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Sierpiński triangle

The Sierpiński triangle, also known as the Sierpiński gasket or Sierpiński sieve, is a self-similar subset of the that approximates the shape of an through an iterative process of subdivision and removal. It is constructed by beginning with a solid and repeatedly excising the interior of the central inverted triangle formed by connecting the midpoints of each side, dividing the original into four smaller s of equal size and removing the middle one, then applying the same procedure to the three remaining triangles . The limit set has zero area () but positive topological complexity, with a of \log_2 3 \approx 1.58496, which lies strictly between 1 (the dimension of a line) and 2 (the dimension of the plane). Named after the mathematician , who first described it in 1915 as an example of a "Cantorian " where every point is a , the structure was introduced in his paper "Sur une courbe cantorienne dont tout point est un point de ramification," published in the Comptes Rendus de l'Académie des Sciences. Although formalized mathematically in the early , patterns resembling the Sierpiński triangle appeared centuries earlier in 13th-century Italian Cosmatesque mosaics, such as those in the Basilica of San Giovanni in Laterano in , predating its interpretation. Sierpiński's work contributed to the foundations of point-set topology and geometry. The Sierpiński triangle exemplifies key fractal characteristics, including exact at every scale and , making it a canonical example in and dynamical systems. It serves as the attractor of an consisting of three contractions, each scaling by a factor of $1/2 toward the vertices of the initial triangle, and can also be generated via probabilistic methods like the chaos game, where random iterations converge to the set. Beyond pure mathematics, the structure arises in cellular automata (e.g., in Wolfram's elementary rules) and has applications in antenna design for its multiband electromagnetic properties, as well as in modeling and porous materials in physics. Generalizations extend to higher dimensions, such as the Sierpiński or , and variants like the Sierpiński curve preserve similar fractal traits.

Constructions

Iterative Triangle Removal

The iterative triangle removal construction of the Sierpiński triangle, first described by the Polish mathematician in 1915, begins with an initial of side length 1 and area 1. In the first , the midpoints of the sides of this initial triangle are connected to form a middle inverted with side length \frac{1}{2} and area \frac{1}{4}; this central triangle is then removed, leaving three smaller upright s, each with side length \frac{1}{2} and area \frac{1}{4}. The process is then repeated recursively on each of the three remaining triangles: in the second , the midpoints of the sides of each are connected to form three middle inverted triangles, each with side length \frac{1}{4} and area \frac{1}{16}, which are removed, leaving nine even smaller upright equilateral triangles, each with area \frac{1}{16}. In general, at the nth , $3^{n-1} inverted triangles, each of area \left( \frac{1}{4} \right)^n, are removed from the current figure. Continuing this removal process infinitely yields the Sierpiński triangle as the limiting set. After a finite number of iterations, say n, the figure consists of $3^n small equilateral of side length \frac{1}{2^n}, arranged in a self-similar that forms the outline of the original , with boundaries that grow progressively more intricate and jagged, foreshadowing the nature of the limit.

Iterative Shrinking and Duplication

The iterative shrinking and duplication method constructs the Sierpiński triangle additively by generating successive levels of smaller, duplicated copies of an initial , rather than subtracting areas from a solid shape. This approach relies on an (IFS) comprising three affine contraction transformations, each scaling by a factor of \frac{1}{2} and translating to position copies at the corners of a larger . The process, popularized in the framework of IFS theory, builds the through repeated application of these mappings to an initial set, converging to the unique known as the Sierpiński triangle. Start with a single small serving as the base unit at level 0. At iteration 1, create three copies of this base , each scaled linearly by a of \frac{1}{2} relative to the side of a larger enclosing , and position them at the three corners of that larger . These placed copies form the level-1 , with the central region left empty, producing a structure resembling a larger outlined by the smaller ones at its vertices. For iteration 2, apply the same to each of the three level-1 triangles: replace each with three even smaller copies, scaled by an additional of \frac{1}{2} (equivalent to \frac{1}{4} relative to the original large triangle's side length), again positioned at the corners of their respective enclosing triangles. This results in $3^2 = 9 tiniest triangles, further refining the pattern with additional gaps. In general, each replaces every existing triangle in the current approximation with three scaled copies using the \frac{1}{2} contraction , multiplicatively increasing the number of smallest triangles to $3^n after n steps. This recursive duplication and shrinking process generates increasingly detailed approximations that accumulate to form the Sierpiński triangle as the , highlighting its construction through positive addition of self-similar components in contrast to subtractive methods.

