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Koch snowflake

The Koch snowflake is a fractal curve and the boundary of a fractal-shaped region in the plane, constructed iteratively from an equilateral triangle by repeatedly replacing the middle third of each line segment with two sides of a smaller equilateral triangle pointing outward.^[1] This process yields a simple closed Jordan curve with an infinite perimeter enclosing a finite area, serving as one of the earliest examples of a mathematical fractal.^[2] Introduced by Swedish mathematician Helge von Koch in his 1904 paper "Sur une courbe continue sans tangente obtenue par une construction géométrique élémentaire," the curve was originally presented to illustrate a continuous but nowhere differentiable function, predating the formal concept of fractals by decades.^[1] The name "snowflake" was later coined by mathematician Edward Kasner in his 1940 book Mathematics and the Imagination, evoking the intricate, symmetrical pattern reminiscent of snow crystals.^[2] The construction begins with an equilateral triangle of side length s, denoted as stage 0; at each subsequent stage n, every existing side is divided into three equal parts, and the middle segment is replaced by two sides of an equilateral triangle of length s/3^n, resulting in $3 \times 4^n sides of length s/3^n.^[1] Key properties highlight its fractal nature: the perimeter at stage n is $3s (4/3)^n, which diverges to infinity as n \to \infty, while the enclosed area approaches \frac{8}{5} times the area of the initial triangle, specifically \frac{2\sqrt{3}}{5} s^2.^[2] The boundary curve has a Hausdorff (or box-counting) dimension of \log_3 4 \approx 1.26186, exceeding 1 to reflect its space-filling tendency despite being a one-dimensional object topologically.^[1] These characteristics make the Koch snowflake a foundational example in fractal geometry, demonstrating paradoxical behaviors like infinite boundary length around a bounded region, and it has influenced studies in analysis, topology, and applications such as modeling irregular natural forms.^[3]

Origins and Development

Historical Context

In the late 19th century, the field of mathematical analysis began to grapple with pathological functions that challenged the intuitive assumptions of continuity and differentiability underlying classical calculus. A pivotal development occurred in 1872 when Karl Weierstrass presented the first explicit example of a continuous function that is nowhere differentiable, constructed as an infinite sum of cosine terms with increasing frequencies.^[4] This function, often expressed as f(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x) where $0 < a < 1 and ab > 1 + \frac{3\pi}{2}, demonstrated that continuity did not imply smoothness, sparking widespread interest in irregular, non-intuitive geometric objects.^[5] Weierstrass's work, initially met with skepticism, laid foundational groundwork for exploring limits of differentiability and inspired subsequent investigations into curves and sets with unusual properties. Building on these ideas, Georg Cantor's development of set theory in the 1880s introduced concepts of infinity and uncountable sets that profoundly influenced geometric thought. Cantor's ternary set, constructed by iteratively removing middle thirds from the unit interval, exemplified a set of measure zero yet possessing the cardinality of the continuum, highlighting paradoxes in dimension and density.^[6] This construction, first described around 1883, emphasized infinite iterative processes in geometry, challenging Euclidean notions of space and paving the way for fractal-like structures by revealing how infinite subdivisions could yield objects with fractional or ambiguous dimensions.^[5] Cantor's ideas permeated European mathematics, encouraging explorations of self-similar sets and the boundaries between one- and higher-dimensional forms during the fin de siècle. By the early 20th century, these precursors converged in the work of Scandinavian mathematicians, particularly in Sweden, where interest in space-filling curves and irregular geometries flourished amid broader philosophical debates on the nature of continuity. Helge von Koch, a prominent Swedish mathematician active from 1900 to 1910, contributed to this milieu by examining curves that embodied infinite complexity within finite bounds, influenced by the pathological examples of Weierstrass and Cantor. His 1904 publication introduced a specific iterative curve—later known as the Koch snowflake—as an illustration of a continuous yet non-differentiable boundary, extending the era's fascination with infinite processes to produce objects that blurred traditional geometric categories.^[6]

