Simplicial Approach to Fractal Structures

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.

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 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.

2. 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 𝑉₀,𝑉₁,…,𝑉_𝑚, is defined [1–3] as the subset of ℝ^𝑛,𝜎=^𝑚def𝑃∈ℝ^𝑛∣𝑃^𝑚_𝑖=0𝜆^𝑖𝑉_𝑖with^𝑚_𝑖=0𝜆^𝑖=1,0≤𝜆^𝑖≤1. (2.1) Let us denote with [𝜎^𝑚]=[𝑉₀,𝑉₁,…,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+1𝑝+1.

The 𝑚-dimensional simplicial complex Σ^𝑚 is defined as the finite set of 𝑝 simplexes (𝑝≤𝑚) such that

(1) for all 𝜎^𝑘∈Σ^𝑚 if 𝜎^ℎ⪯𝜎^𝑘, then 𝜎^ℎ∈Σ^𝑚,

(2) for all 𝜎^𝑘,𝜎^ℎ∈Σ^𝑚, then either [𝜎^ℎ]∩[𝜎^𝑘]=∅ or [𝜎^ℎ]∩[𝜎^𝑘]=[𝜎^𝑗] with 𝜎^𝑗∈Σ^𝑚.

The set of points 𝑃 such that 𝑃∈𝜎^𝑝, 𝑝≤𝑚, and 𝜎^𝑝∈Σ^𝑚 is the geometric support of Σ^𝑚 also called 𝑚-polyhedron ℳ^𝑚. The 𝑝-skeleton of Σ^𝑚 is [Σ^𝑚]=^𝑝def[𝜎^𝑝]forall𝜎^𝑝∈Σ^𝑚. The boundary 𝜕Σ^𝑚 of Σ^𝑚 is the complex Σ^𝑚−1 such that each 𝜎^𝑚−1∈Σ^𝑚−1 is face of only one 𝑚-simplex of Σ^𝑚. 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 the barycentric basis as follows: given the 𝑚-simplex 𝜎^𝑚 with vertices 𝑉₀,…,𝑉_𝑚, the barycentric basis is the set of (𝑚+1) vectors𝐞=_𝑖^def𝑉_𝑖−𝒢^𝑚, (2.2) based on the barycenter 𝒢=^𝑚def𝒢𝜎^𝑚=^𝑚_𝑖=01𝑉𝑚+1^𝑖. (2.3)

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^𝑚_𝑖=0𝐞_𝑖=𝟎. (2.4)

Each point 𝑃∈𝜎^𝑚 can be characterized by a set of barycentric coordinates (𝜆⁰,…,𝜆^𝑚) such that0≤𝜆^𝑖≤1,^𝑚_𝑖=0𝜆^𝑖=1,𝑖=0,…,𝑚, (2.5) and 𝑃−𝒢^𝑚=∑^𝑚_𝑖=0𝜆^𝑖𝐞⇒_𝑖^(2.2),(2.4)𝑃=∑^𝑚_𝑖=0𝜆^𝑖𝑉_𝑖. 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 as𝐞^𝑖,𝐞_𝑘=𝛿^𝑖_𝑘, (2.6) with [14]𝛿=_𝑘^𝑖def𝛿^𝑖_𝑘−1=−1𝑚+1+𝑚𝑚+1,𝑖≠𝑘,𝑚+1,𝑖=𝑘, (2.7)𝛿^𝑖_𝑘 being the Kroneker symbol. According to the definition (2.7), it is^𝑚_𝑖=0𝛿^𝑖_𝑘=^𝑚_𝑘=0𝛿^𝑖_𝑘=0. (2.8)

In addition, the metric tensor in 𝜎^𝑚 is defined as [5]𝑔_𝑖𝑗^def1=−2𝛿^ℎ_𝑖𝛿^𝑘_𝑗ℓ²_ℎ𝑘,𝑖,𝑗,ℎ,𝑘=0,1,…,𝑚 (2.9) being ℓ=²_ℎ𝑘^def(𝑉_𝑘−𝑉_ℎ)²=(𝐞_𝑘−𝐞_ℎ)².

