular, let σ 2 be the simplex so that a map w on K R 2 could be the more general function defined on any face of σ 2 .
Examples.If the skeleton of σ 2 is the set of vertices x 2 , y 2 , and V 3 x 3 , y 3 , the affine map w is defined by the matrix and the constant vector
Some more special maps will be given in the following where, in particular, we consider, without restriction, some special maps on the 1-faces of σ 2 such that 5.9 In this case, the matrix W, acting on σ 2 , follows from the direct sum of lower-order matrices acting on σ 1 simplexes, as follows: a the first vertex V 1 remains fixed, and the map w on σ 2 is a consequence of the transformation of the simplex σ 1 V 2 , V 3 , that is, by defining 12 c the third vertex V 3 remains fixed, and the map w on σ 2 is a consequence of the transformation of the simplex a 11 a 12 a 13 a 14 0 0 a 21 a 22 a 23 a 24 0 0 a 31 a 32 a 33 a 34 0 0 a 41 a 42 a 43 a 44 0 0 0 0 0 0 1 0

5.14
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 matrices W 1 , W 2 , and W 3 , applied to the vectors of coordinates of

Homothety
Let us consider the 2-simplex σ 2 {A, B, C} and the map Figure 1 σ 2 {A, B, C} ⇒ w σ 2 A, B , C , 5.15 such that n C ±n C .This map, according to 5.9 , is obtained as a combination of 3 maps acting on the faces of σ 2 , since This map is a scale invariant, since there results 2 AB λ 2 A B , 0 ≤ λ , 5.17 A C , as well.So that when λ < 1, we have a contraction and a dilation when λ > 1.Moreover, according to 2.9 , the metric g ij of the transformed simplex is given by a conformal transformation g ij λ g ij .

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 system 0xy in R 2 and three homothety maps w 1 , w 2 , and w 3 .Each w i is uniquely and completely determined once we know as it acts on the paired points A x A , y A , B x B , y B , and C x C , y C , vertices of the 2-simplex σ 2  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 of σ 2 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: where M is the matrix

5.19
Once we get the vertices of w σ 2 , we can easily define the map for each point P of the σ 2 convex domain.Comment.In fact, let λ 1 , λ 2 , and λ 3 be the barycentric coordinates of a given point P inside σ 2 , 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, and C as

5.20
Substituting λ 3 1 − λ 1 − λ 2 into the above and rearranging, this linear transformation can be written as where Λ is the vector of barycentric coordinates, and H is the matrix Since H is invertible, we can easily obtain the barycentric coordinates of P x, y :

5.23
According to 5.9 each map w i , i {1, 2, 3} is a contraction dilation of the σ 2 faces, such that the union gives rise to a 2-simplex Figure 2 .Any P ∈ σ 2 is mapped into P ∈ σ w i σ 2 as Moreover, each vertex in the w i σ 2 , i 1, 2, 3 can be expressed as in 3.6 .

5.26
Reiterating this process for each remaining triangle, at the step k, we will obtain the compact set T k given by 3 k triangles whose edges are contracted by 1/2 k .In other words,

5.27
Finally, we note that through the three simplicial maps in R 2 , provided with the natural metric d, we are able to construct the IFS R 2 , d, w 1 , w 2 w 3 that has the well-known Sierpinski gasket T k T k 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 .
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 matrix W .

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.

Translation
Let the translation operator be defined as the operator where v v 1 , v 2 , . . ., v m is a given vector of R m , then the image of a simplex σ under the function T is the translation of σ by T so that any vertex V i is transformed into 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 R 2 , it is being W the zero matrix.

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 .R 2 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 V 0 is the operator where R θ is the matrix R θ cos θ − sin θ sin θ cos θ , 6.7 so that 5.7 , when applied to the simplex With respect to an orthogonal coordinate system with origin O, for any P ∈ σ, we define the rotation as the bijective simplicial map which applies P ∈ σ into P ∈ σ, P def R θ P − v v, 6.9 where v V 0 − O; in particular, the vertices V 0 , V 1 , and V 2 are transformed into 6.10

Von Koch Snowflake
Let 0xy be an orthogonal coordinate system for R 2 , and let σ 2 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 σ 1 A, B , σ 1 B, C , and σ 1 A, C .The coordinates of vertices are A x A , y A , B x B , y B , and C x C , y C , respectively.In the following we will give both the construction of the Koch curve as IFS on σ 1 and the construction of the Koch snowflake as IFS on σ 2 .

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 Figure 5 .
Following the classical construction, we consider the following maps on 1-simplexes: where M is the matrix Hence, w 1 is a factor of the homothety w having A as a fixed vertex, while w 2 leaves B unchanged:

6.13
Moreover, each vertex in the w i σ 1 , i 1, 2, can be expressed as in 3.6 Let us now consider the transformation, on two steps, which first rotates w 1 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 where M is the matrix Finally, let us apply the transformation which first rotates w 1 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 where M is the matrix Since, as previously shown, rotation and translation are isometries, for each w i σ 1 , i 1, 2, 3, 4, we obtain .

6.19
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 w 3 and w 4 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 .
Reiterating this process for each remaining segment, at the step k we will obtain the compact set T k made of 2 2k segments whose sides are contracted by a factor 1/3 k .The IFS map R 1 , d, w 1 , w 2 , w 3 , w 4 gives us the Koch curve L k L k with similarity dimension equal to s log 4 log 1/ 1/3 log 4 log 3 .

6.20
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 fractalshaped curve.

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 the kth step will be 4/3 k of the original triangle perimeter.

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.
12 a 13 a 14 a 15 a 16 a 21 a 22 a 23 a 24 a 25 a 26 a 31 a 32 a 33 a 34 a 35 a 36 a 41 a 42 a 43 a 44 a 45 a 46 a 51 a 52 a 53 a 54 a 55 a 56 a 61 a 62 a 63 a 64 a 65 a 66 34 a 35 a 36 a 43 a 44 a 45 a 46 a 53 a 54 a 55 a 56 a 63 a 64 a 65 a 66 34 a 35 a 36 0 0 a 43 a 44 a 45 a 46 0 0 a 53 a 54 a 55 a 56 0 0 a 63 a 64 a 65 a 66 vertex V 2 remains fixed, and the map w on σ 2 is a consequence of the transformation of the simplex σ 1 V 1 , V 3 , so that 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44