Chaos Game

The Chaos Game is a probabilistic iterative algorithm for generating points that approximate the Sierpiński triangle, introduced by Michael Barnsley in 1988. The algorithm begins with an arbitrary initial point in the plane and an equilateral triangle defined by vertices A, B, and C. Three contraction mappings are specified, each shrinking the current point toward one vertex by a factor of $1/2: the mapping to A is given by P_{n+1} = \frac{P_n + A}{2}, and analogously for B and C. At each iteration, one of the three mappings is selected randomly with equal probability $1/3 and applied to the current point to produce the next point, which is then plotted. After many iterations, typically thousands, the resulting set of points densely approximates the Sierpiński triangle, filling it regardless of the initial point chosen. The method succeeds because the Sierpiński triangle is the attractor—the unique nonempty compact invariant set—of the associated random iterated function system (IFS), where the three contractions each have probability $1/3. In practice, the algorithm can be implemented with the following steps or pseudocode, making it straightforward for computational generation:
  1. Select an initial point P_0 (e.g., the centroid of the triangle or any point inside it).
  2. Define the vertices A, B, C (e.g., A = (0,0), B = (1,0), C = (0.5, \sqrt{3}/2)).
  3. For i = 1 to N (e.g., N = 10000):
    • Choose a vertex V uniformly at random from \{A, B, C\}.
    • Set P_i = \frac{P_{i-1} + V}{2}.
    • Plot or store P_i (optionally skipping the first 50–100 points to avoid transient effects).
This approach is widely used in for rendering fractals, as it efficiently produces detailed approximations through simple random iterations rather than complex geometric computations.

Arrowhead Curve

The Sierpiński arrowhead curve provides a geometric construction of the Sierpiński triangle through iterative replacement of s. It begins with a base curve consisting of a single straight . In the first iteration, the middle third of this segment is replaced by the two sides of an , pointing upward to create a pattern, resulting in a polyline of three equal-length segments. Subsequent iterations apply the same replacement rule to every straight segment in the current curve, but with the pointing alternately downward and upward to maintain the inward-filling pattern. This recursive process, which exhibits in its replacement rule, generates increasingly intricate polylines. As the number of iterations approaches , the arrowhead converges to a continuous path whose image coincides with the Sierpiński triangle. The traces both the boundary and interior points of the in a specific sequential order, filling the set without self-intersections. A key property of this limit is that it is continuous but nowhere differentiable, with an infinite total length due to the tripling of segments at each step while scaling lengths by 1/2.

Cellular Automata

The Sierpiński triangle emerges as a prominent pattern in the space-time evolution of certain elementary cellular automata, particularly Wolfram's Rule 90, which demonstrates how simple local rules can generate complex fractal structures. In a one-dimensional elementary cellular automaton defined on an infinite line of cells, each cell holds a binary state—0 (off or dead) or 1 (on or alive)—and updates synchronously based on its two nearest neighbors. Rule 90 specifies that the next state of a cell is the exclusive OR (XOR) of the states of its left and right neighbors from the previous time step, equivalent to their sum modulo 2. This rule was introduced by Stephen Wolfram as part of his systematic classification of all 256 possible elementary cellular automata in 1983. Starting from an consisting of a single '1' cell amid an infinite sea of '0's, the automaton's evolution over discrete time steps produces a distinctive pattern in the space-time diagram, where the horizontal axis represents cell positions and the vertical axis tracks time progression downward. The resulting figure forms an of '1's that expands linearly with time, punctuated by self-similar triangular gaps of '0's at every scale, visually replicating the Sierpiński triangle's structure. This fractal pattern arises because Rule 90 is linear over the finite field GF(2), allowing the system's evolution to be analyzed using linear algebra modulo 2; the state of each cell at time t and position x corresponds to the binomial coefficient \binom{t}{x} modulo 2, which inherently encodes the Sierpiński triangle's geometry. The self-similarity of the pattern ensures that subsystems behave identically to the whole, with finer details mirroring larger structures after sufficient steps. This configuration connects to the static structure of modulo 2, where rows align with the automaton's time slices. Variations on , such as Rules 60 and 102, generate analogous Sierpiński patterns but with mirrored or skewed orientations, while two-dimensional extensions of linear rules can produce higher-dimensional analogs, though remains the canonical one-dimensional example.