2.2. Measures of the 𝑚-Simplex

Let 𝐋=_𝑖𝑗^def𝑉_𝑗−𝑉_𝑖=𝐞_𝑗−𝐞_𝑖,𝑙=_𝑖𝑗^def𝐋_𝑖𝑗,𝐋_𝑖𝑗, (2.10)

by using the ordinary wedge product of the vectors 𝐞_𝑗1,…,𝐞_𝑗𝑝, we can define the 𝑝-form 𝜔,1𝜔=𝑝!_{𝑗1,…,𝑗𝑚}𝜔^{𝑗1…𝑗𝑝}𝐞_𝑗1∧⋯∧𝐞_𝑗𝑝, (2.11)

whose affine components are 𝜔=^{𝑗1…𝑗𝑝def}⟨𝜔,𝐞_𝑗1∧𝐞_𝑗2∧⋯∧𝐞_𝑗𝑝⟩ [14].

The euclidean measure of the 𝑚-simplex 𝜎 (volume) is [14] 𝜀Ω=^2def1𝐋𝑚!⁰¹∧⋯∧𝐋_0𝑚, (2.12)

from where, it follows thatΩ²=1𝑚!³_{𝑗1𝑚,…,𝑗𝑘1,…,𝑘𝑚}𝜀^{𝑗1…𝑗𝑚}𝜀^{𝑘1…𝑘𝑚𝑚}_𝑎=1𝑙²_{𝑗𝑎𝑘𝑎}, (2.13) being𝜀=^{𝑗1…𝑗𝑚def}±1, (2.14) according to the even/odd permutation 𝑗₀,𝑗₁,…,𝑗_𝑚 of the indices 0,1,…,𝑚.

In particular, the volume of each 𝑝-face 𝜎_{𝑖1…𝑖𝑚−𝑝} (see also [9]) isΩ²_{𝑖1…𝑖𝑚−𝑝}=−12^𝑝1𝑝!³_{𝑗1𝑝,…,𝑗𝑘1,…,𝑘𝑝}𝜀^{𝑗1…𝑗𝑝}𝜀^{𝑘1…𝑘𝑝𝑝}_𝑎=1𝑙²_{𝑗𝑎𝑘𝑎}0<𝑝≤𝑚, (2.15) where 𝑗₁,…,𝑗_𝑝,𝑘₁,…,𝑘_𝑝≠𝑖₁,…,𝑖_𝑚−𝑝.

3. 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] ∀𝐯∈ℐ𝜎_𝑖𝐧^𝑖,𝐯=0(𝑖fixed), (3.1) where the normal vector 𝐧^𝑖 is defined as𝐧=^𝑖def−𝑚ΩΩ_𝑖𝐞^𝑖,ℎ=^𝑖def𝑚ΩΩ_𝑖. (3.2)

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 𝜎 is 𝑚-parallel to 𝜎^⌢ ( 𝜎‖𝜎_𝑚^⌢) if and if only 𝜎_𝑖‖𝜎^⌢_𝑖,thatis,𝐧^𝑖=𝐧^⌢𝑖(𝑖=0,…,𝑚).

Let 𝜑 be a map 𝜑∶ℝ^𝑚⟶ℝ^𝑚,𝜎⟼^𝜑𝜎^⌢ (3.3) 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: ∃𝑂∈ℝ^𝑚∣𝜑(𝑂)≡𝑂, (3.4)

(5) each (𝑚−1)-face 𝜎_𝑖 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,∞)) as𝑃^⌢defΩ=𝑃+𝑡⁰𝑚Ω^𝑚_𝑖=0𝜆^𝑖𝐋_0𝑖; (3.5) in particular, this function acts on any vertex 𝑉_𝑖 as𝑉^⌢_𝑖=𝑉_𝑖Ω+𝑡⁰𝐋𝑚Ω^0𝑖,𝑉^⌢₀=𝑉₀, (3.6) so that we can easily prove that all the previous conditions are easily satisfied [14]. According to the above equations, each edge transforms as𝐋^⌢_𝑖𝑗=Ω1+𝑡⁰𝐋𝑚Ω^𝑖𝑗, (3.7) where 𝐋^⌢_𝑖𝑗=𝑉^⌢_𝑗−𝑉^⌢_𝑖.