Discovery and Initial Description

Niels Fabian Helge von Koch (1870–1924) was a Swedish mathematician born in Stockholm, who earned his doctorate from Uppsala University in 1897 and later succeeded Gösta Mittag-Leffler as professor of mathematics at Stockholm University in 1911.^[7]^[8] His research focused on areas including number theory, infinite linear equations, and the emerging study of pathological curves in analysis.^[7] In 1904, von Koch introduced what is now known as the Koch curve in his paper "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire," published in Acta Mathematica. The curve was presented as an elementary geometric construction demonstrating a continuous function that is nowhere differentiable, challenging classical notions of smoothness in plane curves at the time.^[1] Von Koch described the curve's formation starting with an equilateral triangle as the initial closed path, followed by iterative additions of smaller equilateral triangular protrusions on each side, yielding a limit shape that forms a closed boundary of infinite length enclosing a finite area.^[1] This construction highlighted the curve's paradoxical properties, serving as an early example in the study of non-differentiable yet continuous paths. Although von Koch's work predated the formal development of fractal geometry, the snowflake curve gained recognition as a foundational fractal once Benoit Mandelbrot coined the term "fractal" in 1975 to describe self-similar geometric objects with non-integer dimensions.^[5]

Geometric Construction

Basic Iterative Process

The Koch snowflake is generated through a recursive geometric construction that begins with a simple equilateral triangle as the initial shape, often taken with side length 1 for standardization. This starting figure, denoted as stage 0, serves as the foundation for all subsequent iterations.^[1] The core iteration rule involves modifying every straight line segment present in the current stage. Specifically, at each step n \geq 1, the middle third of each segment is identified and removed, then replaced by two equal segments that form the other two sides of an equilateral triangle protruding outward from the original line. This replacement effectively divides each original segment into four smaller segments, each one-third the length of the predecessor, while introducing a new "bump" or peak. The process is applied uniformly to all segments simultaneously.^[9] The full curve emerges as the limit of this infinite iterative process, where the stage n approximation approaches the final fractal boundary as n tends to infinity. Initially smooth and polygonal, the shape evolves by accumulating these outward triangular additions, resulting in a progressively more detailed and crinkled boundary that maintains closure around the interior region. Visually, early stages reveal a star-like form with subtle indentations, which grow into a highly irregular, symmetrical outline evocative of a crystalline snowflake.^[10]

Detailed Stage-by-Stage Evolution

The construction of the Koch snowflake proceeds iteratively, beginning with a simple equilateral triangle and applying a recursive replacement rule to each side at every stage, as originally described by Helge von Koch.^[11] This process assumes familiarity with basic geometric figures, such as equilateral triangles, where all sides are equal and all angles measure 60 degrees. At stage 0, the snowflake is initialized as an equilateral triangle with side length typically taken as 1 unit, comprising exactly 3 straight sides.^[1] This forms a closed polygonal boundary enclosing a finite area. In stage 1, each of the 3 sides is divided into three equal segments, and the middle segment is replaced by two sides of a smaller equilateral triangle protruding outward, effectively adding a "bump" to the boundary.^[12] This transformation increases the total number of sides to 12, as each original side contributes 4 new segments (the two outer thirds plus the two new sides of the bump).^[1] The resulting figure resembles a triangle with notched edges, maintaining closure while introducing irregularity. For stage 2, the replacement rule is applied uniformly to each of the 12 sides from stage 1, dividing each into three segments and substituting the middle one with two outward-protruding sides of an even smaller equilateral triangle.^[12] This yields 48 sides in total, since the number of segments multiplies by 4 per iteration (12 × 4 = 48).^[1] Visually, the boundary begins to exhibit emergent star-like patterns, with multiple triangular protrusions creating a more intricate, spiky outline around the original triangular shape.^[12] Subsequent stages, such as stage 3, continue this process by applying the rule to all 48 sides, producing 192 sides (48 × 4 = 192), and so on, with the number of sides at stage n given by $3 \times 4^n.^[1] Each iteration refines the boundary further, multiplying the segment count by 4 while reducing individual segment lengths by a factor of $1/3, leading toward the fractal limit as n approaches infinity.^[12] In this limit, the Koch snowflake converges to a continuous, simple closed curve known as a Jordan curve, despite its infinite complexity.

