The Charm Of Geometric Figure

By+Dunelm

Geometry is a most important branch of mathematics, playing vital role not only in wide application of science, but also in arts. Geometric figures has always been an important element in the creating of art since ancient times, take Islam art over one thousand years ago for example, we can see that there were many complicated and delicate geometric shapes in these works. Up to modern times, we are able to enjoy the charm of geometry as the development of geometry itself.

Fractal geometry:

the code of the nature

Mathematists created a series of strange things but very charming geometric shapes at the end of 19th century and the beginning of 20th century. In 1891, Hilbert, one German mathematist, found a curve which could overspread through the whole space, the curve was named Hilbert Curve. In 1904, Koch, a Sweden mathematist, discovered a curve called Koch Curve, for the curve resembled like snowflake, so it is also called Koch Snowflake.

Though appeared complicated, the drawing of Koch Curve can be completed according to some rules,for example, start drawing from an equilateral triangle, and make a tresection of each side, then taking the middle section of the three parts as a side to make another equilateral triangle, thus, a hexagram is formed. Then we draw even smaller equilateral triangle by taking the middle part of the twelve sides of the hexagram as sides, and keep drawing like this. After drawing each time, the total length increased by 1/3, and the shape like a snowflake will be shown after drawing of millions of times. There is one thing needs to be careful, the total length of Koch Curve is endless, but its area is quite limited. At first, the beginning of the creation of such drawing did not catch the attention of the mainstream of mathematists, most of whom regarded it as a strange and unnatural figure that had nothing to do with the research of mathematics. Until 1960, Mandble, an ethnic Polish mathematist, began his research of mathematic value behind these picures, then its great value had been known. In 1975, he created the word—fractal—to refer to things that have self-similarity, namely, refering to the repitition of the same pattern and shape by various specifications, in this way, he was called “the father of fractal field”. In his opinion, the nature is full of complicated yet irregular structures---such as coastline, mountain, cloud, glacier, river system, cluster of galaxies, and others. The traditional geometry is unable to analyze a very common vegetable—broccoli, while the new natural geometry of fractal theory can display its prowess fully. Moreover, fractal geometry can be applied to many fields of our daily life such as rise and fall of the stock market.

One typical example of fractal theory is the system of rivers in our daily life. As is known to all, rivers could change their flowing course, and take Chinas Yellow River for instance, it had underwent five times, of course changed in history. The speed difference between inside and outside bends of the river caused its change in route, thus, rivers would be eroded and the balance of the river would be affected and led to the change of river flow. During their research, scientists found one thing that such a process could be described by fractal theory, thus, we have gained another new approach to study rivers.

Geometirc euation:

the charm of curve

The advent of computers in the 20th century changed mathematic study thoroughly. Computers are not only used as a strong tool in mathematic research, but also make more people appreciate the charm of mathematics. To outsiders of maths, equation is just a combination of alphabets and numbers which is Greek to them. The application of computers, however, the use of computers can make us draw many corresponding curves that match with the equations, which is as plain as daylight, besides, it can bring us great surprise, one example is that mathematists

Just as its name implies, Butterfly curve is a curve whose shape is like a butterfly. The first kind of butterfly curve as shown in Picture1, with a descripion of equation, is plane curve of 6th power. If you think it is too easy, do not worry, there is another description as displayed in Picture 2, it is equation 2, which is a polar equation. Through the change of vaiable θ in the equation, one butterfly curve with different patterns and direction can be formed. With complicated combination and changes, we can see a perfect art.

There is another curved needs to be mentioned, that is cardioid curve. Through polar coordinates, this curve can be expressed by equation 3. That means, when one circle moves around another circle with same radius, then the mark of a point of the circle is cardioid curve.

There is a touching legendary about cardioid curve. It is said that French mathematist Rene Descartes met with Christina—a Sweden princess and they fell in love with each other, but was rejected by the king of Sweden. Later on , Decartes was ill and left a letter to his girlfriend before he died. There was a equation r=a(1-cosθ) in the letter, which could not be understood by others, only the princess knew that the equation stood for the heart of Decartes. The story was widely circulated and one advertisement of a Chinese alkaline mineral water was based on the legend. But as a matter of fact, the story was made up by later generations, there is only one truth in the story is that both of them knew each other.

We will not feel sad for such a beautiful story for we knew that love can be expressed by mathematics. No matter it is butterfly or cardioidi curve, like those famous paitings, these elves of the mathematics world can decorate our hall and life.