3.1. Variation Law of the 𝑝-Faces of 𝝈

The variation law of the edge lengths, resulting from (3.7), is given by the formula^⌢𝑙_𝑖𝑗=Ω1+𝑡⁰𝑙𝑚Ω^𝑖𝑗, (3.8) 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 {𝑙²_𝑖𝑗}_𝑖<𝑗, so that its variation law is^⌢ΩΩ=1+𝑡⁰𝑚Ω^𝑚Ω, (3.9) and for any 𝑝-face,Ω^⌢_{𝑖1…𝑖𝑚−𝑝}=Ω1+𝑡⁰𝑚Ω^𝑝Ω_{𝑖1…𝑖𝑚−𝑝}0<𝑝<𝑚; (3.10) analogously, taking into account the definition (5.5)₂, we have the transformation law of ℎ_𝑖:^⌢ℎ_𝑖Ω=𝑚1+𝑡⁰ℎ𝑚Ω^𝑖. (3.11) There follows, for the fundamental vectors of 𝜎^⌢, that 𝐞=^⌢_𝑖^def𝑉^⌢_𝑖−^⌢Ω𝒢=1+𝑡⁰𝐞𝑚Ω^𝑖_⌢,𝐧^𝑖=𝐧^𝑖,_⌢𝐞^𝑖=^⌢Ω_𝑖/Ω_𝑖⌢𝐞Ω/Ω^𝑖=Ω1+𝑡⁰𝑚Ω⁻¹𝐞^𝑖. (3.12)

4. 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 set𝐾ℝ^𝑛=𝐾⊆ℝ^𝑛∶𝐾isanonemptycompactset. (4.1)

The iterated function system (IFS) 𝑤_𝑖=ℝ^𝑛,𝑑,𝑤₁,𝑤₂,…,𝑤_𝑛 (4.2)

is the finite set of contractions 𝑤_𝑖 on the complete metric space (ℝ^𝑛,𝑑), being the contraction 𝑤 defined as 𝑑𝑤(𝑥),𝑤𝑦≤𝑐𝑑𝑥,𝑦,∀𝑥,𝑦∈ℝ^𝑛, (4.3)

with 𝑐 contraction coefficient.

For each 𝐴∈𝐾(ℝ^𝑛), the (IFS) contracting mapping is 𝑤∶𝐴∈𝐾ℝ^𝑛⟶𝑤₁(𝐴)⋯𝑤_𝑛(𝐴)∈𝐾ℝ^𝑛, (4.4)

with contraction coefficient 𝑐=max{𝑐₁,…,𝑐_𝑛}. Each function 𝑤_𝑖 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 set 𝐴 and by iterating the actions of 𝑤. Hence, 𝐴=_{𝑖1,…,𝑖ℎ=1,…,𝑛}𝑤_𝑖1∘⋯∘𝑤_𝑖ℎ(𝐴), (4.5)

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 𝑠=log𝑛log1/𝑐=−log𝑛log𝑐. (4.6)

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^𝐻𝑤(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 in ℝ². 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), 𝐾ℝ²=𝜎²;𝜎¹;𝜎⁰,𝑤∶𝐾ℝ²⟼𝐾ℝ². (5.1)

In particular, let 𝜎² be the simplex [𝑉₁,𝑉₂,𝑉₃], then it is 𝐾ℝ²=𝑉₁,𝑉₂,𝑉₃;𝑉₁,𝑉₂,𝑉₁,𝑉₃,𝑉₂,𝑉₃;𝑉₁,𝑉₂,𝑉₃, (5.2)

so that a map 𝑤 on 𝐾(ℝ²) could be the more general function defined on any face of 𝜎².

