What Is the Koch Curve?
The Koch Curve is a famous example of fractal geometry. Fractals capture iterations as well as algebraic conditions of angle and scale to original size. Unlike function-based curves of the form y = f(x), a Koch curve is nowhere continuous, and therefore has no tangent anywhere along its length. Because of the Koch curve's unique construction, it has a noninteger dimensionality. Koch curves don't exist in mathematical isolation, but can be combined to form the famous Koch Snowflake.
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Fractal Overview
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Understanding the Koch Curve requires a fractal geometry overview. Fractals are formed by repeating a geometric pattern at continuously smaller scales. A fractal, no matter how apparently complicated, is composed of two themes: a step iteration and geometric repetition at smaller scales. A true fractal is achieved only in the limit of infinite iteration steps. Visual depictions adequately represent the ideal, since continuously smaller scales make it futile to depict fractals such as the Koch Curve beyond six or seven iterations.
Iteration Description
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In the Koch curve, a fractal pattern of 60-degree-to-base line segments one-third the length of the previous line is repeated. The portion of the base line under a newly formed triangle is deleted. Such geometric manipulation, continued indefinitely, forms the Koch Curve.
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No Tangents
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Unlike conventional curves taught in mainstream math classes, the fractal nature of a Koch Curve makes it impossible to define a tangent at any point on the curve. A tangent is a linear approximation of a curve at a point. As the domain of a smooth curve narrows, it progressively resembles a straight line. The Koch curve's fractal formulation--infinitely many iterations at a sharp angle--makes approximation to a straight line impossible. The Koch Curve has a geometric property like that of f(x) = |x| at x = 0, where no tangent is defined.
Curve Dimension
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Koch Curve dimension is illustrative of fractals' counterintuitive length-area properties. Formally named the Hausdorff Dimension, it is a generalization of regular integer dimensionality. Unless by coincidence, each fractal has a unique Hausdorff dimension. The Koch curve has a Hausdorff dimension of log(4)/log(3), or approximately 1.2619. Just as a square area relates to length by an exponent of 2, and cube volume relates to edge length by an exponent of 3, an ideal Koch Curve area relates to length by an exponent of approximately 1.2619.
Koch Snowflake
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The Koch Snowflake is closely related to the Koch Curve. Instead of starting with a single line, begin with an equilateral triangle. Apply same iterative steps. The first iteration looks similar to a Star of David with internal line segments missing. Subsequent iterations on each side of the Koch Snowflake reveal the "snowflake" resemblance.
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