Mandelbulbs: the search for a 3D Mandelbrot Fractal

[@TomRocksMaths]

For the full details of the calculations involved in showing the Koch Snowflake has finite area, but infinite perimeter see here: kzbin.info/www/bejne/fZK7n2awpsupgrs

[@Dozerson2]

Really enjoyed the video! Can you show the original Mandebulb (22:34) rotated in 3d space as well?

[@CosmiaNebula]

Summary: the Koch snowflake stays inside a hexagon, but its perimeter increases by a factor of 1/3 everytime, and (1 + 1/3)^infinity = infinity.

[@nosuchthing8]

Thank you sit! I grew up coding a mandelbrot program on a zx spectrum SO long ago. Whilst watching annihilation I *thought* it was a 3 mandelbrot, but did not know for *sure* until this video.

[@Fine_Mouche]

And for the figure at 10:40, can we make the same sort of calcule ? Like t/making for each step the sum of circumference of smallest/latest size Vs the sum of area of the latest/smallest size.

[@donatodiniccolodibettobardi842]

How can surface length of Koch Snowflake can be infinite if you will eventually reach the Plank length, which is the smallest possible length? You'd end up with very big, but final set of units.

[@CScottJon]

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

[@TomRocksMaths]

Jon

[@tsuchan]

Mine too. He's killing me softly with his loveliness but I end-up learning some maths.

[@TomRocksMaths]

@@tsuchan then I'm doing my job :)

[@CroomMusic]

snowflake koch for the win!

[@mudkip_btw]

I'm surprised he didn't go for the German pronunciation lol

[@AlanZucconi]

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

[@TomRocksMaths]

Thanks Alan! And to Quark Media for the production.

[@renatamachado6312]

Verdade

[@SG-yq7fm]

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

[@sschmachtel8963]

Yeah I guess that as in the video the angles will be multiplied by N and the argument will be taken to the power of N. And for N-> inf this will give you infinity for any argument bigger than 1. Hmm maybe not really good to give away the answer here^ but anyway why not also supply some reasoning that is easy.

[@renatamachado6312]

Entendi. Você explicou de uma forma muito fácil.

[@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.

[@Bethos1247-Arne]

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

[@asklar]

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

[@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

[@jennishaagrawal1622]

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.

[@TomRocksMaths]

Thank you Jennisha!

[@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...

[@Nefariousbig]

Fitting that I just realised this would make the reaction to fractal imagery in itself a fractal 😂

[@abuzzedwhaler7949]

Thanks Tom this video was absolutely sick!!

[@TomRocksMaths]

Glad you enjoyed it :)

[@skasperl]

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

[@jesnoggle13]

That was the best explanation of 3d fractals. Thanks

[@nicburke2031]

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

[@Bashar3A]

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

[@modestorosado1338]

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.

[@sam-ui5lc]

Agreed! Both of them are extremely enthusiastic about maths

[@TomRocksMaths]

Thank you!! But no one can ever replace the legend that is James Grime...

[@JayLikesLasers]

Chaos isn't random. Chaos is deterministic, but sensitive to initial conditions. Come on, Tom!

[@Selen_El_En]

Cool guy. Smart and clever. Gorgeous video!

[@TomRocksMaths]

Thank you Elena!

[@columbus8myhw]

The half-plane is still 2-dimensional, not 1.5. Those Julia sets (not Mandelbrot sets, there's only one of those) you displayed are 2-dimensional, too… but their _boundaries_ are 1.27-dimensional, etc. In general, something with Hausdorff dimension between 1 and 2 will look like a squiggly curve (like the Koch curve or the boundary (not the interior) of the Koch snowflake).

[@columbus8myhw]

Don't think of the Hausdorff dimension as "how much of the surrounding space does it fill up". It's a measure of roughness. Roughly, it's about the speed at which measurements change when you get smaller and smaller rulers.

[@landsgevaer]

Yes, pity of that mistake. Also, a space filling curve does not visit all points in the plane, like the middle-third fractal does not remove all points from the interval, not even eventually.