Examples 5. If the skeleton of 𝜎² 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 𝑎𝑊=¹¹𝑎₁₂𝑎₁₃𝑎₁₄𝑎₁₅𝑎₁₆𝑎₂₁𝑎₂₂𝑎₂₃𝑎₂₄𝑎₂₅𝑎₂₆𝑎₃₁𝑎₃₂𝑎₃₃𝑎₃₄𝑎₃₅𝑎₃₆𝑎₄₁𝑎₄₂𝑎₄₃𝑎₄₄𝑎₄₅𝑎₄₆𝑎₅₁𝑎₅₂𝑎₅₃𝑎₅₄𝑎₅₅𝑎₅₆𝑎₆₁𝑎₆₂𝑎₆₃𝑎₆₄𝑎₆₅𝑎₆₆ (5.3) and the constant vector 𝑈=𝑢₁,𝑢₂,𝑢₃,𝑢₄,𝑢₅,𝑢₆. (5.4) The function 𝑤 maps a 2-simplex into a 2-simplex whereas, by a matrix product, the vector 𝑋=𝑥₁,𝑦₁,𝑥₂,𝑦₂,𝑥₃,𝑦₃ (5.5) is mapped into the vector 𝑊𝑋+𝑈, (5.6) 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 𝑊=10000001000000𝑎₃₃𝑎₃₄0000𝑎₄₃𝑎₄₄000000𝑎₅₅𝑎₅₆0000𝑎₆₅𝑎₆₆, (5.7) with 𝑎₃₃𝑎₄₄−𝑎₃₄𝑎₄₃=±1,𝑎₅₅𝑎₆₆−𝑎₅₆𝑎₆₅=±1, (5.8) 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 of 𝜎² such that 𝑤𝜎²=𝑤₁𝜎²₁∪𝑤₂𝜎²₂∪𝑤₃𝜎²₃,#𝑤𝜎²=3. (5.9) 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 vertex 𝑉₁ remains fixed, and the map 𝑤 on 𝜎² is a consequence of the transformation of the simplex 𝜎₁=[𝑉₂,𝑉₃], that is, by defining 𝐼=1001,𝑊₁=𝑎₃₃𝑎₃₄𝑎₃₅𝑎₃₆𝑎₄₃𝑎₄₄𝑎₄₅𝑎₄₆𝑎₅₃𝑎₅₄𝑎₅₅𝑎₅₆𝑎₆₃𝑎₆₄𝑎₆₅𝑎₆₆, (5.10) it is 𝑊=𝐼⊕𝑊₁=10000001000000𝑎₃₃𝑎₃₄𝑎₃₅𝑎₃₆00𝑎₄₃𝑎₄₄𝑎₄₅𝑎₄₆00𝑎₅₃𝑎₅₄𝑎₅₅𝑎₅₆00𝑎₆₃𝑎₆₄𝑎₆₅𝑎₆₆, (5.11)

(b) the second vertex 𝑉₂ remains fixed, and the map 𝑤 on 𝜎² is a consequence of the transformation of the simplex 𝜎₁=[𝑉₁,𝑉₃], so that 𝑊₂=𝑎₁₁𝑎₁₂𝑎₁₅𝑎₁₆𝑎₂₁𝑎₂₂𝑎₂₅𝑎₂₆𝑎₅₁𝑎₅₂𝑎₅₅𝑎₅₆𝑎₆₁𝑎₆₂𝑎₆₅𝑎₆₆,𝑎𝑊=¹¹𝑎₁₂00𝑎₁₅𝑎₁₆𝑎₂₁𝑎₂₂00𝑎₂₅𝑎₂₆𝑎001000000100₅₁𝑎₅₂00𝑎₅₅𝑎₅₆𝑎₆₁𝑎₆₂00𝑎₆₅𝑎₆₆, (5.12)

(c) the third vertex 𝑉₃ remains fixed, and the map 𝑤 on 𝜎² is a consequence of the transformation of the simplex 𝜎₁=[𝑉₁,𝑉₂], that is, 𝑊=𝑊₃𝑎⊕𝐼=¹¹𝑎₁₂𝑎₁₃𝑎₁₄𝑎00₂₁𝑎₂₂𝑎₂₃𝑎₂₄𝑎00₃₁𝑎₃₂𝑎₃₃𝑎₃₄𝑎00₄₁𝑎₄₂𝑎₄₃𝑎₄₄00000010000001, (5.13) being 𝑊₃=𝑎₁₁𝑎₁₂𝑎₁₃𝑎₁₄𝑎₂₁𝑎₂₂𝑎₂₃𝑎₂₄𝑎₃₁𝑎₃₂𝑎₃₃𝑎₃₄𝑎₄₁𝑎₄₂𝑎₄₃𝑎₄₄. (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 𝑊₁,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-simplex 𝜎²={𝐴,𝐵,𝐶} and the map (Figure 1)𝜎²={𝐴,𝐵,𝐶}⟹𝑤𝜎²=𝐴,𝐵^′,𝐶^′, (5.15)

Figure 1 

Homothety map.

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 𝜎², since𝑤𝜎²=𝑤₁([𝐴,𝐵])∪𝑤₂([𝐵,𝐶])∪𝑤₃([𝐴,𝐶]). (5.16) This map is a scale invariant, since there results ℓ²_𝐴𝐵=𝜆ℓ²_{𝐴′𝐵′},(0≤𝜆), (5.17)ℓ²_𝐵𝐶=𝜆ℓ²_{𝐵′𝐶′},andℓ²_𝐴𝐶=𝜆ℓ²_{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=𝜆𝑔_𝑖𝑗.

