Fractals

Fractals exhibit self-similarity and complex patterns that emerge from simple geometric rules, often involving recursive processes. Geometric sequences, characterized by a constant ratio between successive terms, can manifest in the scaling properties of fractals, where each iteration of the fractal pattern can be seen as a geometric transformation. For example, in the construction of fractals like the Koch snowflake, each stage involves multiplying or scaling by a fixed ratio, reflecting the principles of geometric sequences in their iterative growth. Thus, both concepts explore the idea of infinite complexity arising from simple, repeated processes.

Fractal time is a theory that suggests time is not linear but instead repeats in patterns at different scales. This concept relates to the perception of time by proposing that events in the past, present, and future are interconnected and can be seen as repeating patterns or cycles.

The concept of time being a fractal suggests that patterns repeat at different scales. This idea can help us see the interconnectedness of events and how they influence each other. It can also challenge our linear perception of time and make us consider our place in a complex and interconnected universe.

No, not every triangle is half of a rectangle. A rectangle has four sides and four right angles, while a triangle has three sides and three angles. The area of a triangle is not necessarily half of the area of a rectangle with the same base and height. Triangles and rectangles are different geometric shapes with distinct properties.

Well, let's take a moment to appreciate the beauty of simplifying fractions. If we divide both 36 and 126 by their greatest common factor, which is 18, we get 2 over 7. Just like that, we've simplified the fraction and created a little masterpiece of math!

Counting the whole square as iteration 0, there are 46 = 4096 segments after iteration 6.

They show self-similarity. When you look at a more detailed (or zoomed-in) version of the fractal you have the same image.

No, it is not.

Traditional geometric figures have dimensions which are integers: 0 for a point, 1 for a line or Mobius strip, 2 for a plane figure or Klein bottle, and 3 for a solid. Fractals have dimensions which are not integers.

put your pencil to the paper and begin drawing a line, and never stop... (you are going need an infinitely large sheet of paper)

* * * * *

You do not need an infinitely large piece of paper. But you will need infinitely many pencils and an infinite amount of time!

Fractal Analytics was created in 2000.

Fractals are generated from recursive mathematical equations, this is why you can zoom-in on them infinitely and they will continue to repeat themselves (this is also why they are so computationally intensive)

I would suggest using a 3d printer and thingiverse.

your dad

There is an element of fractal property in the manner in which an artery divides into smaller vessels and these in turn to still smaller vessels and so on until you reach capillaries. And then, you have the reverse process of capillaries joining together to form veins which join up to form larger veins and so on. Many branching processes are approximately fractal.

By their very nature fractals are infinite in extent.

Maple leaf, peacock

7.4

Yes.

Crystals are usually not fractals.

When the middle third of a line segment is removed and repeated infinitely on the resulting line segments the result is the Cantor Set.

When shifting to 2 dimensions, starting with a triangle, dividing it up into 4 similar smaller triangles and removing the middle triangle results in the Sierpinski Gasket; the limit of colouring Pascal's triangle with the even numbers as black and the odd numbers as white, as the number of rows tends to infinity is the Sierpinski Gasket.

Shifting to 3 dimensions, starting with a cube, dividing it up into 27 smaller cubes and removing the middle cube of each face and the centre cube results in the Menger Sponge.

The Sierpinski Gasket and Menger Sponge are 2 and 3 dimensional analogues respectively of the Cantor Set.

It need not affect it in any way whatsoever.

yes i think so because a fractal is an object that is self-similar

all squares are similar; so are all cubes

Numbers are not fractal so it is not possible to answer the question.