[@columbus8myhw]

Dave Langers No, he got that right, actually. Space-filling curves like the Hilbert curve _do_ hit every point. The Hilbert curve is usually given as the limit of "iterations" made of straight line segments. Each iteration misses some points, it's true, and there are points that no iteration will hit. But the _limit_ of the iterations does hit every point. In fact, it hits some points _multiple times!_ Writing H(x) for the point of the Hilbert curve that's x along the curve, H(1/6)=H(1/2)=H(5/6) is the center of the square. Writing H_n for the nth iteration (remembering it's made by connecting finitely many vertices with line segments), we define H(1/6) by H(1/6) = lim_{n->infinity} H_n(1/6).

[@OMNS777]

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

[@MathRocks]

how always, good video

[@TomRocksMaths]

Thanks Marcos!

[@chacharles31]

I'm discovering Hausdorff dimension, I"m watching every video on youtube around the subject to know more about it. It's the best video so far. By any chance, do you have a paper or a video on how to compute the hausdorff dimension on a mandelbrot and julia ?

[@TomRocksMaths]

The calculation involves logarithms, so maybe start there?

[@deinauge7894]

but are you sure that the part about dimensions is correct in this video? the half plane has just dimension 2 (not 1.5 as stated in the video)... and having dimension 1.8 does not mean that it fills 80% of the plane. in fact, everything with dimension less than two has an area of zero!

[@thespuditron9387]

I feel like you’ll get a perfect sphere when n goes to infinity. 👍🏻👍🏻

[@kristinaswanson6063]

Indeed Tom rocks maths

[@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.)

[@rezafathoni03]

9:33 - 10:00 I don't know why but the way it zooms just made me feel anxious and sick. Anyway, great video dude 👍👌

[@shubhsharma150]

for n=inf, it produces the unit Sphere

[@AFastidiousCuber]

I think it would be interesting to see how this could be generalized even further. There was no reason you had to define "multiplication" of 3-vectors in an analogous way to multiplication of complex numbers, and it would be interesting to see what kinds of fractals different rulesets give rise to. In fact, you could do something like this for 2-vectors as well.

[@TomRocksMaths]

Absolutely - it's a very fun concept!

[@davec.6456]

Thank you for a clear explanation. If it is possible to get a perfect four-dimensional Mandelbrot Fractal, has anyone tried rendering it as a three-dimensional shadow of what that four-dimensional object looks like? Like how we show the three-directional shadow of a Hypercube. Of course, then we would have to render that shadow as a two-dimensional object to show it on KZbin, but still, I think that would be beautiful.

[@yeplipyep3437]

Fantastic video

[@sp277]

Great video. Subscribed!

[@TomRocksMaths]

Welcome :)

[@bammam5988]

I haven't seen Annihilation yet but they also used it in Doctor Strange. You should do a video about the mandelbox! It's even more intricate and fascinating.

[@rokronroff]

Now you have to 3d print them for merch

[@TomRocksMaths]

Ooooooo I like this idea, A LOT.

[@brucesekulic5443]

Your step size can be considered an e approximator and you don’t actually accurately map the actual coast perimeter when walking around U.K. you approximate it in some places underestimating the actual length, in some places over estimating it, depending whether you are on a beach or a cliff top... you have a lower limit to the size of your step...hence you are able to ‘walk the coast’ of the U.K.

[@BetzalelMC]

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

[@wandrespupilo8046]

Couldn't you just take the 4d quaternion hypermandelbulb, and show the 3-dimensional crossection where the value for the fourth dimension axis is 0? like having the axis be x, y, z, w, and restricting w to be 0 if someone knows how to do that, please do it

[@richskater]

I participated in the business card Menger Sponge project when I was in school. Why did no one tell us it was a 3D Sierpiński Carpet...

[@TomRocksMaths]

They never tell you the cool stuff...

[@ehtikhet]

Grammar and literature is analogous. You have to learn all the conjugations before the profound meaning of the poetry becomes clear, unfortunately STEM teaching forgets to show you the poems along the way. It pays to have inspirational teachers like Tom to show you the beauty of the structures such grammatical tools can create and describe.

[@KIdMikie56]

Ive seen this before but came here after rewatching a seen from the movie Annihilation, and reading comments about mandelbulbs.

[@josephmerrill2686]

Only a small fraction of fractals are strongly self similar. Fractals are not generally self similar. They are generally infinitely detailed as iterations goes to infinity, fractional dimensional and jagged in some way. The self similar fractals popular in the 90s are actually less interesting artistically to me than less-self-similar ones.

[@JulioMartinez-hy2mu]

HELLO, Tom. Greetings from México.

[@TomRocksMaths]

Hello Julio!

