Everything is a fractal of triangles

Project: Math is Everywhere / Fractals

Broken dimensions [edit]

Mathematics obviously finds “broken” interesting. So it has withdrawn its solid basis from one of its old foundations, the numbers, and introduced “broken bases” - and now it also wants to rob the geometry of its dimensions.

In 1967 the French mathematician Benoît Mandelbrot asked exactly this question. The answer brought mathematics to two new areas. Chaos Theory and Fractals.

This also provides a direct reference to the Koch snowflake, because if its circumference is provided with more and more "spikes", it becomes longer and longer. So it is only a question of how detailed coastlines or other boundary lines are measured.

Other areas were quickly examined from these new perspectives. Even blood vessels, nerve plexuses (brain), tree tops, etc. demanded new "measurement" possibilities.


There is a great deal of clear material available in this area of ​​mathematics. One of the richest treasure troves is the University of Bremen.

Limited but infinite [edit]

Often the first reaction to this heading is to say, "This is a contradiction". Well, this is about math. Therefore, extreme caution should be exercised when using the word “contradiction”. Another variant of the statement overwriting this section reads:

“Infinite, but Not limitless. "

And indeed: it is no Contradiction. Objects like Koch's snowflake have such properties.

Enlargement of one side of the triangle

They are interesting because it is so easy to prove this property. Each page has the same structure and therefore the same development.

This process is repeated in every subsequent generation on every straight section of the circumference. The proof is now very simple. The scope is determined by the addition all Side lengths. With each step, each side is lengthened by the factor . If with the initial state (equilateral triangle) of the snowflake is denoted and each side has the length 1, then:

Then it results for the n-th step and the subsequent border crossing

Fractals obviously have properties that are different from normal Objects are very different. This area of ​​math is still very young and the effects on other sciences can hardly be foreseen. The following sections will show "self-similarity" in areas that are rarely the subject of in-depth research because they so easy appear. Before that, however, a few considerations and circumstances must be briefly addressed.

The mother of all fractals

There is no such thing as the “original fractal”! But the equilateral triangle is so often found that it lends itself to further considerations like no other object.

Even the oldest traditional “arrangements” of numbers were based on the shape of a triangle. This was shown by the Chinese mathematician as early as 1303 Zhu Shijie that after today Blaise Pascal named triangle.

This arrangement of numbers has been known for centuries and continues to deliver surprising things. One reason for the frequent occurrence of the equilateral triangle in this context could actually be in the Pascalschen Triangle itself. The numbers arranged are Tetrahedral numbers. The triangle is just a two-dimensional tetrahedron. Viewed in this way, the triangle is taken from the (Pascal's) triangle. As the figure shows, are equal in Pascal's triangle all Dimensions for tetrahedra included. So the “1” cells correspond to “zero-dimensional” tetrahedra.

Geometry or pattern? [Edit]

The numbers in Pascal's triangle take on large values ​​very quickly. That makes it difficult to look for common traits. Fortunately, there are now computers that can be entrusted with this task.

A property of natural numbers is parity, i.e. the determination of whether a number is even or odd. It is known that this property changes from number to number.

Triangles in triangles [edit]

The illustration opposite shows the tetrahedral numbers (binomial coefficients) as a "mini triangle" if the property odd is available.

This shows a self-similar Template. It is now triangles that alternate. They have their base sometimes at the bottom and sometimes at the top. At the same time, they are in turn made up of triangles. It is actually a "fractal" pattern.

Tetrahedra are not visible, however, are they? Maybe not in height (which is missing), but as a geometric grid. With something good will four triangles can be seen: One light, surrounded by three darker ones. If the three dark ones are "folded up", a tetrahedron is present. There is a corresponding grid for each side surface of the tetrahedron. The tetrahedron can therefore be "folded up" into smaller and smaller tetrahedra.

In the figure opposite are the blue So the triangles Base areas the tetrahedron, which can be "folded up" from the side surfaces. It turns out that the tetrahedra are also self-similar. Will the side faces after Inside (in the direction of the center of the body) "folded", there are "penetrations" of several terahedron points. These "bodies" in turn turn out to be those Projections the "higher dimensional" tetrahedron to the lower dimension.

From the patterns that the individual "points" form, a geometric figure is simply "explained" by the fact that connected points form a "line". The for stretch The required points then simply result from the location at which there is no longer a “line point”. In this way it is possible to “explain” a geometric figure from a pattern.

