Omnipotent Data

In fact, fractals are quite common in our daily life.

give a chestnut

snowflake!

It's not snow beer, it's snowflakes!

A snowflake looks like a hexagon when you look at it with the naked eye.

When you put it under a microscope and magnify it hundreds of thousands of times, the shape of the details you see is also hexagonal.

In other words, a snowflake is a larger hexagonal crystal composed of n extremely tiny hexagonal crystals!

Of course, there are also sperm, which also conform to the fractal principle.

So people use mathematical methods to represent these fractal phenomena.

After hundreds of years of research, fractal theory has three very important models in the field of mathematics.

They are: three-point Cantor set, Koh curve, Julia set.

The project challenged by the two contestants this time is related to the Julia set (Julia set).

The definition of the Julia set sum is simple: Z(n+1)=Z(n)^2+ (is a constant)

The definition is so simple that an ordinary high school student can understand its meaning.

But the magic of the Julia set is that its mathematical definition is very simple, but the images he generates are incredibly complex, which contains profound mathematical principles or philosophies that we humans have imagined.

Well, philosophical issues have been touched upon.

A Julia set, in simple terms, is formed by iterating the formula Z(n+1)=Z(n)^2+.

Most people should know about iteration.

For example: consider the function f(z)=z^2-0.75. After fixing the value of z0, we can calculate a series of z values ​​through continuous iteration: z1=f(z0), z2=f(z1), z3=f(z2),…. For example, when z0=1, we can iterate sequentially:

z1=f(1.0)=1.0^2–0.75=0.25

z2=f(0.25)=0.25^2–0.75=-0.6875

…………

z5=f(-0.6731)=(-0.6731)^2–0.75=-0.2970

………

As can be seen,

Z(n) function, after continuous iteration, the result will gradually tend to a certain value.

Of course, this is just a variation of Z(0)=1.

After a series of indescribable studies on Julia sets, mathematicians found that not all Z(0) values ​​can form bounded fractal graphics.

The value of Z(n) is finite only if Z(0) is in the range [-1.5, 1.5].

That is to say, only within [-1.5, 1.5], the Julia set can form a bounded fractal figure.

But this time, the program group fixed the value of Z(0), and made questions based on the changes of the parameters.

parameter, which can be written as (x, y)=x+iy.

The value of is determined by a real part x and an imaginary part y.

Change the values ​​of x and y, and the corresponding fractal diagram will also change.

Moreover, the changes of x and y are non-linear, fast and slow.

Guests will randomly select 7 of the 100 fractal animations generated by changing x and y within a certain interval (accurately [-1, 1]).

Cut 50 fractal images from each fractal animation.

Cheng Nuo and Li Shiye can choose 2 pieces each, and display the values ​​of x and y corresponding to the fractal figure.

Then the two deduced the generation logic from formulas to graphics through on-site learning.

Then judge the specific values ​​of x and y according to the generated logic pushed to out, accurate to 3 digits after the decimal point. The error is between [-0.001, 0.001]!

Seven questions, seven fractal animations, seven production logics, 175 fractal graphics, 28 million possible values ​​of x and y.

What the contestants need to do is to find the only correct one among the 28 million possibilities!

There are seven questions, and there is a rush-answer mode.

One point for a correct answer and one point for a wrong answer.

Whoever gets four points first wins!

Rules, it's over.

All the audience, you look at me, and I look at you.

Confused!

Dumbfounded!

...

Everyone is dumbfounded!

Did you understand what was said?

Barely understand... 0.0001%.

...

After reading this question, I feel that I didn't bring my brain today!

Haha... me too... my brain has already been flushed away by me in the toilet!

...

It's too nerve-wracking to talk about this topic, let's change the subject. What are you going to have for lunch today?

I think I need to supplement with Shenbao. Shenbao, a bottle of refreshment!

What a topic...

What the hell?

Is it because my Mandarin Chinese is not up to standard or what?

I know all these words. But why am I confused when they are connected together?

Is your strongest brain gone, or are we the audience unable to hold the knife?

It’s okay to use some brain-burning projects to insult our IQ, we can still understand a little bit.

But this question, to be honest, really... didn't understand it at all!

It was hard for them to imagine that two teenagers in their twenties wanted to challenge him for a project that they didn't even understand the rules of the topic.

really……

I wait for the scum, the only meaning of being born is to make up for human beings.

Maybe sometimes when the bully learner starts to act aggressive, just be a salted fish called 666!

Teacher Jiang also saw the bewilderment in the audience's eyes, and said with a smile, Maybe there are many audiences who don't understand the challenge rules of this project. It's okay, let's do an animation demonstration.

First of all, these fractal animations are after the complex value of the iterative function f(z)=z^2+ on the complex plane changes continuously and similarly, we...

Give up, give up completely...

I knelt down, I really knelt down...

Teacher Jiang, are you sure that what you are talking about is not a bible?

I thought we could understand something after you said it?

But... the more I talk, the more confused I become!

The audience no longer had much hope of understanding the topic.

He just looked forward to the start of the match, and quietly watched Cheng Nuo and Li Shiye pretend to be aggressive.

Melon seeds, beer, and Mazza are all ready.

Two big guys, please start your performance!

I wait for salted fish, I have no other skills, but I have practiced the ability to call 666!

…………

Now, four guests are invited to choose seven of the 100 fractal animations as the contestant's topic!

Finally, in the anticipation of the salted fish audience, the competition officially began!

Cheng Nuo and Li Shiye sat side by side on the challenge seat.

In front of everyone, there is a display screen for uploading questions.

The guests quickly selected 7 fractal animations.

Seven fractal animations, corresponding to seven different x, y values ​​and laws of fractal graphics.

Okay, next, change the x and y values ​​of these seven fractals.

On the big screen, the values ​​of seven fractal animation imaginary numbers (x, y) can be seen, starting from [1, 1] and changing at a step of 0.001.

Next, randomly capture 50 fractal images on each fractal animation.

In fact, according to 0.001 step, each fractal animation will have 1,000,000 change maps.

If only 50 of them are captured, there will be many fractal graphics in the middle interval.

It will also have a great impact on the judgment of the two players!