Pascal's Triangle Modulo 2

Pascal's triangle is a triangular array where each entry in row n (starting from n = 0) is the \binom{n}{k} for k = 0 to n, constructed such that each number is the sum of the two numbers directly above it in the previous row. When the entries are considered modulo 2, the pattern of odd values—those congruent to 1 modulo 2—reveals a striking structure equivalent to the Sierpiński triangle. Specifically, shading the cells where \binom{n}{k} \equiv 1 \pmod{2} and leaving even entries blank produces a figure that, when rotated 180 degrees or viewed upside down, matches the self-similar geometry of the Sierpiński triangle. The mathematical foundation for determining which entries are odd relies on Lucas' theorem, a result in that computes coefficients modulo a prime p. For p = 2, Lucas' theorem simplifies to the condition that \binom{n}{k} \equiv 1 \pmod{2} the binary expansion of k is a "subset" of the binary expansion of n—meaning that for every bit position, the bit in k is 1 only if the corresponding bit in n is also 1 (equivalently, there is no carry-over when adding the binary representations of k and n - k). This bitwise criterion ensures that the odd entries cluster in a highly structured, recursive manner, mirroring the iterative removal process in the standard Sierpiński triangle construction. To illustrate, consider the first few rows of modulo 2:
Row nBinaryEntries modulo 2
001
111 1
2101 0 1
3111 1 1 1
41001 0 0 0 1
51011 1 0 0 1 1
61101 0 1 0 1 0 1
71111 1 1 1 1 1 1 1
In row 0, the single entry is . Row 1 has two odds: and . Row 2 shows odds at the ends ( 0 ), creating a . Row 3 fills completely ( , since 3 ≡ 1 mod 2). Higher rows exhibit gaps that widen in powers-of-two patterns, such as the central blank triangle in rows 4–7. Extending to the infinite , the modulo 2 pattern displays the full self-similar structure of the Sierpiński triangle at every scale: each upward-pointing triangle of odds is composed of three smaller copies of the same pattern, separated by inverted blank triangles, recursively . This fractal repetition arises directly from the binary nature of Lucas' theorem, where the pattern over $2^m rows consists of three scaled versions of the pattern over $2^{m-1} rows, surrounding a central even region. The resulting figure has \log_2 3 \approx 1.585, consistent with the Sierpiński triangle's properties.

Towers of Hanoi

The Towers of Hanoi puzzle involves three vertical pegs and a set of n disks of distinct decreasing sizes, initially stacked in order of size on one peg, with the largest at the bottom. The objective is to transfer the entire stack to another peg, adhering to two rules: only one disk may be moved at a time, and no larger disk may be placed atop a smaller one at any point. The state graph of this puzzle, known as the graph H_n^3, models all possible legal configurations as , totaling $3^n for n disks, since each disk can independently occupy any of the three pegs while maintaining strictly decreasing order on each peg. Edges in the graph represent valid single-disk moves between configurations. This graph is undirected and connected, with each having 3 except for the three terminal states (all disks on one peg), which have 2. The graph possesses a recursive structure that parallels the construction of the Sierpiński triangle. Specifically, H_n^3 comprises three disjoint copies of H_{n-1}^3, each representing the configurations of the top n-1 disks while the largest disk is fixed on one peg. These subgraphs are interconnected by three additional edges forming a triangle, which correspond to the possible moves of the largest disk between pegs once the smaller disks are appropriately positioned on the auxiliary peg. This subdivision repeats at every level, yielding a self-similar pattern. For small values of n, the graphs illustrate this progression clearly. When n=1, H_1^3 is a K_3—a with three vertices (one for each peg position of the single disk) and three edges. For n=2, the graph has nine vertices arranged as three such , one for each position of the largest disk, connected at their corners by the three edges for moving the largest disk, forming a larger triangular outline with internal structure. At n=3, with 27 vertices, the pattern expands to include nine smaller grouped into three medium-sized ones, connected similarly, resembling a of the Sierpiński triangle. In the limit as n \to \infty, the Hanoi H_n^3 converges geometrically and topologically to the infinite Sierpiński gasket , a object whose vertices and edges densely approximate the continuous Sierpiński triangle in the . Paths within this infinite encode sequences of legal moves for puzzles with arbitrarily many disks, with the from one terminal state to another requiring $2^n - 1 steps, corresponding to an optimal traversal. A key property is that the shortest path length between any two states in H_n^3 can be determined efficiently via a finite-state that simulates the recursive movement of disks, accounting for the positions differing across the representations of the states. For the classic problem of moving the from one to another, this is exactly $2^n - 1, and the sequence of moves aligns with a reflected , where successive configurations differ by exactly one disk position.

