Mandelbulbs: the search for a 3D Mandelbrot Fractal

Follow Tom on his journey to Delft in the Netherlands in his quest to find a 3D Mandelbrot Set, otherwise known as a 'Mandelbulb'.

We begin with a discussion of the definition of a fractal, with examples from the natural world, as well as generating our very own in the form of the Koch Snowflake - a shape with infinite perimeter, but finite area. We then define the 2D Mandelbrot Set and introduce the concept of the Hausdorff Dimension which can take non-integer values, as exemplified by the Sierpinski Carpet Fractal. After a brief diversion via the Coastline Paradox and Space-filling Curves, we move onto 3D fractals in the form of the Menger Sponge, before finally using the White-Nylander Formula to generate our 3D renderings of the Mandelbrot Set, otherwise known as Mandelbulbs.

Produced by Quark Media - Jamie Halliday and Joseph Brooks Shuttleworth

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Attribution:
Paramount Pictures. ANNIHILATION. YouTube Movies. March 25, 2019.

Credits:
Cwgreene
Google Maps
Desmos
StudyWorld
WorldAtlas
Zbigniew Fiedorowicz
Byron Mayfield

[SG-yqfm]

As N goes to infinity, it will approximate a sphere, just like the generalized Mandelbrot set will approximate a circle as N reaches infinity

[CScottJon]

This guy is my personal hero! Also, didn't expect to hear you say "Koch" so much, but I'm not complaining :p

[Bethos-Arne]

I learnt more in this 25 minutes than in 2.5 years in university.

[AlanZucconi]

That was really nice!
Such a good production value as well!
🤩

[MrZoomZone]

I experimented with all this in 1990-2000 initially inspired by James Gleicks book Chaos and Mandelbrots own book The Beauty of Fractals. This video finally made me understand fractal dimension. The first example showing d=1.5 made it crystal clear. Down hill all the way from there. Thank you.

[Nefariousbig]

There's something in the formation of curves and superstructures within mathematics that really powerfully emphasizes the 'life' dormant in such seemingly simple things. It's like bouncing between seeing the code behind the universe, and the magic behind the code. The more you dive into the numbers the more geometry and structure and all these crazy forms start to loop back into things and lead you to more numbers...

[asklar]

Tom's videos and a good "Koch curve"...two of my favorites

[OMNS]

Love your channel! Stumbled upon it by accident! I was so bad at Math.. Youtube has made it easy :)

[jesnoggle]

That was the best explanation of 3d fractals. Thanks

[FanTazTiCxD]

I completely lost it when he described the different formulas, but I still kept watching and felt fascinated by what I understood when seeing the images and imagining that certain formulas can generate very interesting shapes. One day, though, one day I will be able to watch a video like this, and understand everything as it is explained. Maybe in a few years

[abuzzedwhaler]

Thanks Tom this video was absolutely sick!!

[jennishaagrawal]

I have seen your complete video on this where you derived formulas for perimeter and area that was awesome.
Really your topics works like windows to see the beauty of maths.

[BetzalelMC]

Closest thing I can think of to what an approximation of the entirety of spacetime/universe would look like from without..

[Selen_El_En]

Cool guy. Smart and clever. Gorgeous video!

[nicburke]

As n=infinity, all the available space is filled so the object tends to a sphere. I think.

[skasperl]

Maths, Kochs and Boolbs in one video. All you need.

[zenithparsec]

The Mandelbot set is z(n+1) = z(n) * z(n), for points which do not diverge.
The Julia Set is what you showed, z(n+1) = z(n)*z(n) + c, for all points which do not diverge.
Both start with z(0) = c.

And the fractional dimension of a rectangle is 2.

It might have been helpful to show where the 8/5ths came from, but as a hint, think about the area of a 3 similar triangles to the original, but each having a side length 1/3 of the large one, and what the sum of the fractions of that (imagine the triangle was a square and you drew a square with sides 1/3 the length, and you can easily see the area is 1/9th for each square. The triangles have the same ratio side lengths, so they must also have an area of 1/9th the parent length. and there are 3 of them... hope that helps you start.)

[Corniel]

I still try to understand why Delft was relevant in this story, but nice that visited this nice city. :)

[modestorosado]

Am I the only one who kind of sees a resemblance between Tom and James Grime? Not necessarily a physical resemblance. It feels like they have the same "playful" approach to mathematics.

[BasharA]

I started watching this to learn about Mandlebrot. But now I miss the Netherlands