Generalizations of Koch Curve and Its Applications Xinran Zhu

At the beginning of 20 th century, when studying the continuous and non-differential curve, a Sweden mathematician H.Von. Koch discovered a curve that can describe the snowflake ’ s construction --- Koch curve. Von. Koch

Remove one third in the center of each side of an equilateral triangle and place a small equilateral triangle at the location where the line segments are removed. Repeat the transform, the figures of Koch snowflake can thus be obtained.

That is to say, its perimeters tends to be infinite, and its areas goes to a fixed value.

The situation presented above can be generalized by taking the initial figure as regular m – gon with side length a, and further to three-dimensional space.

Theorem 1. Theorem 1. Let initiator be a regular m –gon with side length a instead of the original equilateral triangle. By conducting the same transform of classical Koch Curve. We get a sequence of polygon

The perimeters and the areas are respectively 、 While

The figure of an equilateral triangle after 4,7,10 times of transforms. The figure of a square after 4,7,10 times of transforms.

Remark 1. Remark 1. Keep a regular m -gon with side length a as initiator, but replace the generator equilateral triangle with square. Similarly to Theorem 1, are obtained.

The perimeters and the areas are respectively 、 While

The figure of an equilateral triangle after 4,7,10 times of transforms. The figure of a square after 4,7,10 times of transforms.

Remark 2. Remark 2. Regular m-gon with side length a is still the initiator, but take a mixing generator, equilateral triangle at first, then square, use them alternatively. Thus we obtain.

The perimeters and the areas are respectively 、 While

The figure of an equilateral triangle after 4,7,10 times of transforms. The figure of a regular decagon after 4,7,10 times of transforms.

Remark 3 Remark 3: Initiator is still a regular m -gon with side length a, but we use multiple equilateral triangles as generators.Let 、 keep the same meaning. While

Remark 4 Remark 4: Initiator keeps unchanged as a regular m -gon with side length a, and multiple squares as generators. 、 have the same meaning. While

The figure of a regular pentagon after 1,3 times of transforms.

Theorem 2 Theorem 2: Keep regular m - gon as initiator with side length a and, but this time the generator is an isoceles triangle. The set is obtained by removing a segment of length While in the center of each side of and replacing it by the other two sides of length of the isoceles triangle based on the removed segment. Repeat similar transform, we got 、 ( ) The figure of each side is

In the following, we consider some Koch- type polyhedrons in three-dimensional space.

Theorem 3 Theorem 3: Let be a regular tetrahedron with the edge length a as initiator; Add small regular tetrahedrons as generators on each face of initiator. Repeat the transform on the new polyhedron we obtained, we get. We denote by and the perimeter and area of respectively. While

Remark 5 Remark 5: Take initiator as a regular octahedron with the edge length a this time; Still use small regular tetrahedrons as generators. Similarly we obtained. Denote by and the perimeter and area of respectively. While

Remark 6 Remark 6: Take initiator as a regular icosahedron with the edge length a this time; Keep small regular tetrahedrons as generators. We got. Denote by and the perimeter and area of respectively. While

Remark 7 Remark 7: Take initiator as a cube with the edge length a this time; Similarly to before, add a small cube as generator on each face of initiator. Denote by and the perimeter and area of respectively. While

Potential Applications of Generalized Koch Curves

Education Through the simulation program developed, it greatly enriches the demonstration of fractal and related geometry lessons in teacher ’ s teaching, and also make the teaching more convenient and more interaction in the class.

The Program Interface 2-D Mode 3-D Mode

Potential Art Value The figures that expanded from original Koch snowflake, can be widely applied in designing and producing TV advertisement, popular craft, pattern design, etc.

The Generalized-Koch-Curve Handkerchief

Coffee Mat

A Suit using Generalized-Koch- Curve Pattern

Generalized-Koch-Curve Jewelry

Hat and glove

Potential Applications in the Biochemistry Redo the antenna according to its ultrastructure and fractal theory.

Reconstruct the grain of activator based on fractal rules

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