Intrinsic Properties

Perimeter Calculation

The Koch snowflake begins with an equilateral triangle of side length s, yielding an initial perimeter at stage 0 of P_0 = 3s.^[1] In each subsequent iteration, every straight side of length s_k from the previous stage is replaced by four segments, each of length s_k / 3: the two outer segments retain the original direction, while the middle third protrudes outward to form an equilateral triangular bump, adding two new segments that together increase the total length per side by a factor of $4/3.^[2] This replacement effectively multiplies the perimeter by $4/3 at every stage, as the number of sides quadruples while their lengths are one-third as long.^[13] The general formula for the perimeter at stage n is thus P_n = 3s \left(\frac{4}{3}\right)^n.^[1] This recursive growth arises from the iterative process described in the geometric construction, where the scaling factor exceeds unity.^[14] As n approaches infinity, the perimeter P = \lim_{n \to \infty} P_n diverges to infinity because \left(\frac{4}{3}\right)^n grows exponentially without bound.^[2] Specifically, the perimeter can be expressed as P = P_0 \sum_{k=0}^{\infty} \left(\frac{4}{3}\right)^k, and this series diverges because the ratio r = 4/3 > 1.^[1] This infinite boundary length exemplifies a key property of fractal curves, where the curve becomes arbitrarily intricate yet remains continuous.^[13]

Area Determination

The Koch snowflake is constructed iteratively starting from an equilateral triangle of side length s, which encloses an initial area of

A_0 = \frac{\sqrt{3}}{4} s^2.

This serves as the area at stage 0. At each subsequent stage n \geq 1, the construction adds new equilateral triangles to the existing sides. Specifically, stage n introduces $3 \times 4^{n-1} such triangles, each with side length s / 3^n and individual area \frac{\sqrt{3}}{4} (s / 3^n)^2 = A_0 / 9^n. The total area added at stage n is thus

\delta A_n = 3 \times 4^{n-1} \times \frac{A_0}{9^n} = \frac{1}{3} \left( \frac{4}{9} \right)^{n-1} A_0.

Summing these contributions from stage 1 to infinity yields the total added area:

\sum_{n=1}^{\infty} \delta A_n = \frac{1}{3} A_0 \sum_{k=0}^{\infty} \left( \frac{4}{9} \right)^k = \frac{1}{3} A_0 \cdot \frac{1}{1 - 4/9} = \frac{1}{3} A_0 \cdot \frac{9}{5} = \frac{3}{5} A_0,

where the sum is a geometric series with ratio $4/9 < 1. The limiting area of the Koch snowflake is therefore finite:

A = A_0 + \frac{3}{5} A_0 = \frac{8}{5} A_0 = \frac{2 \sqrt{3}}{5} s^2.

This contrasts with the infinite perimeter, highlighting the fractal's paradoxical properties.^[1] The finite area A also implies that solids formed by rotating the Koch snowflake around an external axis in its plane have finite volume. By Pappus's centroid theorem, this volume is V = 2\pi \bar{r} A, where \bar{r} is the distance from the area's centroid to the axis of rotation. For a symmetry axis from a vertex to the midpoint of the opposite side, direct application is complicated by the axis intersecting the interior, requiring decomposition or integration to account for the figure's structure relative to the axis; nevertheless, the resulting volume remains finite, while the generated surface area is infinite due to the boundary's infinite length.^[15]

Fractal Dimension and Self-Similarity

The Koch snowflake exhibits self-similarity, a defining property of fractals, where the entire curve can be decomposed into four smaller copies of itself, each scaled by a factor of $1/3.^[1] This structure satisfies the mathematical definition of a self-similar set, as the union of these scaled copies precisely reproduces the original curve at every stage of its iterative construction.^[1] The fractal dimension of the Koch snowflake, specifically its similarity dimension, is calculated using the formula d = \frac{\log N}{\log (1/r)}, where N=4 represents the number of scaled copies and r=1/3 is the scaling factor.^[1] Substituting these values yields d = \log_3 4 \approx 1.26186.^[1] For this quasi-self-similar set, the Hausdorff dimension equals the similarity dimension, confirming that the curve's effective dimensionality is non-integer and lies strictly between that of a one-dimensional line (dimension 1) and a two-dimensional surface (dimension 2). These dimensional properties underscore the Koch snowflake's intricate geometry, which is continuous everywhere but nowhere differentiable, as originally demonstrated by its creator.^[7] At finite stages of construction, the approximations are rectifiable curves with well-defined lengths, but the limiting curve is not rectifiable due to its infinite perimeter.^[1] Despite its pathological smoothness, the Koch snowflake forms a Jordan curve—a simple closed curve that divides the plane into an interior and exterior region—enclosing a finite area.^[1]