[@ahzobo]

minor issues: - modulus of z=a+i b is |z| = sqrt(a^2+b^2), square root is missing in one of the hand-written formulas - the generalisation to 3D is given for *exponentiation* (White-Nylander): z^n: (r, theta, phi) -> (r^n, n*theta, n*phi), you could guess a definition for "multiplication", but this is not given in the video. Also, It should be noted that these operations lead to nice fractals, but do not follow the rules that we normally expect for operations called "exponentiation" and "multiplication". This is the issue with there being no 3D vector "multiplication", mentioned in the video.

[@RussSwanson]

I don’t know how I landed here. Confusing, and beautiful

[@wybird666]

The coastline paradox is not a paradox because you compared apples to pears: "If you walk around the coastline, you will get back to where you started." The fractal coastline was a result of using shorter and shorter measuring sticks. By the same vein, one would need to take smaller and smaller steps to walk around the coast, and by the same argument would take longer and longer to walk round (increasing to infinity as the step size approached zero). The actual reason why the coastline is not fractal is a physical constraint and not mathematical one: the coastline is built out of matter, and hence atoms (or molecules), which have a finite size and thus putting a lower bound on the step size one can take. Even neglecting that, space is quantized, but at that point one needs to ask what is it being measured (the isocontour of electron density?)

[@baoboumusic]

Delft is an awesome little town. I know pretty much every place you show :) Btw is this pre-pandemic footage?

[@TomRocksMaths]

Delft is indeed awesome. This was filmed in Feb 2019.

[@erniesulovic4734]

I am also wondering, instead of drawing triangles when we divide the line by 3, what would it look like if we used hemispheres instead?

[@chimetimepaprika]

The stereotype is that math nerds are not cool, but most math nerds are, in fact, pretty cool. Except for me:(

[@TomRocksMaths]

You are exceptionally cool, don't let anyone else tell you otherwise :)

[@daanvberkel1980]

Nice, you are in the Netherlands. The country that I live in. Are you still there? Or back home again?

[@TomRocksMaths]

This was filmed in Feb 2019, but (COVID permitting) I'll be back next June!

[@daanvberkel1980]

@@TomRocksMaths Will there be a meet and greet?

[@TomRocksMaths]

@@daanvberkel1980 I hadn't been planning on it, but I'm sure we can arrange something :)

[@Fine_Mouche]

22:32 : and what we can see when we infinite zoom in ? And for 3D we can have different values of n, but why for 2D it seem not ? And in reverse for 2D we can start at different values of C and for 3D not.

[@user-vn9ld2ce1s]

If n goes to infinity... But what happens when u use non-integer values of n? It should work, at least on paper, does it just make a mess?

[@jamaican234]

As N goes towards infinity, it will flow like a superfluid forever flowing more of the pattern out while shifting the already occupied space into a compressed section along the opposite point of origin...or, back into itself (Infinity origin points - like a zip on the boundary of infinity) I suppose it would be pine cone shaped... bulbed dendrites forming off the potrusions with pockets of void in the cracks around the bigger origin void points. I'm finding my thoughts quite hard to describe at this depth.

[@TomRocksMaths]

It would in fact be a sphere I believe

[@Kantirant]

i do not understand why the area at 6:26 should be 8/5, should'n it technically be infinite , since it adds on infinitely ? even if its small?

[@AriaBreath]

Thanks, I don’t understand maths but still really enjoyed the video!

[@TomRocksMaths]

awesome :)

[@soldenstoll8495]

As N goes to infinity, it would approach the the open sphere |z| < 1 where z is our 3D vector.

[@davidmakin6984]

Please - the ones with a single value of C are Julia Sets !!

[@mrmanning6098]

My first instinct is to assume it is easier to make a 4d mandelbulb than a 3d one, partly because of the quaternions being a nice stand-in for the complex numbers in the original mandelbrot set

[@mrmanning6098]

I just got to the part where you mention the White Nylander formula. I am thankful to you for showing me something new. This will be so fun to explore.

[@TomRocksMaths]

Awesome - let me know how you get on :)

[@mrmanning6098]

@@TomRocksMaths I found a great (and accessible) paper called Expanding the Mandelbrot Set into Higher Dimensions by Javier Barrallo. It answered a lot of the questions I had after this video. Quick question(s): What courses should one take to learn more about fractals in general? I imagine the typical undergraduate courses are on the list, but which of them are most directly related to analyzing and categorizing fractals of all natures?