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 system 0𝑥𝑦 in ℝ² and three homothety maps 𝑤₁, 𝑤₂, and 𝑤₃. 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 [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 of 𝜎² 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:𝑤₁∶𝑥_𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵𝑥_𝐶𝑦_𝐶𝑥⟹𝑀⋅^𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵𝑥_𝐶𝑦_𝐶,𝑤₂∶𝑥_𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵𝑥_𝐶𝑦_𝐶𝑥⟹𝑀⋅^𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵𝑥_𝐶𝑦_𝐶+000,𝑤1/21/21/2₃∶𝑥_𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵𝑥_𝐶𝑦_𝐶𝑥⟹𝑀⋅^𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵𝑥_𝐶𝑦_𝐶+000,1/21/21/2 (5.18) where 𝑀 is the matrix𝑀=1/20000001/20000001/20000001/20000001/20000001/2. (5.19)

Figure 2 

Fundamental maps.

Once we get the vertices of 𝑤(𝜎²), we can easily define the map for each point 𝑃 of the 𝜎² convex domain.

Comment
In fact, let 𝜆₁, 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 𝑥=𝜆₁𝑥_𝐴+𝜆₂𝑥_𝐵+𝜆₃𝑥_𝐶,𝑦=𝜆₁𝑦_𝐴+𝜆₂𝑦_𝐵+𝜆₃𝑦_𝐶. (5.20)

Substituting 𝜆₃=1−𝜆₁−𝜆₂ into the above and rearranging, this linear transformation can be written as 𝐻⋅Λ=𝑃−𝐶, (5.21)

where Λ is the vector of barycentric coordinates, and 𝐻 is the matrix 𝑥𝐻=^𝐴−𝑥_𝐶𝑥_𝐵−𝑥_𝐶𝑦_𝐴−𝑦_𝐶𝑦_𝐵−𝑦_𝐶. (5.22)

Since 𝐻 is invertible, we can easily obtain the barycentric coordinates of 𝑃=(𝑥,𝑦): 𝜆₁=𝑦_𝐵−𝑦_𝐶𝑥−𝑥_𝐶+𝑥_𝐶−𝑥_𝐵𝑦−𝑦_𝐶𝑦_𝐵−𝑦_𝐶𝑥_𝐴−𝑥_𝐶+𝑥_𝐶−𝑥_𝐵𝑦_𝐴−𝑦_𝐶,𝜆₂=𝑦_𝐶−𝑦_𝐴𝑥−𝑥_𝐶+𝑥_𝐴−𝑥_𝐶𝑦−𝑦_𝐶𝑦_𝐶−𝑦_𝐴𝑥_𝐵−𝑥_𝐶+𝑥_𝐴−𝑥_𝐶𝑦_𝐵−𝑦_𝐶,𝜆₃=1−𝜆₁−𝜆₂. (5.23)

According to (5.9) each map 𝑤_𝑖,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^⌢=𝑤_𝑖([𝜎²]) as^⌢1𝑃=𝑃−2^𝑚_𝑖=0𝜆^𝑖𝐋_0𝑖. (5.24) Moreover, each vertex in the 𝑤_𝑖([𝜎²]),𝑖=1,2,3 can be expressed as in  (3.6)𝑉^⌢_𝑖=𝑉_𝑖−12𝐋_0𝑖, (5.25) so that𝐋^⌢_𝑖𝑗=12𝐋_𝑖𝑗,^⌢𝑙_𝑖𝑗=12𝑙_𝑖𝑗,_⌢1Ω=4Ω. (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,𝐋^⌢_𝑖𝑗=12^𝑘𝐋_𝑖𝑗;^⌢𝑙_𝑖𝑗=12^𝑘𝑙_𝑖𝑗,_⌢1Ω=2^2𝑘Ω. (5.27)

Finally, we note that through the three simplicial maps in ℝ², 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=⋂_𝑘𝑇_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).

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 matrix 𝑊).