Advanced Mathematical Representations

De Rham Curve Formalism

The de Rham curve formalism provides a parametric framework for representing the Koch curve as the limit of a sequence of affine transformations defined by a functional equation. Introduced by Georges de Rham in 1957, this generalization defines plane curves using affine maps that compose to form self-similar structures. The Koch curve arises as a special case, where the curve f: [0,1] \to \mathbb{R}^2 is constructed via iterated function systems with four contractions, corresponding to the four subsegments in each iteration.^[16] For the Koch curve, the affine transformations include scalings by $1/3, translations, and rotations by \pm 60^\circ to account for the straight portions and the protruding sides of the equilateral triangle bump. The parametrization uses the base-4 expansion of t = \sum_{k=1}^\infty e_k / 4^k (where e_k \in \{0,1,2,3\}), iteratively applying the map corresponding to each digit e_k: digits 0 and 3 for straight segments, and 1 and 2 for the rotated bump sides. This yields a continuous curve through uniform convergence of the approximations.^[16] This formalism establishes the nowhere differentiability of the Koch curve by analyzing the variation implied by the functional compositions and connects to group representations via the monoid generated by the transformation matrices, aiding in analytic studies of self-similarity and fractal dimension. The approach allows theoretical investigations of limit properties without computing finite iterations, useful in fractal geometry proofs.^[16]

Lindenmayer System Encoding

The Koch snowflake can be generated using a parallel Lindenmayer system (L-system), a formal grammar that iteratively rewrites strings to produce self-similar fractal structures through simple production rules. This approach adapts L-systems, originally developed for modeling plant growth, to discrete geometric constructions, enabling computational simulation of the snowflake's iterative process. The system uses turtle graphics interpretation, where symbols direct a virtual "turtle" to draw lines and turns: F denotes moving forward by a fixed length while drawing, + indicates a left turn by a specified angle, and - indicates a right turn by the same angle; non-drawing symbols like + and - are ignored in rewriting but affect orientation.^[17]^[18] For the Koch snowflake, the axiom is the initial string F++F++F, representing the three sides of an equilateral triangle with double turns (++) equivalent to 120° to close the shape. The sole production rule is F → F+F--F+F, which replaces each forward segment with a sequence that adds an equilateral triangular protrusion: forward, left turn, forward, right double turn (120°), forward, left turn, forward. Other symbols (+, -) remain unchanged. This rule effectively subdivides each line into four segments of one-third the length, introducing the characteristic bump at the middle third, mirroring the geometric construction.^[1]^[17] To generate the curve, the rules are applied in parallel to all F symbols in the axiom, iterated n times to yield a string of drawing commands. The turtle interprets this string starting from an initial position and heading, with a turn angle of 60° for each + or -, ensuring the protrusions form equilateral triangles pointing outward; the forward length scales by 1/3 at each iteration to maintain proportionality. This produces an exact approximation of the nth stage of the Koch snowflake, with the string length growing exponentially as 4^n, facilitating efficient rendering in computer graphics despite the increasing complexity.^[1]^[19] The encoding employs a deterministic 0L-system (D0L-system), where rules apply uniformly without context or stochasticity, highlighting its roots in formal language theory for generating fractal languages. Such systems connect to broader applications in theoretical computer science, where the Koch snowflake exemplifies how context-free grammars produce non-differentiable curves with infinite perimeter in finite iterations.^[17]