Self-like coincidence

Can random events be self-similar? No, but their sequence. More precisely, the pattern of places associated with a random event. Sounds complicated, but is quite simple. All you need is a cube, a drawn triangle, a ruler and a pencil.

The sequence
  1. Any starting point choose.
  2. Roll the dice, and if there are more than 3 points, just subtract 3
  3. The ruler from the point place on the corner with the number rolled.
  4. Exactly on the half Stretch the new to mark.
  5. That point is that new Starting point and the whole process is repeated.

Tovery many Repetitions result in a pattern similar to the one shown here.

How can that be? Chance knows no patterns. It is due to the "addressing" of the triangles. It takes place indirectly via the Cantor set. A detailed discussion would go beyond the scope here. Here we only refer to the work of Peitgen, Juergens, Saupe and Barnsley referenced.

Self-similar logic

Logic has something to do with “true” and “false”. How should here something similar be and to what similar? In the “Logical! Or ?! ”was made by the Predictability the logic spoken and towards the course Syllogisms referenced. The conditions discussed there are used here to show the "self-similarity" of logic.

Logic is predictable; that fact is already a Self-reference. The common Computers work with the dual system. With so-called logic circuits the calculations are carried out based on the logic. So if the circuits calculate with logic and the logic is calculable, then the logic with logic can calculated become.

Such references on yourself are an indication of Self-likeness. In order to provide evidence, must at the bottom to be started. Mathematics means something like that literally. It has to be Operations For a logical statement ("true", "not true") and the possible Results all Operations are started. All operations ?! How many are there anyway?

How many logical operations? [Edit]

  statement
(or. )
Result
(or. )
1. Nothing or “NA_UND?” gate "true""true"
"not correct""not correct"
 
2. On the other hand or "NEGATIONS" gate "true""not correct"
"not correct""true"
 
3.Nevertheless or "ALWAYS" gate "true""true"
"not correct""true"
 
4. decline or "NEVER" gate "true""not correct"
"not correct""not correct"

So I can imagine a lot of operations. Really? Mathematics even helps with "excessive imagination". Imagination is good, proof of existence is better. The circuits - from now on called “gates” - are now derived from the available options and given names (not the correct ones).

What can be done with "true" or "false" to get a result? First of all, the result must be either “true” or “false”, otherwise the value range of the logic would be left. After that is cleared up; what can with one logical statement employed become?

Well, more surgeries than the opposite for a logical statements do not exist. If now “true” with 1 and “false” with 0 rated and the "statement" with and the "result" with is designated, the math can begin.

Of the four operations, the negation be best known. It is simply called linear function viewed, with the graph:


The function has the general form . This form is well known. It makes the whole thing easier when we take place m and b use other identifiers here:


The values ​​for x and y taken.

xy
  0   1  
  1   0  

Well have to just nor the as to be determined. There are two unknowns, but also two equations; one for x = 0 and a x = 1 with known y. In the general Equations are the row numbers of the truth table on the variables x and y available as "bracketed exponents".

The Unknown are with and determined quickly. For the equation of negation surrendered:

All at once

The procedure just shown cannot be accepted by mathematicians. You will be contacted immediately Linear Algebra because that's what they learned in the first few semesters. This branch of mathematics is about as important to mathematicians as a computer is to computer scientists.

Therefore the system of equations is also different seen. For mathematicians, instead of the two equations easy

This means that all other operations can also be determined. Using matrices and vectors is just as easy as using normal Numbers. It can now all four Operations in one Equation can be written to the appropriate to find.

This results in the individual components of the vector

The exponent “-1” in the matrix simply means “inverted matrix”. It is important for further considerations and it is usually quite difficult to determine. But here Not. Math loves it short and sweet. The matrix is ​​given the name

The superscript "(1)" denotes the number of statements (xVariables) that are affected by the operations. The general notation is now

More statements [edit]

The previous operations were on a Limited statement. It becomes more extensive with several statements. Links like "OR", "AND", as well as relations like "equal", "not equal" have to be considered. Two statements are sufficient to start with. Start again with the table and equation (s).

The equation is:

The matrix for the system of equations is now

Again, the superscript “(2)” indicates the number of variables. The inverted Matrix is ​​also now required to use general notation. This matrix is ​​here just like that shown.

The similarities to the "small" matrices are unmistakable - would find mathematicians with a penchant for linear algebra. Because this “slope” is not so is widespread, in the next section the simple explanation.

Self-generation [edit]

The two large matrices generate themselves from the little ones. They are last extensive formulas. you are just present to show the requirements. It is also possible much easier, but the mathematical background would no longer be recognizable.