[@mateusmachadofotografia8554]

What about the mandelbox ?

[@sheshagirigh]

Very nice..

[@TomRocksMaths]

Thanks :)

[@keldrean]

I want to see n=18 it might confirm a theory I have, wow this is cool! Oh, n = infinity is like the fourth dimension. The cube becomes a sphere

[@AnymMusic]

ok legit question tho, how and where can I find more about how all these formulas work. I've gotten more interested, but never had anything like it at school, and as much as I love fractals, this is a bit too high level for me lol

[@TomRocksMaths]

I'd suggest my other video on fractals where I explain how to construct your own without using any formulas: kzbin.info/www/bejne/fZK7n2awpsupgrs

[@VirtualTurtle]

Search up Mandelbulb3D, there are some tutorials and it's a really expansive software

[@clarariachi9294]

Are all coastlines fractal or just the ones that exhibit self-similarity at smaller scales

[@utilizator1701]

20:21 wrong formula for |z|. Edit: he corrected the formula at 20:35.

[@raptor9514]

10:40 VVhat? 2-dimensional Hausdorff measure of a halfplane is infinite, so its Hausdorff dimension cannot be smaller than 2! And as a subset of a 2-dimensional plane, it connot be bigger than 2, thus its Hausdorff dimension is 2.

[@erniesulovic4734]

Tom, walking around the coastline being finite compared to the diameter of fractal being infinite, would that not come under Set Theory? Or isn't it the same or similar to the paradox, a man walks halfway across a room, then half again and half again, he would never reach the other end of the room yet in practice, he easily walks from one side to the other........

[@TomRocksMaths]

I’d go with the second description - it’s very similar to Zeno’s paradox

[@erniesulovic4734]

@@TomRocksMaths Cool. Thanks

[@bryceforrest210]

3b1b doesn't like that your definition of fractals at 0:45 relies on self-similarity!

[@regulus2033]

Hello! I have a question. What does the picture at 11:07 describe? And also: I see that the set on this picture is finite, and the Hausdorff dimension of a finite object should equal 0, so how can it equal 1.27?

[@TomRocksMaths]

The image is of the Mandelbrot set generated using the complex number iteration I talk about in the video, using a value of c=-1.

[@kidredglow2060]

0:45 thats wrong, its just a thing with infinite detail, it doesnt NEED to be self similar

[@davethesid8960]

At least you found a cute coot.

[@zfloyd1627]

0:53 Cliff? I actually thought it looked like poop at first.

[@ZedaZ80]

Would the 4D version use quaternions?

[@ZedaZ80]

(I still don't understand quaternions, I just remember them using 4 components)

[@TomRocksMaths]

Absolutely, nice spot!

[@ZedaZ80]

@@TomRocksMaths oh cool, thanks!

[@katiebarber407]

is there a way to make a chart that shows all of the math involved? for assistance in creating them into nodes in blender

[@artyommoxid6233]

I don't get it

[@fistandantillusflinx5041]

In 2D planer polar 2-vector is equivalent to complex number Mandelbrot. In 3D there is no complex number analog but spherical polar 3-vector work. Question: In 4D are the 4-vector and quaternion Mandelbrot once again equivalent?

[@TomRocksMaths]

4D Mandelbrot's defined via quaternions are indeed possible.

[@fistandantillusflinx5041]

@@TomRocksMaths I was asking about the equivalence between n-vector and complex number analog in those dimensions (2, 4, 8, 16, ?) that have a complex number analog. The 3D mandelbulb is interesting in and of itself, but it is 'more' interesting if the equivalence relationship always holds. Coincidence versus possibly deeper understanding.

[@factsopinionsandinterestin6832]

Practical applications are all well and good, but when am I going to not use this?

[@grproteus]

our Koch curve, you say?

[@festusmaximus4111]

could you also take a 3d projection of a true 4d mandelbulb? Would they look the same as the n=2 case?

[@TomRocksMaths]

Interesting idea - I'll have to investigate!

[@yandyyay]

Yeah this is why I love math and infinite sets.... Whilst it's well accepted that the area of a Koch snowflake converges onto 8/5 of the original area how can it really be so? Yes the amount of area added at each successive stage gets smaller and smaller but never quite reaches zero.... When does infinity stop?

