In geometry what is an orthocenter

What did it start with?

The classic case: the Siepinski triangle.

Waclaw Sierpinski a Polish mathematician and philosopher (1882-1969) dealt mainly with set theory. It was so significant that even a crater on the far side of the moon bears his name.

The following construction of an equilateral triangle goes back to the name Siepinski:

Take an equilateral triangle and color it in (for a better idea).

In the second step we divide each side of the triangle in the middle and connect the points to form a "new" triangle.

Then we imagine that we are cutting this “new” triangle out of the original triangle. This gives us three “leftover” triangles that are still colored in. We cut out the triangle in the middle. In each of the three painted triangles, we halve all sides again and connect the points to form a triangle, which we again “cut out”.

It should look like this:

If we repeat dividing all the triangles we have drawn in a few more times, it will look like this:

If we repeat this procedure very, very often, it happens that at some point in a triangle that is still being painted the sides are so small that we can only see points. Nevertheless, mathematically this very, very short page can still be divided as often as you like. It is then impossible for us to connect the points to form a new triangle, since the length of the side looks like a point to us. Sharing this works wonderfully mathematically, but we fail to see it through our perception.

Mathematical:

The following definition applies:

The Sierpinski triangle is the set of all points on the plane that are left over after repeating the procedure infinitely.

Definition of terms:

The triangle to be divided becomes initiator called.

The cut out triangle will be generator called.

The constant repetition of the process of sharing becomes iteration called.

How many triangles are there?

To this we count the iterations (repetitions). One often speaks of dimensions or degrees.

By dividing a triangle, we get three new triangles.

N = number of triangles

n = iteration, dimension, degree (how often have I already divided my original triangle)

If I split my original triangle once, I get three new triangles. If I divide these three new triangles (n = 2), I get 3² triangles = 9. In the second pass of the division, I get nine new triangles. In the third run I get 3³ = 27 new triangles. Feel free to count the second figure above - it's true.

In the fourth run (iteration, dimension, degree) we come to 81 triangles. In the fifth iteration (iteration, dimension, degree) we are at 243 triangles.

You will surely realize that such a construct can no longer be realized with a pencil and ruler on a A4 sheet of paper.

This is where the transition from mathematics to computer science comes into play. This basis, which is based on purely mathematical principles, is digitized and graphically reproduced on the PC.

Graphic design

In a very spontaneous experiment it would look something like this:

Do you recognize it again? Yes, this graphic still clearly has the Sierpinski triangle as a distinguishing feature.

Another spontaneous attempt:

 

 

Would you still guess that this graph is based on the Sierpinski triangle? I took the graphic above, which consists of three identical, overlapping Siepinski triangles, and changed the direction and extent in the plane of each “large” triangle. Thus the whole structure changes.

In a further step I could add various other structures, move them, change their size, design the color scheme.

This is called fractal design.