Turtle Graphics with Thue-Morse Sequence

The turtle graphics approach to rendering the Koch snowflake involves a cursor that interprets a sequence of commands: moving forward (denoted F) by a fixed unit length while maintaining its current orientation, turning left (+) by 60 degrees, or turning right (-) by 60 degrees, with the initial position at the origin and heading upward to form a closed equilateral triangle base.^[20] This setup ensures the curve remains simple and closed at each finite iteration, as the total rotation sums to a multiple of 360 degrees.^[21] The Thue-Morse sequence, a binary cube-free sequence defined recursively by t_0 = 0, t_{2k} = t_k, and t_{2k+1} = 1 - t_k for k \geq 0, serves to dictate the turn directions without recursive subdivision.^[20] In this context, each bit maps to a turn angle: 0 corresponds to a left turn of +60 degrees, and 1 to a right turn of -60 degrees, interspersed with forward moves to trace the polygonal approximations.^[21] For implementation at stage n, the first $3 \times 4^n bits of the Thue-Morse sequence determine the orientations of the $4^n bumps across the three initial sides, with forward moves between each turn to construct the non-intersecting protrusions outward from the base triangle.^[20] This generates a polygonal curve of $3 \times 4^n segments, converging to the Koch snowflake as n increases.^[21] The cube-free property of the Thue-Morse sequence—no substring of the form www where w is a nonempty block—prevents three consecutive identical turns that could cause self-intersections or inward folds, ensuring the curve remains Jordan-simple at every stage.^[20] Additionally, the sequence's recursive doubling construction enables efficient computation and visualization, as only the relevant prefix needs to be generated for each iteration, offering an alternative to L-system string expansions.^[22]

Applications and Generalizations

Plane Tessellation

The Koch snowflake originates from an equilateral triangle at stage 0, enabling a basic tiling of the plane where six such triangles fit seamlessly around any vertex, as their 60-degree internal angles sum to a full 360 degrees.^[2] This arrangement leverages the geometric compatibility of equilateral triangles in a hexagonal packing, forming a foundational tessellation without gaps or overlaps.^[23] The tiling property persists through each subsequent iteration of the Koch construction, owing to the consistent introduction of 60-degree angles in the protruding segments and the inherent self-similarity that ensures boundaries align precisely at every scale.^[23] As the fractal evolves, the modified edges of adjacent triangles deform in a complementary manner, maintaining edge-to-edge contact and vertex alignment across stages.^[24] In the infinite limit, Koch snowflakes tile the Euclidean plane to form a fractal tessellation, where each individual snowflake encloses a finite area while possessing an infinite perimeter, collectively covering the infinite plane without voids or redundancies.^[25] This results in a boundary network of unbounded total length, contrasting sharply with the well-defined areal coverage.^[2] The tessellation's mathematical underpinnings stem from the snowflake's invariance under 60-degree rotations and lattice translations, which facilitate periodic arrangements in a hexagonal grid.^[23] Examples include tilings using Koch snowflakes of two different sizes, achieved by scaling factors like $1/\sqrt{3} to fill interstitial spaces between larger snowflakes.^[24] These tessellations have practical applications in computer graphics, such as procedural generation of rugged terrains that mimic natural fractal landscapes, and in modeling crystal growth, particularly dendritic ice structures that replicate the snowflake's branching morphology.^[26]^[27] In higher iterations, the tilings exhibit increasingly non-periodic characteristics, enhancing their utility for simulating organic, irregular patterns in visual simulations.^[23]

Curve Variants

The quadratic Koch curve represents a modification to the original Koch construction where the middle third of each line segment is replaced by two segments forming a right-angled protrusion, resulting in five segments per iteration at one-third the scale, rather than the four segments of the standard version. This alteration yields a fractal dimension of \log 5 / \log 3 \approx 1.465, compared to approximately 1.262 for the original Koch curve, and leads to a perimeter growth factor of 5/3 per iteration. The curve, also known as the Minkowski sausage in some contexts, produces a more irregular, sausage-like boundary in its closed form.^[28] The Lévy C curve, a variant introduced by Paul Lévy, adapts the iterative replacement rule with 90° turns instead of the 60° angles in the Koch curve, beginning with a horizontal line and generating an L-shaped protrusion in subsequent steps. This construction results in a self-similar fractal that has a Hausdorff dimension of 2. Unlike the non-filling Koch curve, the Lévy C curve's orthogonal turns contribute to its space-filling properties, making it useful for modeling plane-filling patterns.^[29] Other variants extend the Koch construction into twisted, wave-like, or higher-dimensional forms. The Kochawave curve, for instance, introduces asymmetry by alternating protrusion directions, creating a wavy appearance while maintaining self-similarity and a dimension close to the original. In three dimensions, extensions such as the 3D Koch curve or Koch island generalize the iteration to surfaces, replacing edges with polyhedral protrusions and yielding dimensions around 1.5 for boundary surfaces, with applications in modeling complex terrains. Comparisons across variants highlight differences in growth rates: the quadratic version's 5/3 factor produces faster perimeter divergence than the standard 4/3, altering compactness and visual roughness. Post-Koch developments in the 1970s, notably by Richard Voss, incorporated random perturbations to these constructions for simulating natural forms like coastlines and landscapes, bridging deterministic fractals with stochastic models observed in geophysics.^[30]^[31]^[32]

