[Extra Visual] All orbits of the Mandelbrot Set shown together.

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This visual shows a series of balls on the complex plane. The equation z=z²+c is applied to every ball once per iteration. Those balls outside the Mandelbrot will escape, and head towards infinity. Those inside the Mandelbrot Set will remain bounded within the radius two circle.
This video is extra footage associated with the explainer video "The Mandelbrot Set Explained". • The Mandelbrot Set Exp...
If you look carefully you can see the period motion associated with the periods 2 and 3 bulbs. For example you will notice that the balls return to the period 3 bulb every third iteration. The colour of the balls is graduated for each ball, so that you can get a rough idea of where it started. This graphic shows the first 400 iterations.
I will attempt to upload extra graphics associated with the main sequence of explainer videos, you will find them by exploring the channel playlists. Generally I'll set these videos so that subscribers aren't notified, so that subscribers aren't pestered by these uploads. Join the Maths Town Twitter feed if you would like to be notified about everything I am doing (including the Maths Town channels). / mathstown
In this series:
1 - • The Mandelbrot Set Exp... The Mandelbrot Set Explained
2 - • Julia Sets, and how th... Julia Sets, and how they relate to The Mandelbrot Set
3 - • How to Build a Julia Set How to Build a Julia Set
Extra Visuals (No commentary):
• [Extra Visual] All orb... - All orbits of the Mandelbrot
• [Extra Visual] All Per... - Period 1 orbits of the Mandelbrot
• [Extra Visual] All Per... - Period 2 orbits of the Mandelbrot
• [Extra Visual] Buildin... - Building a Mandelbrot step-by-step
• [Extra Visual] Period ... - Period 2 orbits of a Julia Set
• [Extra Visual] Period ... - Period 3 orbits of a Julia Set
• [Extra Visual] Period ... - Period 4 orbits of a Julia Set
• [Extra Visual] Period ... - Period 5 orbits of a Julia Set
• [Extra Visual] Period ... - Period 6 orbits of a Julia Set

Опубликовано:

12 сен 2024

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Комментарии : 15

@asherwilkins465 4 года назад

Seeing how at the end of the video all the negative numbers iterate by going back and forth really open my eyes to exactly what's going on when you go in that direction of the mandelbrot set. I just think that's so cool man

@asherwilkins465 4 года назад

No channel has more rewatchability per video than you I swear. Especially the mathemagicigian videos like this one that give you infinite things to learn and observe each time you watch it. It's just so cool to me, I really appreciate you putting these out there

@TheMathemagiciansGuild 4 года назад

Thank-you. I really appreciate the kind words. You'd be amazed at how inspiring nice comments are. :-)

@THE_ONLY_GOD 2 года назад

Great visual way to show that! Thanks!

@mnada72 Год назад

If this beauty is within 2d, I wonder if there is 3d version and how intriguing it would be. Thank you.

@YourPrerogative 4 года назад

Very helpful! I noticed that there's one period 3 bulb at the far left end of the needle -- it has one ball going back and forth from it, to a point around 1.8, back into the main cardioid (I think), and then back to the original bulb. If you were to zoom in around that seemingly invisible point at 1.8, do you know if you would you see anything particularly interesting?

@YourPrerogative 4 года назад

More like 1.4 sorry

@TheMathemagiciansGuild 4 года назад

There is a period 3 orbit within the cardiod of the biggest mini-Mandlebrot on the real number-line. It occurs at about -1.75 from memory. I've animated it in this video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-7MotVcGvFMg.html%3A00 (at 23:00 or so).

@TheMathemagiciansGuild 4 года назад

Actually that orbit that lands in the period 3 mini-Mandelbrot looks like it hits the circle of the mini every 6th iteration, so I think it might even be a period 6 orbit. (The cardioid of that mini-brot is certainly period 3, so the circle would be double that 6). (There are no orbits perfectly on the real number line, just as a result of the way I programmed the animation, otherwise you would see many more on this line.).