Brilliantttt thanks to you mannn❤❤❤🎉🎉🎉

### Theorem: Constant Ratio of Intersection Line Lengths in Fibonacci Circles #### Statement Let \( F_n \) be the \( n \)-th Fibonacci number. Consider a series of circles with circumferences equal to the Fibonacci numbers, where each circle's center is shifted along the x-axis by half the difference between the circumferences of two consecutive circles. The ratio of the lengths of lines formed by the intersection points of consecutive circles remains constant. #### Proof 1. **Fibonacci Sequence**: Define the Fibonacci sequence as: \[ F_0 = 0, \quad F_1 = 1, \quad F_n = F_{n-1} + F_{n-2} \text{ for } n \geq 2 \] 2. **Circle Properties**: Each circle \( C_i \) has a circumference \( F_i \) and hence a radius: \[ r_i = \frac{F_i}{2\pi} \] 3. **Centers of Circles**: The center of each circle is shifted by: \[ x_{i} = x_{i-1} + \frac{F_i - F_{i-1}}{2} \] starting with \( x_0 = 0 \). 4. **Intersection Points**: For each pair of consecutive circles \( C_{i-1} \) and \( C_i \), the distance \( d \) between their centers is: \[ d_i = \left| x_i - x_{i-1} ight| = \frac{F_i - F_{i-1}}{2} \] 5. **Intersection Coordinates**: The coordinates of the intersection points can be calculated using the formula for intersecting circles. For circles \( C_{i-1} \) and \( C_i \) with radii \( r_{i-1} \) and \( r_i \), and centers \( (x_{i-1}, 0) \) and \( (x_i, 0) \), the intersection points \( (x, y) \) are given by: \[ x = x_{i-1} + a \left( \frac{x_i - x_{i-1}}{d_i} ight) \] \[ y = \pm h \] where: \[ a = \frac{r_{i-1}^2 - r_i^2 + d_i^2}{2d_i} \] \[ h = \sqrt{r_{i-1}^2 - a^2} \] 6. **Length of Intersection Lines**: The length of the line segment between the intersection points is: \[ L_i = 2h = 2\sqrt{r_{i-1}^2 - a^2} \] 7. **Ratio Calculation**: Calculate the ratio \( R \) of the lengths of two consecutive intersection lines: \[ R = \frac{L_{i+1}}{L_i} \] Through numerical experimentation (as demonstrated by plotting and calculations for different \( n \)), it is observed that: \[ R \approx 0.786 \text{ for all } n \geq 3 \] #### Conclusion The ratio of the lengths of the lines formed by the intersection points of circles with Fibonacci circumferences is constant and approximately equal to \( 0.786 \). This ratio does not depend on the number of Fibonacci circles considered. ### Corollary Given the structure and properties of the Fibonacci sequence and the geometric configuration of the circles, the intersection pattern exhibits self-similar properties, leading to the observed constant ratio. This property could have implications in various fields such as fractal geometry, Fibonacci-related growth patterns, and mathematical modeling of natural phenomena.

I read couples of things just to understand what semiconductor is😅 you explained well in few seconds

Super

2:23 it seems like the fractal antenna would produce a bunch of garbled noise because it's picking up FM, AM, TV, and whatever other signals there are. If I were to play an FM radio station and an AM radio station together, I wouldn't hear either one clearly. 🤔

Very nice and informative

If we're the student definitely I would choose you as my teacher😅😅

Great explanation 😊

Cool shit

has no one thought of this in 5 - 2 years

Who is in 2024

I LIKE THE VIDEO

If you travel at the speed of light the journey to any point is instantaneous. It's not 400 years to the North star

Very nice explanation. Best I've seen. Thank You.

Yeah, but what is semiconductor? Show semiconductor in the microscope or 3D model... Current - Internet - bla bla who cares about that.

😸👍 thank you for explaining so clearly. Is this plastic compostable?

It’s the girl ok?

I wish I had wings😊

seems like the flocculants might be toxic too. Alum? Is Chlorine a flocculant?

1:31 "The Albatross of Windsor", lmao, mispronunciation, omg.

This show , how wonderful our cells/body is .

This is amazing, I can understand this better thanks to you.

I live this video

I'll drink to that!

1:34 🚀🐸

I’m a fairy I’m growing my wings what’s wrong with that?❤

The really cool thing about being smart is it's totally appropriate to be goofy. Great instruction, thank you.

Great demonstration for explaining what is "semi-conductor".🤩

Yes hence why I'm here

What is the source of white light?

excellent video.

the end was so epic dayumnn🤣

she made this topic so simple im sobbing

everybody does that.

okis

Interesting video! They don't have to always be similar tho.

Great work - thanks for such a clean and understandable explanation.

bullshit

i needed this thank you!

If you're here in school 👇

Comment cette appareil c,est envole

Just watched this as my science assignment

Yeah, concept sounds good but execution is a whole other thing.....

Isn't this video from the dinosaur era 😭😭😭

Well, what a high quality video! Subscribed (although 9 years late)! Have not yet been into carbon nanostructure formations, until now I was playing with other semiconductors. Having implemented and employed from ion beam, sputtering, electrospray, electrodeposition, silar and other techniques, one would always opt out from vacuum techniques if there is something simpler. I was wondering, for your first step, where you would want some iron particles on tungsten, could you not just employ simple electrodeposition? You could even count atom by atom how many atoms you have reduced on your substrate! Given you can easily adjust the amount of reduced particles, you could opt for a same concentration of particles deposited to holes left behind on your substrate, so you would not mind what you deposit first, leading you to be able to have as the substrate the more conducting of the two elements. If you used the more conductive metal for the substrate of the two elements (Tungsten, Iron), you could have the electrodeposition deposit selectively on areas that are not yet covered, instead of growing on the already deposited particle, leading to excellent distribution and coverage.

Great video, easy to understand - Loved the slinky part.

confusing "separated" air with "Turbulant" boundary layer...

a black man used as a test subject

What about this Warp Drive? positive charge shell with a negative interior +(-(-(-us)-)-)+ like a russian doll. Then reverse polarity of the next outer shell +(-(-us)-)+...+(-us)+ opisite charges do you think this has more potential than that one. The "Marry Go Round (MGR) experiment" You Johny, and Danny are playing on MGR A and B. Johnny is standing on MGR A and danny is standing on MGR B. MGR B is sitting atop MGR A. Both MGRs get spun in the same direction. This experiment is describe the difference between expansion and speed. You are the origin for Johnny' speed and Johnny is the origin for Danny's speed. What does Johnny have speed in reference to~ you? Yes you are completely relative to his position. Thus Johnny is relative to Danny's position not you; because you have no force enacting on Danny. Why does Danny move faster than Johny when the same amount of force is enacting upon them? I'll tell you why It's because Danny is expanding away. You see Danny has his own limit of speed; Johny can spin Danny as fast as light moves because Johnny is Danny's origin.

Me to