Functional Extensions

The functional Koch curve generalizes the classical Koch curve to a function f: [0,1] \to \mathbb{R}, where the iterative construction is applied to the graph of an initial simple function, such as f(x) = 0, by successively adding triangular waves to each segment, resulting in a fractal interpolation function (FIF) that serves as the attractor of an iterated function system (IFS).^[33] These FIFs on the Koch curve employ non-constant harmonic scaling functions, which are Hölder continuous but not Lipschitz, enabling the modeling of irregular, self-similar graphs with fractal dimension greater than 1.^[33] Parameterizations of the Koch curve can be extended using de Rham-like functionals, which define continuous curves through iterative corner-cutting processes on polygonal arcs, allowing for non-planar or time-varying representations suitable for applications in signal processing, such as modeling turbulent or irregular waveforms.^[16] These functionals provide a parameterized family of curves where the limit set forms a continuous path, adaptable to dynamic systems beyond the plane.^[34] Multifractal variants of the Koch curve introduce varying Hölder exponents along the path, creating irregular roughness profiles where the local Hölder exponent h_f(x_0) = \liminf_{x \to x_0} \frac{|f(x) - f(x_0)|}{|x - x_0|^\alpha} fluctuates, yielding a local dimension of [\log 4 / \log 3](/page/Log) \approx 1.2619 at typical points but a globally variable multifractal spectrum.^[35] In parametrized families like the von Koch functions F_\lambda for \lambda \in (\sqrt{2}/6, 5/6), the pointwise Hölder exponent h_{F_\lambda}(x) ranges from a minimum \alpha_{\lambda,\min} = \gamma_\lambda - [\log](/page/Log)(6\lambda + 1)/\log 6 to 1, with the multifractal spectrum d_{F_\lambda}(\alpha) = \tau^*_{\mu_\lambda}(\alpha + \gamma_\lambda - 1) describing the Hausdorff dimension of level sets, peaking at \alpha_{\lambda,L} = 1 - \log(36\lambda^2 - 1)/(4\log 3 + 2\log 6).^[35] This variability captures heterogeneous scaling behaviors not present in uniform fractals.^[36] Such functional extensions find applications in antenna design, where Koch fractal geometries enable compact structures; for instance, a second-order Koch-type wire dipole antenna achieves a reflection coefficient of -16 dB at 2.45 GHz for Wi-Fi with a radius of just 20 mm, demonstrating significant miniaturization compared to linear dipoles.^[37] Similarly, Koch snowflake antennas for deep space CubeSat constellations operate in X-band frequencies, leveraging the fractal's space-filling properties for broadband performance in constrained volumes. In chaos theory, these curves model deterministic chaotic dynamics, as seen in fractal billiards on the Koch snowflake, where orbits exhibit sensitive dependence on initial conditions within the bounded domain.^[38] Post-2000 computational implementations have also incorporated functional Koch variants in generative art, using procedural algorithms to produce self-similar patterns in digital visuals.^[33]

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Aug 6, 2025 · De Rham curves are obtained from a polygonal arc by passing to the limit in repeatedly cutting off the corners: at each step, the segments ...
[35]
[PDF] THE MULTIFRACTAL NATURE OF A PARAMETRIZED ... - HAL
Abstract. In a famous paper published in 1904, Helge von Koch introduced the curve that still serves nowadays as an iconic representation of fractal shapes.
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[PDF] Multifractal analysis of a parametrized family of von Koch functions
Sep 22, 2024 · In our case all exponents are less than one, so : Definition. The pointwise Hölder exponent of a locally bounded function f at x0 is. Hf (x0) ...Missing: varying | Show results with:varying
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https://doi.org/10.3390/fractalfract4020025
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[PDF] arXiv:0912.3948v1 [math.DS] 19 Dec 2009
Dec 19, 2009 · Abstract. In this paper, we attempt to define and understand the orbits of the Koch snowflake fractal billiard KS.