Fractals: A Different Type of Geometry

The Dimensions in Between

The mathematics that we have learned since elementary school is a bit beautiful. A line is perfectly straight, the path of the ball thrown upward is a parabola  and the shape of the earth we see in books is a sphere. The truth is, the lines, parabolas, and spheres are “flawless models” of the real world.  A straight line that can be drawn using a ruler or a meter stick is not perfectly straight (try using a magnifier). The shape of the path of the ball thrown upward is not all the time a symmetric parabola. Lastly, Earth is not a perfect sphere, it bulges in the equator, it has mountains, valleys, and trenches.

Clearly, the perfect mathematics that we know is an ideal notion and it does not have a very close resemblance of reality. It also appears that the mathematics, particularly the Geometry that we have learned in school is not enough to describe the real world. The dimensions that we have talked in the last post, cannot simply apply to these types of “irregularities.” Continue reading

The Seven Bridges of Königsberg

The city of  Königsberg, Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel river.  There were two islands on the river and there were seven bridges connecting them and the main land as shown in Figure 1.

Figure 1

Residents observed that using the bridge at the southern part of the city (Bridge 1 in Figure 2) as starting point, they could not stroll around crossing all the bridges only once. They had to skip one bridge or cross some bridges twice. Some of them conjectured that it was impossible to cross the seven bridges once and only once, but they could not explain why.  Continue reading

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