Properties

Fractal Dimension

The provides a measure of the irregularity and space-filling properties of fractals like the Sierpiński triangle, which is a self-similar set constructed as the union of three smaller copies of itself, each scaled by a of 1/2. For such self-similar sets satisfying the condition, the Hausdorff dimension d is the unique solution to the scaling relation $3 \left(\frac{1}{2}\right)^d = 1, yielding d = \frac{\log 3}{\log 2} \approx 1.58496. This value lies strictly between the dimension of a line (d=1) and that of the plane (d=2), quantifying the fractal's intermediate complexity: it is more intricate than a one-dimensional yet does not fill the two-dimensional area. The Sierpiński triangle has zero, consistent with its being less than 2, while its d-dimensional is positive and finite. The can be confirmed using the box-counting method, where the number of boxes N(\epsilon) of side length \epsilon needed to cover the set scales as N(\epsilon) \approx 3^{\log(1/\epsilon)/\log 2} = \epsilon^{-\log 3 / \log 2}, so the box-counting equals \frac{\log 3}{\log 2}.

Self-Similarity

The Sierpiński triangle is a self-similar set, meaning it can be expressed as the of scaled and translated copies of itself. Formally, if S denotes the Sierpiński triangle, it satisfies the equation S = f_1(S) \cup f_2(S) \cup f_3(S), where each f_i is a —a that contracts the by a factor of $1/2 toward one of the three vertices of an initial . These transformations ensure that each subs copy is geometrically similar to the original set with a scaling ratio of exactly $1/2, and the three resulting copies are disjoint, with no overlap between them. The self-similar structure enables a recursive of the Sierpiński triangle, in which every sub-triangle mirrors the overall form through appropriate scaling and positioning. This property underscores the fractal's invariance under magnification within its boundaries. Within the (IFS) framework, the Sierpiński triangle serves as the unique for the IFS defined by the three contractions f_1, f_2, f_3, as guaranteed by the contraction mapping theorem applied to complete metric spaces. A striking visual manifestation of this occurs when zooming into any corner of the triangle, revealing an identical copy of the entire structure at a reduced scale.

Measure and Boundary

The of the Sierpiński triangle is zero. Consider an initial of area 1. In the first , the central subtriangle with vertices at the midpoints of the sides is removed, accounting for an area of \frac{1}{4}. In the second , three such subtriangles, each of area \frac{1}{16}, are removed from the remaining pieces, for a total removed area of \frac{3}{16}. Continuing iteratively, the total removed area is the infinite sum \sum_{n=1}^{\infty} \frac{3^{n-1}}{4^n} = \frac{1/4}{1 - 3/4} = 1, leaving a set of area 0. The perimeter of the Sierpiński triangle is . Starting with an initial of side 1, the perimeter is 3. Each iteration introduces new boundary segments around the removed subtriangles: at step n, $3^n segments of $2^{-n} are added, contributing a of (3/2)^n. The total added is thus \sum_{n=1}^{\infty} \left( \frac{3}{2} \right)^n, a divergent since the common ratio $3/2 > 1, yielding an total perimeter. The Sierpiński triangle is a compact set with empty interior in \mathbb{R}^2, so its topological coincides with the entire set. The complement of the set in the plane has full outside the initial triangle and positive measure equal to 1 within the initial triangle. Although the approximating finite-stage sets are bounded by simple closed curves ( curves), the itself is and nowhere differentiable, yet its complement maintains positive area. The Hausdorff measure of the Sierpiński triangle at its Hausdorff dimension d = \log 3 / \log 2 \approx 1.585 is positive and finite, providing a non-zero "packing measure" that quantifies its size in this fractional dimension. This follows from the self-similar structure, where the measure satisfies a scaling relation across the three subtriangles, ensuring $0 < \mathcal{H}^d(S) < \infty. Estimates place this measure between approximately 0.77 and 0.82.

