Fractal Geometry

Fractal Geometry

Length: 1384 words (4 double-spaced pages)

Rating: Excellent

Open Document

Essay Preview

More ↓
Fractal Geometry


In the past, mathematics has been concerned largely with sets and functions to which the methods of classical calculus could be applied. Sets or functions that are not sufficiently smooth or regular have tended to be named as " pathological" and not worthy of study. They were regarded as individual curiosities and only rarely were thought of as a class to which a general theory might be applicable. However, in recent years this attitude has changed. Irregular sets provide a much better representation of many natural phenomena than do the figures of classical geometry. Fractal geometry provides a general framework for the study of such irregular sets. (Falconer) The word ‘fractal’ was coined in 1975 by mathematician Benoit B. Mandelbrot to describe an intricate-looking set of curves, many of which were not yet seen before the creation of the computer. Fractals often exhibit self-similarity, which means that various copies of an object can be found in the original object at smaller size scales. This continues for many magnifications – like an endless nesting of Russian dolls within dolls. (Pickover) Fractals appear everywhere in nature, in galaxies and landscapes, in earthquakes and geological cracks, in aggregates and colloids, and even in the human body. Fractal geometry is an important tool in the analysis of phenomena, ranging from rhythms in music melodies to the human heartbeat and DNA sequences. Many professions including, mathematics, astronomy, physics, chemistry, engineering, and biology use fractal geometry. (Bunde)

Waclaw Sierpinski was born on March 14, 1882, in Warsaw, Poland. Sierpinski attended the University of Warsaw in 1899, when all classes were taught in Russian. He graduated in 1904 and went on to teach mathematics and physics at a girl's school in Warsaw. He left teaching in 1905 to get his doctorate at the Jagiellonian University in Cracow. After receiving his doctorate in 1908, Sierpinski went on to teach at the University of Lvov. During his years at Lvov, he wrote three books and many research papers. These books were The Theory of Irrational numbers (1910), Outline of Set Theory (1912), and The Theory of Numbers (1912). In 1919, Sierpinski accepted a job as a professor at the University of Warsaw, and this is where

Waclaw Sierpinski (The Mactutor)

he would spend the rest of his life. Throughout his career, Sierpinski wrote 724 papers and an amazing 50 books. Sierpinski studied many areas of mathematics, including, irrational numbers, set theory, fractal geometry, and theory of numbers.

How to Cite this Page

MLA Citation:
"Fractal Geometry." 123HelpMe.com. 19 Oct 2019
    <https://www.123helpme.com/view.asp?id=32055>.

Need Writing Help?

Get feedback on grammar, clarity, concision and logic instantly.

Check your paper »

Essay about Fractal Geometry

- Fractal Geometry The world of mathematics usually tends to be thought of as abstract. Complex and imaginary numbers, real numbers, logarithms, functions, some tangible and others imperceivable. But these abstract numbers, simply symbols that conjure an image, a quantity, in our mind, and complex equations, take on a new meaning with fractals - a concrete one. Fractals go from being very simple equations on a piece of paper to colorful, extraordinary images, and most of all, offer an explanation to things....   [tags: essays papers]

Free Essays
1448 words (4.1 pages)

Fractal Geometry Essay

- Fractal Geometry In the past, mathematics has been concerned largely with sets and functions to which the methods of classical calculus could be applied. Sets or functions that are not sufficiently smooth or regular have tended to be named as " pathological" and not worthy of study. They were regarded as individual curiosities and only rarely were thought of as a class to which a general theory might be applicable. However, in recent years this attitude has changed. Irregular sets provide a much better representation of many natural phenomena than do the figures of classical geometry....   [tags: Mathematics Math Geometric Essays]

Free Essays
1384 words (4 pages)

Evaluation of the Fractal Dimension of a Crystal Essays

- Evaluation of the Fractal Dimension of a Crystal Abstract The purpose of this experiment was to determine the effects of voltage and molarity changes on the fractal dimension of a Cu crystal formed by the re-dox reaction between Cu and CuSO4. Using the introductory information obtained from research, the fractal geometry of the Cu crystals was determined for each set of parameters. Through the analysis of data, it was determined that the fractal dimension is directly related to the voltage....   [tags: Chemistry Chemical Papers]

Free Essays
1956 words (5.6 pages)

Essay about Fractals: The Organization of Chaos

- Fractals: The Organization of Chaos Please ignore the references to pictures or figures. I no longer have them, so I could not include them on this page. Thanks. Fractals are a relatively new concept in geometry. Most concepts for Euclidean geomtery, the division of geometry which deals with lines, circles, triangles, and other standard shapes, stem from the Late Greek and Early Rioman times. Considering the age of mathematics, the study of fractals is new becasue it dates to the beginning of this century....   [tags: Mathematics Geometry Essays]

