If you wish to understand The Nature of Universe, we can do it! We, ourselves, are small copies of Universe, we have got this answer!

J. Boivin

 A fractal is a mathematical object that is self-similar, where each part resembles to the whole. Most fractals are generated by a relatively simple equation where the results are fed back into the equation until it grows larger than a certain boundary. Some fractals are just a graph of an equation using complex numbers.
 

Who discovered fractals?

The mathematicians kept on asking themselves about some paradoxes, since 100 years ago. Thus, Sierpinski, a Polish mathematician, created some fractals, without knowing their meaning: The Curve, The Triangle and The Carpet. In the same time, in Sweden, Herge Von Koch invented “The Snowflakes’ Curve” or “Coast Line”.
Fractals were not discovered in a single instant, but knowledge of them grew quickly in the computer age. The first real fractal were discovered by a French mathematician named Gaston Julia. In his time there were no computers, so serious study of fractal objects was not practical at all.
In March 1980 the French mathematician Mandelbrot saw appearing on his computer screen something that would change his life completely. Many compare his discovery to Newton's discovery of the universal laws of mechanics. This discovery introduced a completely new field in Mathematics: Fractal Geometry.

The application of fractal geometry is a subject of study in many scientific fields: medical science, meteorology, Biology and telecommunication benefit from this new science. Mandelbrot looks back at his discovery: “The beauty of the Mandelbrotset was extraordinary especially because it came so unexpectedly. That day in May 1980 my life was enlighten by an intellectual and esthetic revelation.”

Many scientists grouped fractals in 2 great categories: artificial, the fractals drawn by the help of the computer, and natural fractals: trees, snowflakes, clouds, mountains. We present you soon some fractals’ trees drawn in Logo.

Fractals in Logo?

"Logo is the name for a philosophy of education and a continually evolving family of programming languages that aid in its realization." (Harold Abelson, 1982) The Logo programming environments that have been developed over the past 30 years are sources in constructivist educational philosophy, associated with Jean Piaget, the Swiss psychologist. Him’ theory is most closely to the study and documentation in the learning processes of children; Logo helps the kids to create interaction with other people and the world around them.
The most popular Logo environment has involved the Turtle, originally a robotic creature that moved around on the floor, but in present is an electronic cursor. The traditional Euclidean geometry is construct on abstractions: a point that has no size, a line who has length but no thickness. This is difficult for young children to understand. The turtle is a concrete object that may be seen and handled. The turtle moves around as you do and you can recognize with it and understand what it is doing. Turtle geometry is fit for young children as well as adults; it wasn’t intended to be a replacement for traditional geometry but rather, as an alternative entry point into geometry and mathematics in general. So, the Logo Programming Language was designed as a tool for learning. Extending beyond the typical perception of mathematics as a body of sterile formulas, fractal geometry mixed art with mathematics to demonstrate that equations are more than just a collection of numbers.  With we can visually model much of what we witness in nature, the most recognized being coastlines and mountains. Fractals are used to model soil erosion and to analyze seismic patterns as well.
The natural beauty of the fractal, in the classroom gives students incentive to explore coordinate systems, counting schemes, pattern development, integer arithmetic, the concept of infinity, and other topics in the mathematics and science curriculum. The teaching of fractals must be in an intuitive way; some complex problems became accessibly, opening a new extend of reader’s knowledge in school.
"The first meeting with fractals is a new experience, because fractal geometry is human geometry while the linear geometry is moving geometry.” (Mandelbrot)

Witnesses in the Middle of Nature

How can we understand Math’s nature and her beauty? How did Nature design the beautiful trees? Its forces are unlimited. Can we try it help of the computer? We want to offer an extremely useful and spectacular tool for learning and studying. Technical speaking, the most powerful facilities offered by the actual programming languages are explored by natural using of recursive method. We try to make a correlation between technique, art and nature. Our contribution is focussing on important pedagogical objectives, develops the procedural thinking, the recursive approach of the problems. We hope you will enjoy our new experiment!

We propose you to discover the real nature. We are presenting to you our school's garden and our method used with the fractals in Logo. Any fractal is the best method for watch the infinity!
Our group practiced Math in nature, because we had to use many calculations for our demonstrations. Mathematics is working in a human scale and is founding a new order in our lives, as life itself is a result of the theory of fractals.
There is a secular oak in our school garden. When our teacher, Petru Dumitru began to work in our school, 20 years ago, he planted an apple tree. We also planted some little trees, 7 years ago, when we started the school. So, the garden of our school is in fact the garden of several generations. We are very proud of it!
The first level of each fractal engendering’ is the line, a simple line, but in a special algorithm: each part resembles to the whole. We used all mathematics operations, division mostly, because the branch is ½ or 1/3 of trunk. We repeated this rule 5 to 6 times for each Logo draw.

The Garden of Our School
 

Raluca's Tree * Levels 1-5