Topological Features

The Sierpiński triangle is a compact metric space embedded in the Euclidean plane. It arises as the intersection of a decreasing sequence of closed and bounded sets obtained through iterative removal of open triangles from an initial equilateral triangle, rendering it closed and bounded and thus compact by the Heine-Borel theorem. The set is path-connected, meaning that for any two points in the Sierpiński triangle, there exists a continuous path connecting them that lies entirely within the set; this property distinguishes it from totally disconnected fractals such as the Cantor set. It is also locally connected, with every neighborhood of a point containing a connected neighborhood basis in the subspace topology. As a dendrite, the Sierpiński triangle is a locally connected continuum containing no simple closed curves, exhibiting a tree-like structure without cycles. Its homotopy type is that of a contractible space, with trivial fundamental group due to the absence of loops, though its branching structure—featuring points of order 3 where three arcs meet—prevents simple arc-connectedness between all pairs of points. Removing a branch point of order 3 disconnects the space into exactly three connected components, reflecting its ternary self-similarity that preserves these topological features.

Generalizations

Other Moduli

The construction of the Sierpiński triangle using Pascal's triangle 2 can be generalized to other prime moduli p > 2, where entries C(n, k) congruent to 0 p are shaded, leaving the non-zero residues to form a self-similar pattern within the triangular array. This pattern arises from Lucas' theorem, which states that C(n, k) \not\equiv 0 \pmod{p} if and only if each base-p digit of k is at most the corresponding digit of n. The resulting non-zero set exhibits : at each iteration, the large of side length p contains \frac{p(p+1)}{2} smaller copies of the , each scaled by a factor of \frac{1}{p}. The of this set is given by d_H(p) = \frac{\log \left( \frac{p(p+1)}{2} \right)}{\log p}, which lies between 1 and 2, increasing toward 2 as p grows, reflecting a denser filling of the compared to the modulo-2 case. For instance, when p=3, the dimension is \log_3 6 \approx 1.631, and the pattern consists of 6 scaled copies, displaying triangular symmetry with denser clustering along the diagonals and boundaries than the sparser Sierpiński . These structures are gasket-like embedded in the of the , with the non-zero positions forming a Cantor dust of higher dimension rather than a simple curve.

Higher Dimensions

The Sierpiński tetrahedron, also known as the tetrix, is the three-dimensional analogue of the Sierpiński triangle, constructed iteratively from a regular . The process begins with a solid tetrahedron, after which the midpoints of each edge are connected to form four smaller tetrahedra at the corners and a central octahedral region; the central portion is removed, leaving the four corner tetrahedra, each scaled by a factor of 1/2 relative to the original. This removal process is repeated recursively on each remaining tetrahedron, resulting in a self-similar structure composed of increasingly smaller tetrahedral components. This construction can be formalized using an (IFS) consisting of four contraction mappings, each scaling by 1/2 and translating to one of the four vertices of the original . The resulting set is the unique nonempty compact of the IFS, invariant under the contractions, and exhibits inherited from the two-dimensional case, where subsets are scaled copies of the whole. The of the Sierpiński tetrahedron is given by Hutchinson's formula as \log_2 4 = 2, reflecting the four self-similar copies at scale 1/2. In higher dimensions, the analogue is the Sierpiński d-simplex (or d-tetrahedron), a generalization defined for d \geq 2 using an IFS with d+1 contraction mappings, each scaling by 1/2 toward one of the d+1 vertices of a d-dimensional simplex. The iterative removal process mirrors the three-dimensional case: starting from a solid d-simplex, connect the midpoints of the faces to isolate d+1 smaller simplices at the vertices and remove the interior, repeating ad infinitum. The Hausdorff dimension is \log_2 (d+1), which increases logarithmically with d but remains strictly less than d for all finite d. For example, in four dimensions, the Sierpiński 4-simplex (hyper-tetrahedron) has dimension \log_2 5 \approx 2.322, while in higher dimensions such as d=7, it reaches \log_2 8 = 3. These higher-dimensional Sierpiński simplices possess zero d-dimensional due to the iterative removal of interior volume, concentrating the structure on a set of lower-dimensional measure while maintaining positive (d-1)-dimensional measure in certain projections. Their self-similar nature facilitates analytical study in areas like and random walks, with the attractor serving as the of the IFS iterations. In applications, these fractals model porous media, such as in elements where Sierpiński tetrahedral perforations enhance fluid flow and thermal conductivity through hierarchical , as demonstrated in numerical simulations of gyroid-based structures.