Free Essays
1958 words (5.6 pages)

Fractals: A New-Age Mathematics to Explain Our World Essay

- Fractals: A New-Age Mathematics to Explain Our World Fractal art is a new-age art that tantalizes the eyes and mind with patterns, shapes, colors, and abstract imagery. Artists have once again found a way to harness the abstractedness of mathematics and integrate it into their work. So where does this new art form of fractal design stem from. The reality is that fractals themselves are relatively young in the mathematical world. Of course since the beginning of art and history and mathematics, self-similar objects have existed and been intriguing to the human mind....   [tags: Fractals Mathematics Math Papers]

Free Essays
1852 words (5.3 pages)

Four Geometry Formulas Essay

- As you begin the course of geometry students are generally familiarized with frequently used formulas in mathematics. These formulas include finding the perimeter and area of two-dimensional figures and finding the volume and surface area of three-dimensional figures. For every diverse shape there is a related formula for finding its perimeter, area, volume, or surface area. Therefore, we will only focus on four formulas for four singular shapes or figures. We will find the perimeter of a square, the area of a triangle, the volume of a right circular cylinder and the total surface area of a sphere....   [tags: Geometry]

Research Papers
649 words (1.9 pages)

Essay about Conic Sections in Taxicab Geometry

- In this essay the conic sections in taxicab geometry will be researched. The area of mathematics used is geometry. I have chosen this topic because it seemed interesting to me. I have never heard for this topic before, but then our math teacher presented us mathematic web page and taxicab geometry was one of the topics discussed there. I looked at the topic before and it encounter problems, which seemed interesting to explore. I started with a basic example, just to compare Euclidean and taxicab distance and after that I went further into the world of taxicab geometry....   [tags: Mathematics, Geometry, Taxicab Geometry]

Research Papers
1769 words (5.1 pages)

The Importance of Geometry in the Construction Industry Essay

- According to J. Dee, (1608), an English scholar, “There is nothing which so much beautifies and adorns the soul and mind of man as does knowledge of the good arts and sciences. ... Many ... arts there are which beautify the mind of man; but of all none do more garnish and beautify it than those arts which are called mathematical, unto the knowledge of which no man can attain, without perfect knowledge and instruction of the principles, grounds, and Elements of Geometry.” Geometry was derived from the Greek word meaning earth measurement which focuses on the study of shapes, sizes, relative configuration, and spatial properties....   [tags: geometry, construction, topography]

Research Papers
1227 words (3.5 pages)

Fractals: A Mathematical Description of the World Around Us Essay

- Fractals, a Mathematical Description of the World Around us In being characterized with fractional dimensions, Fractals are considered to be a new division of math and art, which is perhaps why the common man recognizes them as nice-looking and appealing pictures that are valuable as background on computer screens and art patterns. But they are more meaningfully understood by way of the recognition that many of nature’s physical systems and a lot of human works of art are not standard geometry forms....   [tags: Mathematics ]

Research Papers
1677 words (4.8 pages)

Geometry Essay

- Geometry Geometry was actually first used in ancient Egypt and Babylon at around 2000 BC in both cases. In order for the Egyptians to build such massive structures as the pyramids they had to have made plans for them prior to the actually building, in these plans geometry had to be used. On ancient Babylonian tablets there is evidence that they understood the Pythagorean theorem. The so-called "father" of Geometry is Euclid a Greek mathematician. He wrote The Elements, books of postulates and theorems, which paved the way for modern Geometry....   [tags: Papers]

Free Essays
362 words (1 pages)

Related Searches

Sierpinski is viewed as one of the greatest Polish mathematicians ever. He is noted for his construction of the Sierpinski gasket. (The Mactutor)

In 1915, Sierpinski thought of a great variation for the ternary tree. The Sierpinski gasket is obtained by starting with an equilateral triangle thought of as a solid object. You then divide it into four smaller equilateral triangles, of which the middle one is removed. The middle one is removed so a triangular hole is produced. With the three remaining solid equilateral triangles, you proceed in the same way so three smaller triangular holes appear, and so on. The fractal dimension of the Sierpinski gasket is df= log3/log2. (Lauwerier) We denote by L the length of a fixed side, say the side of the triangle. M(L) denotes the mass of the gasket when the length of the side is L. So M(L) is the mass of the whole gasket. Because of self-similarity the whole gasket is again in one of the sub-triangles. Its mass is now M(L/2). Since 3 of the smaller triangles make up the whole gasket we get M(L) = 3 M(L/2). On the other hand the relation between the linear measure L and the mass of any geometric object is M = A L^d where A is some constant depending on the shape of the object, and d is the dimension. Combining the equations we get A L^d = 3A (L/2)^d, which simplifies to 1 = 3(1/2)^d or 2^d = 3. Taking logarithms on both sides and solving for the dimension we get d = log(3) / log(2) =1.5850….(Daepp)