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 𝑇∶ℝ^𝑚⟶ℝ^𝑚,𝜎⟼^𝑇𝜎^⌢ (6.1)

such that 𝑇(𝑃)=𝑃+𝐯, (6.2)

where 𝐯=(𝑣₁,𝑣₂,…,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 into𝑉^⌢_𝑖=𝑉_𝑖+𝐯. (6.3) 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 𝑈=𝑣₁,𝑣₂,𝑣₁,𝑣₂,𝑣₁,𝑣₂,𝑣₁=𝐶𝑛𝑠𝑡.,𝑣₂=𝐶𝑛𝑠𝑡., (6.4)

being 𝑊 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 𝑅∶ℝ²⟶ℝ²,𝜎⟼^𝑅𝜎^⌢ (6.5)

such that 𝑅(𝑃)=𝑉₀+𝑅_𝜃𝑃−𝑉₀, (6.6)

where 𝑅_𝜃 is the matrix𝑅_𝜃=cos𝜃−sin𝜃sin𝜃cos𝜃, (6.7) so that (5.7), when applied to the simplex 𝜎²=[𝑉₀,𝑉₁,𝑉₂] with one fixed vertex, becomes𝑊=10000001000000cos𝜃−sin𝜃0000sin𝜃cos𝜃000000cos𝜃−sin𝜃0000sin𝜃cos𝜃. (6.8)

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=^⌢def𝑅_𝜃(𝑃−𝐯)+𝐯, (6.9) where 𝐯=𝑉₀−𝑂; in particular, the vertices 𝑉₀,𝑉₁,and𝑉₂ are transformed into𝑉^⌢₀=𝑉₀,𝑉^⌢_𝑖=𝑅_𝜃𝐋_0𝑖+𝑂+𝑉₀−𝑂,(𝑖=1,2). (6.10)

6.3. Von Koch Snowflake

Let 0𝑥𝑦 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=(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 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 Figure 5).

Following the classical construction, we consider the following maps on 1-simplexes:𝑤₁∶𝑥_𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵𝑥⟹𝑀⋅^𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵,𝑤₂∶𝑥_𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵𝑥⟹𝑀⋅^𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵+00,2/32/3 (6.11) where 𝑀 is the matrix𝑀=1/300001/300001/300001/3. (6.12) Hence, 𝑤₁ is a factor of the homothety 𝑤 having 𝐴 as a fixed vertex, while 𝑤₂ leaves 𝐵 unchanged: 𝑤₁(𝑃)2=𝑃−3𝜆^𝑖𝐋_0𝑖,𝐋_0𝑖=𝑉_𝑖𝑤−𝐴,₂(𝑃)2=𝑃−3𝜆^𝑖𝐋_0𝑖,𝐋_0𝑖=𝑉_𝑖−𝐵. (6.13)

Moreover, each vertex in the 𝑤_𝑖([𝜎¹]),𝑖=1,2, can be expressed as in(3.6)𝑉^⌢_𝑖=𝑉_𝑖−23𝐋_0𝑖. (6.14)

Let us now consider the transformation, on two steps, which first rotates 𝑤₁([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 obtain𝑤₃∶𝑥_𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵𝑥⟹𝑀′⋅^𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵+001/31/3, (6.15) where 𝑀′ is the matrix√𝑀′=1/6−√3/600√3/61/600001/6−√3/6003/61/6. (6.16)

Finally, let us apply the transformation which first rotates 𝑤₁([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 is𝑤₄∶𝑥_𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵⟹𝑀^′′⋅𝑥_𝐴𝑦_𝐴𝑥_𝐵𝑦_𝐵+002/32/3, (6.17) where 𝑀^′′ is the matrix𝑀^′′=−1/6−√√3/600√3/6−1/60000−1/6−√3/6003/6−1/6. (6.18)

Since, as previously shown, rotation and translation are isometries, for each 𝑤_𝑖([𝜎¹]),𝑖=1,2,3,4, we obtain𝐋^⌢_𝑖𝑗=13𝐋_𝑖𝑗,^⌢𝑙_𝑖𝑗=13𝑙_𝑖𝑗,_⌢1Ω=3Ω. (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 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 of 𝑤([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 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 (ℝ¹,𝑑,𝑤₁,𝑤₂,𝑤₃,𝑤₄) gives us the Koch curve 𝐿=⋂_𝑘𝐿_𝑘 with similarity dimension equal to 𝑠=log4()=log1/1/3log4log3. (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 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 the 𝑘th 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.