Koch tetrahedron

KOCH TETRAHEDRO

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TETRAHEDRON In geometry , the tetrahedron

compose of four triangular faces.

Three of which meets at each vertex.

It has six edges and four vertices.

PICTURES OF TERAHEDRON

Tetrahedron is a kind of pyramid.

The base of tetrahedron is a triangle (any of the four triangles are considered as the base) so tetrahedron is also known as “TRIANGULAR PYRAMID”.

“ Tetrahedra ” is used for plural form.

KOCH TETRAHEDRONOne the most symmetric and

easy to understand the fractal.

It is named after the Swedish Helge Von Koch (1870-1924) .

Koch snowflake is also known as Koch star or Koch island.

He showed that it is possible to have figures that are continuous everywhere but differentiable nowhere.

HISTORYo Helge von koch described the

shape of snowflake in a paper in 1904.

It is one of the earliest fractal curves to have been described in the paper titled ”On a continuous curve without the tangents, constructible from elementary geometry”.

FRACTAL GEOMETRY Fractals are typically self-

similar patterns, where self-similar means they are "the same from near as from far“.

As mathematical equations, fractals are usually nowhere differentiable, which means that they cannot be measured in traditional ways.

The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over.

MATH BEHIND KOCH ’S SNOWFLAKE

NUMBER OF SIDES LENGTH OF SIDES

PERIMETER AREA

NUMBER OF SIDES For each iteration, one side of the

figure from the previous stage becomes four sides in the following stage. Since we begin with three sides, the formula for the number of sides in the Koch Snowflake is

n = 3*4a

in the ath iteration. For iterations 0, 1, 2 and 3, the number

of sides are 3, 12, 48 and 192, respectively.

LENGTH OF SIDE In every iteration, the length of a

side is 1/3 the length of a side from the preceding stage. If we begin with an equilateral triangle with side length x, then the length of a side in iteration a is

length = x*3-a

For iterations 0 to 3, length = a, a/3, a/9 and a/27.

PERIMETER Since all the sides in every iteration of

the Koch Snowflake is the same the perimeter is simply the number of sides multiplied by the length of a side

p = n*length p = (3*4a)*(x*3-a) for the ath iteration. Again, for the first 4 iterations (0 to 3)

the perimeter is 3a, 4a, 16a/3, and 64a/9.

As we can see, the perimeter increases by 4/3 times each iteration so we can rewrite the formula as

p = (3a)*(4/3)a

So as a-->infinity the perimeter continues to increase with no bound.

Also, as a-->infinity the snowflake is made up of sharp corners with no smooth lines connecting them. Therefore, while the perimeter of the snowflake, which is an infinite series, is continuous because there are no breaks in the perimeter, it is not differentiable since there are no smooth lines.

AREA

APPLICATIONS OF KOCH TETRAHEDRON

It is used in HOOP SNAKES.

It is geometry that produces a fractal of regular tetrahedron based on the Koch division.

It is used in colour plates.

FUTURE APPLICATIONS

Among the many applications of Fractal Geometry also described as a n understanding model for the distribution of matter in galaxies, Molecular Brownian motion and a number of various researches.

Implementation of these ideas resulted in a much more general version of the earlier program where transposition patterns could be described precisely and non-Statistically.

REFERENCES en.wikipedia.org/wiki/

Tetrahedron http://www.daviddarling.info/e

ncyclopedia/K/Koch_snowflake.html

http://www.3dvision.com/wordpress/2008/10/30/3d-koch

snowflake/http://en.wikipedia.org/wiki/Koch_snowflake