Sierpinki gasket (The MacTutor)

There are two similar fractal concepts that Sierpinski created, which are extremely fascinating. The Sierpinki carpet is generated in close analogy to the Sierpinski gasket. We first pick a positive integer n and a second one k with k < n^2.(Daepp) Out of those squares, k squares are chosen and removed. This procedure is then repeated by dividing each of the smaller squares left into n^2 smaller squares and removing the k squares that are located at the same positions as in the first iteration. This procedure is repeated again and again. (Bunde)

Sierpinski Carpet (Bunde)

The Sierpinski carpet for the above example has n = 5 and k = 9. It is possible for the carpets to look different when k is changed. Using the same idea as for the calculation of the dimension of the gasket, we denote by L the length of a square side and by the M mass of the carpet. Considering that n^2 – k smaller squares with side length L/n make up the whole carpet with side length L we get M(L) = (n^2 – k) M (L/n). Combining this with the general formula M = A L^d for some constant A we get (n^2 – k) A (L/n)^d = A L^d which simplifies to n^2 – k = n^d. Taking logarithms on both sides and solving for the dimension we get d = log(n^2 –k) / log n. In the example above, the fractal dimension is log(16) / log(5) = 1.7227….(Daepp)

The Sierpinski sponge is constructed by starting from a cube, and subdividing it into 3*3*3 = 27 smaller cubes. Using the same arguments as for the gasket and the sponge, we get the dimension df = log(20) / log(3) = 2.7268….(Daepp) You take out the central cube and its six nearest neighbors. Each of the remaining twenty small cubes are processed in the same way. After each iteration, the volume of the sponge is reduced by a factor of 20/27, while the total surface area increases. M(1/3L) = 1/20M(L), which makes the fractal dimension for this example, df = log20/log3. (Bunde)

Sierpinski Carpet (Bunde)

Today, trees are very important to the field of fractal geometry. Pythagoras (569 – 500 BC) studied the famous figure in which squares are placed on the sides of a right triangle. He proved that the sum of the areas of the squares on two of the sides equaled the area of the square on the hypotenuse. In our present day, the figure that Pythagoras created has grown into the fractal term "tree." (Lauwerier)

In the Pythagoras tree, you find that a square with a given square number n supports an isosceles right triangle from which two smaller squares sprout. The one on the left has a number 2n, an even number, while the one on the right, 2n + 1, is an odd number. Together, these three form a geometrical representation of the Pythagoras Theorem, which states that the smaller squares together, have the same area as the larger square. The smaller squares are equal to the larger square, which makes each one half the size of its predecessor. If the area of the original square is 1, then the total of squares 2 and 3 will also equal 1. The same holds true for each of the smaller squares in the series. In the example below, the eight squares, 8,9,10,…,15, together have the same area as the original base square. (Lauwerier)

Pythagoras Tree (Lauwerier)

Fractal geometry is not only fun to play around with, but it is also amazing to look at. Almost everywhere you look, you find a fractal of some sort. The Sierpinski gasket and Pyhtagoras’ tree interest me the most because they are very easy to work with. The only problems that I would have with fractal geometry, is trying to create them on the computer! To conclude, fractals are a big part of everything around us, such as the human body and the galaxy. Where would we be if people like Sierpinski and Archimedes hadn’t made their discoveries?

Works Cited:

1] Bunde, Armin and Havlin, Shlomo, Fractals in Science, NewYork:Springer-Verlas, 1994.

2] Falconer, Kenneth, Fractal Geometry, Chichester, Englend: Jonh Wiles anf Sons, 1990.

3] Lauwerier, Hans, Fractals, Princeton, NJ: Princeton University Press, 1996.

4] Pickover, Clifford, Fractal Horizons, New York: St. Martin’s Press, 1996.

5] The MacTutor History of Mathematics, Biography of Waclaw Sierpinski, Dec. 6,1999, http://www-groups.dcs.st-and.ac.uk/history/Mathematics/Sierpinski.html
Return to 123HelpMe.com