What are fractals?

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Question by: Laura Bianchi | Last updated: February 1, 2022

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A fractal is a geometric object with internal homothety: it is repeated in its form in the same way on different scales, and therefore by enlarging any part of it, a figure similar to the original is obtained.

How are fractals born?

To generate a fractal of this type, we can start with a triangle lying on an arbitrary plane. The midpoints of each side of the triangle are connected to each other and the triangle is thus divided into four smaller triangles. Each midpoint is then raised or lowered by a randomly chosen amount.

What does it mean to speak in fractal?

Sometimes I hear about fractals. That of 1982 is technical and strictly mathematical, reserved for those who chew a lot of advanced mathematics and therefore must be left aside: it is however the only one that allows to define “operationally” a set, a shape, a curve, a surface, a fractal volume . …

Who Invented Fractals?

Benoît Mandelbrot (Warsaw, 20 November 1924 – Cambridge, 14 October 2010) was a French naturalized Polish mathematician, known for his works on fractal geometry.

How is a fractal calculated?

FRACTAL DIMENSION – SECOND PART

  1. If n is the number of linear enlargements, we denote by f (n) the number of copies of the object. (figures 1 – 5)
  2. We have that f (n) is represented by the power of base n and exponent of the dimension.
  3. So we can write f (n) = n. d
  4. We therefore have d = logn[f(n)] = log[f(n)]/ logn.

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How is the von Koch curve constructed?

Generation of the curve

  1. divide the segment into three equal segments;
  2. delete the central segment, replacing it with two identical segments that make up the two sides of an equilateral triangle;
  3. go back to step 1 for each of the current segments.

Who invented the shapes?

The origins of geometry in Egypt

According to the Greek historian Herodotus (5th century BC) the origin of geometry (a word composed of two Greek terms, the meaning of which is “measurement of the Earth”) can be traced back to the ancient Egyptians.

What is geometry for in life?

Galileo and Descartes attribute to geometry the ability to understand a large part of the phenomena of the world and observe how it allows us to formulate and solve many problems and produce knowledge.

How many axioms are there in Euclidean geometry?

Its geometry consists in the assumption of five simple and intuitive concepts, called axioms or postulates and, in the derivation from said axioms, of other propositions (theorems) that have no contradiction with them.

What are the 5 axioms of geometry?

Plane partition axiom. Axioms congruence. Transport axioms. Axioms of the parallel.

What are the axioms of the straight line?

Axiom of belonging to the line: – at least two distinct points belong to each line – given two distinct points there is one and only one line to which both belong. Axiom of belonging to the plane: Each plane contains at least three non-aligned points.

How many ordering axioms are there?

Of any three points on a line, only one is located between the other two. Axioms II, 1-3 are called linear ordering axioms. They allow the following definition to be given: A pair of points A and B is called a segment and indicated by AB or BA.

What are the limits in real life for?

The limit of a function or sequence is useful for studying the behavior of a function in an inaccessible section starting from the analysis of the neighborhood, i.e. of the data in the immediate vicinity or of the tendential ones.

What are mathematical formulas for?

Mathematics formulas are essential for solving problems and exercises. They provide a synthesis of the definitions and theorems that are studied in the theory and constitute the bridge between words and calculations.

What is math for?

Mathematics therefore teaches reasoning, opens the mind and helps develop the ability to communicate and discuss, to argue correctly, to understand the points of view and arguments of others.

Who invented the mathematical formulas?

Greek mathematics is believed to have begun with Thales of Miletus (c. 624-546 BC) and Pythagoras of Samos (c. 582 – 507 BC).

Who Invented Mathematical Expressions?

The expressions and their evaluation were formalized by Alonzo Church and Stephen Kleene in the 1930s in their lambda calculus. Lambda calculus has had important implications in the development of modern mathematics and computer programming languages.

Who invented pasta?

An ancient origin

A very suggestive story has it that pasta was invented by the Chinese and brought to Europe by Marco Polo in 1295, on his return from the empire of the Great Khan.

What are the limits for?

In mathematics, the concept of limit is used to describe the course of a function as its argument approaches a given value (limit of a function) or the course of a sequence as the index grows unlimitedly (limit of a sequence ).

What does it mean to have a limit?

In these terms, the concept of limit assumes a mostly negative interpretation because it is associated with something that must be overcome, that must be eliminated, something that is lacking as in disabilities.

What does it mean to cross the line?

In the sense fig., O. the limits, or. the measure, to go beyond the limits set by convenience, beyond the degree of endurance.

How is the line defined?

A straight line is defined as an infinite set of points.

A straight line is therefore a line positioned on a plane, it is a line that never changes direction and that has neither a beginning nor an end.

What does the axiom of partition of the plane by a line state?

partition of the plane, axiom of in Euclidean geometry, establishes that every line r of a plane α divides the plane into two disjoint and non-empty subsets α1 and α2 (half-planes) such that: … b) if two points A and B both belong to the same subset α1 or α2then the segment AB does not intersect the line r.

Why is a line a proper subset of the plane?

Since the lines contain part of the points of a plane, they are a subset of that plane.

What are the axioms of congruence?

Axioms of congruence. If A, B are two points of a line a and furthermore A ‘is a point on the same line or on another line a’, we can always find a point B ‘, from a given part of the line a’ with respect to A ‘, such that segment AB is congruent, that is, equal to segment A’B’.

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