[@TomRocksMaths]

There are lots of examples of infinite series that converge to a finite number. For example, (1/2)^n. If you add together all of the terms from n=0 to infinity the answer is 2.

[@yandyyay]

@@TomRocksMaths thanks for the reply, surely i just wasn't thinking straight ☺.. Its obvious now with that simple example on each successive iteration the sum is only getting bigger by half the distance left to reach 2

[@sakhawat8092]

Bruh u cool man ..

[@X22GJP]

Fractals...interesting things, and the agreed definitions seem to be a bit dividing. I guess it depends if you are talking mathematically or physically, and how pedantic you are. For me I've always thought of it as: "A true fractal is a mathematical construct where no matter how far you zoom in, the pattern is indistinguishable from when you first started" Examples of the above would be a Koch snowflake or Sierpinski Triangle. The former having an infinitely long perimeter but a finite area, and the latter having zero area - all very strange! Venture outside of that type of mathematics and "fractals" become less defined. For example, the Mandelbrot Set in my view is NOT a fractal, but it definitely displays fractal-like properties and varying degrees of self-similarity as you zoom in. But, not always identical. Same with Julie Sets. Venture outside mathematics and into the real world, and all we can ever do is say that things exhibit fractal-like properties. A fern leaf has more self-similarity than a coastline for example, and so is more like a true-fractal in that sense, but neither can ever be true fractals because you'll eventually get down to sub-atomic details that look completely different. So yeah, it really depends on how pedantic we are being when it comes to labelling something as "fractal" or "fractal-like". Edit: "Koch" is generally pronounced "Coke" in English, not "Cock", lol Although still incorrect, the real pronunciation is as per the name "Bach", with a soft "ch" sound, like in a Scottish "Loch". I don't think "Bach" would take too kindly to being called "Back" would he?!

[@ragnkja]

The vowel in both “Koch” and “Bach” is short, though in the latter case that short vowel is one that doesn’t occur in English. By the way, Koch was Swedish, which implies that his surname is more likely to end in /k/, rather than /x/ as in “Bach”.

[@qwadratix]

Why not quaternions? I've always thought of them as a 3D complex number. (Not that I'm going to attempt it. I can barely count my change in the newsagent's these days)

[@TomRocksMaths]

Quaternions are in fact a 4-dimensional analogue to complex numbers.

[@qwadratix]

@@TomRocksMaths Ah yes, true. One real, three 'imaginary', still...

[@dontaskformyname6096]

who did these captions? I just wanna talk

[@TomRocksMaths]

I think they were auto-generated, unless I'm missing something...

[@davidbriscoe5745]

If 3d is impossible couldnt you use pi d (an imaginary dimension) and thus pi fibonacci numbers would produce the golden paradigm

[@emacipadung8054]

Circle

[@LucaIlarioCarbonini]

Please consider to pronunciate "Peano" like pe-AH-know

[@HarveyHirdHarmonics]

And "Koch" with a "clearing throat" sound.

[@ragnkja]

@@HarveyHirdHarmonics Why? That sound does not occur in Swedish as far as I’m aware, and Helge von Koch was Swedish.

[@davidbriscoe5745]

Thats integer pi

[@SzeptyTechniki]

Well... there is one more question... do you have 3d printer and some time? :D wink wink

[@keltonfoster]

This is freaking ridiculous, what the hell is wrong with wanting to see videos about mandelbrot that is such a big deal they hide the good ones.

[@renatamachado6312]

Legal

[@SNdynasty]

waaar delft

[@TomRocksMaths]

sure is!

[@UnyPhi]

NGL kinda mad you spend so long going over fractals then Mandelbrot.

[@TomRocksMaths]

It's meant as an introduction to the idea of fractals, and to explain how the Mandelbulbs are generated. There's lots of great stuff already out there on the Mandelbrot set eg: kzbin.info/www/bejne/hHiwg3VqhJ5laKs

[@bachirblackers7299]

Still there s too much to reveal about x^

[@rat_king-]

British empire... i disagree. The british coastline is infinite.

[@MarkusDarkess]

No ideal about this

[@hugoeriksson6524]

your definition of a fractal is terrible. a fractal is a shape with a definite area but infinite perimeter.

[@overheadthealbatross7963]

this is as ugly as the 2d mandelbrot set is beautiful

[@jamesbugbee6812]

Runs like a girl.