Related Fractals

The serves as the two-dimensional square-based analogue to the . Its construction begins with a , which is divided into nine congruent subsquares of side length one-third; the central subsquare is then removed, leaving eight subsquares. This removal process is repeated iteratively on each of the remaining subsquares, resulting in a self-similar composed of eight copies scaled by a factor of 1/3 at each stage. Key properties of the Sierpiński carpet include a (area) of zero, despite being constructed within the , and a of \log 8 / \log 3 \approx 1.8928. Unlike the Sierpiński triangle, the carpet remains connected at every finite stage of its construction, developing enclosed holes but with no separation of its components. Additionally, it exhibits universality for continua: every in the can be embedded homeomorphically into the carpet, a property established in its original description. The Menger sponge extends this iterative removal concept to three dimensions as the cubic analogue of the . Starting from a unit cube, each face is divided into nine equal squares, the central square on each face is removed along with the central cubic region, yielding 20 smaller cubes of side length one-third; the process recurses on these subcubes. This results in a with zero volume but infinite surface area. Another related structure is the Vicsek fractal, a cross-shaped variant proposed as a deterministic model for . It begins with a central square and four adjacent squares forming a plus sign, then iteratively replaces each square's corners with scaled versions of the same cross pattern, scaled by 1/3. Like the others, it has zero area and an infinite perimeter due to its non-integer dimension. These share through iterative constructions that remove interior portions while preserving boundary complexity, analogous to the Sierpiński triangle's midpoint connection rule but adapted to square, cubic, or cross base shapes.

History and Naming

Early Developments

The concept of iterative removal processes, central to the construction of the Sierpiński triangle, found an early precursor in Georg Cantor's 1883 introduction of the , a one-dimensional obtained by successively removing middle thirds from a , serving as a foundational analogue for higher-dimensional removal-based fractals like the Sierpiński triangle. Patterns resembling the Sierpiński triangle appeared centuries earlier in 13th-century Cosmatesque mosaics, such as those in the of . John in . These pre-1915 developments lacked a unified framework or analysis of non-integer dimensions, existing instead as disparate examples in and art. Combinatorial patterns in , observed since the 17th century, reveal self-similar triangular formations akin to the Sierpiński structure when entries are highlighted by parity.

Sierpiński's Contribution and Etymology

In 1915, formally introduced what is now known as the Sierpiński triangle in his paper "Sur une courbe cantorienne dont tout point est un point de ramification," published in Comptes Rendus de l'Académie des Sciences 160: 302–305. There, he defined the set starting with a closed and iteratively removing the open central triangle formed by connecting the midpoints of each side's current triangles. Sierpiński proved that the limiting set has zero—meaning zero area—while its boundary has infinite length, establishing paradoxical properties that challenged classical notions of dimension and measure. Sierpiński's key contributions included connecting the construction to sets, highlighting its role in extending one-dimensional pathological sets to the plane and demonstrating topological universality properties. The of the set is \log_2 3 \approx 1.585, arising from the self-similar structure where each iteration replaces one triangle with three smaller ones scaled by a factor of $1/2. The figure is commonly called the Sierpiński triangle, though it is also referred to as the Sierpiński , with the latter term evoking the shape's resemblance to mechanical gaskets or seals used in . This naming was popularized in the through the burgeoning field of fractal geometry, notably in works drawing on Sierpiński's ideas. In his 1916 paper "O krzywych, które zawierają podobieństwo każdego danej krzywej" ("On Curves Which Contain the Image of Any Given Curve"), published in Matematicheskii Sbornik 30: 267–287, Sierpiński extended the iterative removal process to squares, yielding the —a two-dimensional analog with similar zero-area and infinite-perimeter traits—and further generalized the approach to higher dimensions, influencing subsequent constructions like the . These innovations positioned Sierpiński's work as foundational to theory, predating Benoit Mandelbrot's 1975 coinage of the term "fractal" by six decades and inspiring developments in and .